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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