Nuprl Lemma : name-morph-inv-eq
∀[I,J:Cname List]. ∀[f:name-morph(I;J)]. ∀[x:nameset(I)].
(name-morph-inv(I;f) (f x)) = x ∈ nameset(I) supposing ↑isname(f x)
Proof
Definitions occuring in Statement :
name-morph-inv: name-morph-inv(I;f)
,
name-morph: name-morph(I;J)
,
isname: isname(z)
,
nameset: nameset(L)
,
coordinate_name: Cname
,
list: T List
,
assert: ↑b
,
uimplies: b supposing a
,
uall: ∀[x:A]. B[x]
,
apply: f a
,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x]
,
member: t ∈ T
,
uimplies: b supposing a
,
name-morph-inv: name-morph-inv(I;f)
,
name-morph: name-morph(I;J)
,
uiff: uiff(P;Q)
,
and: P ∧ Q
,
all: ∀x:A. B[x]
,
or: P ∨ Q
,
sq_type: SQType(T)
,
implies: P
⇒ Q
,
guard: {T}
,
subtype_rel: A ⊆r B
,
nameset: nameset(L)
,
coordinate_name: Cname
,
int_upper: {i...}
,
bfalse: ff
,
band: p ∧b q
,
ifthenelse: if b then t else f fi
,
so_lambda: λ2x.t[x]
,
so_apply: x[s]
,
prop: ℙ
,
istype: istype(T)
,
iff: P
⇐⇒ Q
,
rev_implies: P
⇐ Q
,
cand: A c∧ B
,
not: ¬A
,
false: False
,
cons: [a / b]
,
top: Top
,
name-morph-range: name-morph-range(f;I)
,
exists: ∃x:A. B[x]
,
assert: ↑b
,
bnot: ¬bb
,
btrue: tt
,
it: ⋅
,
unit: Unit
,
bool: 𝔹
Lemmas referenced :
assert-isname,
isname_wf,
bool_cases,
subtype_base_sq,
bool_subtype_base,
eqtt_to_assert,
band_wf,
btrue_wf,
eq_int_wf,
bfalse_wf,
istype-assert,
nameset_wf,
name-morph_wf,
list_wf,
coordinate_name_wf,
list-subtype,
member_filter_2,
subtype_rel_dep_function,
bool_wf,
l_member_wf,
l_member-settype,
iff_transitivity,
assert_wf,
nameset_subtype_extd-nameset,
member_wf,
iff_weakening_uiff,
assert_of_band,
equal_wf,
assert_of_eq_int,
istype-int,
filter_wf5,
list-cases,
member-implies-null-eq-bfalse,
null_nil_lemma,
btrue_neq_bfalse,
product_subtype_list,
hd_member,
assert_elim,
null_wf3,
subtype_rel_list,
top_wf,
istype-void,
null_cons_lemma,
set_subtype_base,
le_wf,
int_subtype_base,
set_wf,
subtype_rel_self,
name-morph-inv_wf,
assert-bnot,
bool_cases_sqequal,
eqff_to_assert,
false_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
sqequalRule,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
applyEquality,
setElimination,
rename,
because_Cache,
hypothesis,
productElimination,
independent_isectElimination,
lambdaEquality_alt,
dependent_functionElimination,
unionElimination,
instantiate,
independent_functionElimination,
equalityTransitivity,
equalitySymmetry,
inhabitedIsType,
isect_memberEquality_alt,
axiomEquality,
isectIsTypeImplies,
universeIsType,
setEquality,
lambdaFormation_alt,
setIsType,
dependent_set_memberEquality_alt,
independent_pairFormation,
productEquality,
closedConclusion,
intEquality,
promote_hyp,
productIsType,
equalityIsType1,
voidElimination,
hypothesis_subsumption,
dependent_pairFormation_alt,
equalityIsType3,
natural_numberEquality,
lambdaEquality,
lambdaFormation,
cumulativity,
dependent_pairFormation,
equalityElimination,
functionEquality,
impliesFunctionality,
addLevel,
dependent_set_memberEquality
Latex:
\mforall{}[I,J:Cname List]. \mforall{}[f:name-morph(I;J)]. \mforall{}[x:nameset(I)].
(name-morph-inv(I;f) (f x)) = x supposing \muparrow{}isname(f x)
Date html generated:
2019_11_05-PM-00_24_32
Last ObjectModification:
2018_11_08-PM-00_19_56
Theory : cubical!sets
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