Nuprl Lemma : fpf-rename-cap3
∀[A,C,B:Type]. ∀[eqa:EqDecider(A)]. ∀[eqc,eqc':EqDecider(C)]. ∀[r:A ⟶ C]. ∀[f:a:A fp-> B]. ∀[a:A]. ∀[z:B]. ∀[c:C].
  (rename(r;f)(c)?z = f(a)?z ∈ B) supposing ((c = (r a) ∈ C) and 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 : 
uall: ∀[x:A]. B[x]
, 
member: t ∈ T
, 
uimplies: b supposing a
, 
prop: ℙ
, 
so_lambda: λ2x.t[x]
, 
so_apply: x[s]
, 
all: ∀x:A. B[x]
, 
implies: P 
⇒ Q
Lemmas referenced : 
equal_wf, 
fpf-cap_wf, 
inject_wf, 
fpf_wf, 
deq_wf, 
fpf-rename-cap2, 
fpf-rename_wf
Rules used in proof : 
sqequalSubstitution, 
sqequalTransitivity, 
computationStep, 
sqequalReflexivity, 
isect_memberFormation, 
introduction, 
cut, 
hypothesis, 
thin, 
hyp_replacement, 
equalitySymmetry, 
applyLambdaEquality, 
extract_by_obid, 
sqequalHypSubstitution, 
isectElimination, 
cumulativity, 
hypothesisEquality, 
sqequalRule, 
lambdaEquality, 
because_Cache, 
applyEquality, 
functionExtensionality, 
isect_memberEquality, 
axiomEquality, 
equalityTransitivity, 
functionEquality, 
universeEquality, 
independent_isectElimination, 
lambdaFormation, 
dependent_functionElimination, 
independent_functionElimination
Latex:
\mforall{}[A,C,B:Type].  \mforall{}[eqa:EqDecider(A)].  \mforall{}[eqc,eqc':EqDecider(C)].  \mforall{}[r:A  {}\mrightarrow{}  C].  \mforall{}[f:a:A  fp->  B].  \mforall{}[a:A].
\mforall{}[z:B].  \mforall{}[c:C].
    (rename(r;f)(c)?z  =  f(a)?z)  supposing  ((c  =  (r  a))  and  Inj(A;C;r))
Date html generated:
2018_05_21-PM-09_27_19
Last ObjectModification:
2018_02_09-AM-10_22_29
Theory : finite!partial!functions
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