Nuprl Lemma : fpf-rename-cap
∀[A,C,B:Type]. ∀[eqa:EqDecider(A)]. ∀[eqc:EqDecider(C)]. ∀[r:A ⟶ C]. ∀[f:a:A fp-> B]. ∀[a:A]. ∀[z:B].
  rename(r;f)(r a)?z = f(a)?z ∈ B supposing Inj(A;C;r)
Proof
Definitions occuring in Statement : 
fpf-rename: rename(r;f)
, 
fpf-cap: f(x)?z
, 
fpf: a:A fp-> B[a]
, 
deq: EqDecider(T)
, 
inject: Inj(A;B;f)
, 
uimplies: b supposing a
, 
uall: ∀[x:A]. B[x]
, 
apply: f a
, 
function: x:A ⟶ B[x]
, 
universe: Type
, 
equal: s = t ∈ T
Definitions unfolded in proof : 
member: t ∈ T
, 
uall: ∀[x:A]. B[x]
, 
subtype_rel: A ⊆r B
, 
so_lambda: λ2x.t[x]
, 
so_apply: x[s]
, 
uimplies: b supposing a
, 
all: ∀x:A. B[x]
, 
top: Top
, 
fpf-cap: f(x)?z
, 
implies: P 
⇒ Q
, 
bool: 𝔹
, 
unit: Unit
, 
it: ⋅
, 
btrue: tt
, 
uiff: uiff(P;Q)
, 
and: P ∧ Q
, 
ifthenelse: if b then t else f fi 
, 
bfalse: ff
, 
prop: ℙ
, 
not: ¬A
, 
false: False
, 
iff: P 
⇐⇒ Q
, 
exists: ∃x:A. B[x]
, 
cand: A c∧ B
, 
guard: {T}
, 
inject: Inj(A;B;f)
, 
rev_implies: P 
⇐ Q
Lemmas referenced : 
fpf-dom_wf, 
subtype-fpf2, 
top_wf, 
istype-void, 
fpf-rename-ap, 
equal-wf-T-base, 
bool_wf, 
assert_wf, 
bnot_wf, 
not_wf, 
eqtt_to_assert, 
uiff_transitivity, 
eqff_to_assert, 
assert_of_bnot, 
inject_wf, 
fpf_wf, 
deq_wf, 
istype-universe, 
equal_wf, 
fpf-rename_wf, 
fpf-rename-dom
Rules used in proof : 
sqequalSubstitution, 
sqequalTransitivity, 
computationStep, 
sqequalReflexivity, 
equalityTransitivity, 
hypothesis, 
equalitySymmetry, 
cut, 
introduction, 
extract_by_obid, 
sqequalHypSubstitution, 
isectElimination, 
thin, 
hypothesisEquality, 
applyEquality, 
sqequalRule, 
lambdaEquality_alt, 
inhabitedIsType, 
independent_isectElimination, 
lambdaFormation_alt, 
isect_memberEquality_alt, 
voidElimination, 
because_Cache, 
baseClosed, 
isect_memberFormation_alt, 
unionElimination, 
equalityElimination, 
productElimination, 
independent_functionElimination, 
equalityIstype, 
dependent_functionElimination, 
universeIsType, 
axiomEquality, 
isectIsTypeImplies, 
functionIsType, 
instantiate, 
universeEquality, 
voidEquality, 
isect_memberEquality, 
lambdaFormation, 
functionExtensionality, 
lambdaEquality, 
cumulativity, 
hyp_replacement, 
applyLambdaEquality, 
productEquality, 
independent_pairFormation, 
dependent_pairFormation
Latex:
\mforall{}[A,C,B:Type].  \mforall{}[eqa:EqDecider(A)].  \mforall{}[eqc:EqDecider(C)].  \mforall{}[r:A  {}\mrightarrow{}  C].  \mforall{}[f:a:A  fp->  B].  \mforall{}[a:A].
\mforall{}[z:B].
    rename(r;f)(r  a)?z  =  f(a)?z  supposing  Inj(A;C;r)
Date html generated:
2019_10_16-AM-11_26_08
Last ObjectModification:
2019_06_25-PM-03_26_15
Theory : finite!partial!functions
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