Nuprl Lemma : fpf-rename-ap
∀[A,C:Type]. ∀[B:A ⟶ Type]. ∀[eqa:EqDecider(A)]. ∀[eqc:EqDecider(C)]. ∀[r:A ⟶ C]. ∀[f:a:A fp-> B[a]]. ∀[a:A].
  (rename(r;f)(r a) = f(a) ∈ B[a]) supposing ((↑a ∈ dom(f)) and Inj(A;C;r))
Proof
Definitions occuring in Statement : 
fpf-rename: rename(r;f)
, 
fpf-ap: f(x)
, 
fpf-dom: x ∈ dom(f)
, 
fpf: a:A fp-> B[a]
, 
deq: EqDecider(T)
, 
inject: Inj(A;B;f)
, 
assert: ↑b
, 
uimplies: b supposing a
, 
uall: ∀[x:A]. B[x]
, 
so_apply: x[s]
, 
apply: f a
, 
function: x:A ⟶ B[x]
, 
universe: Type
, 
equal: s = t ∈ T
Definitions unfolded in proof : 
fpf-ap: f(x)
, 
fpf-rename: rename(r;f)
, 
fpf: a:A fp-> B[a]
, 
pi2: snd(t)
, 
pi1: fst(t)
, 
uall: ∀[x:A]. B[x]
, 
member: t ∈ T
, 
all: ∀x:A. B[x]
, 
so_lambda: λ2x.t[x]
, 
deq: EqDecider(T)
, 
so_apply: x[s]
, 
implies: P 
⇒ Q
, 
subtype_rel: A ⊆r B
, 
uimplies: b supposing a
, 
top: Top
, 
prop: ℙ
, 
exists: ∃x:A. B[x]
, 
and: P ∧ Q
, 
cand: A c∧ B
, 
eqof: eqof(d)
, 
uiff: uiff(P;Q)
, 
rev_uimplies: rev_uimplies(P;Q)
, 
fpf-dom: x ∈ dom(f)
, 
iff: P 
⇐⇒ Q
, 
guard: {T}
, 
inject: Inj(A;B;f)
, 
squash: ↓T
, 
true: True
, 
rev_implies: P 
⇐ Q
Lemmas referenced : 
hd-filter, 
assert_wf, 
fpf-dom_wf, 
subtype-fpf2, 
top_wf, 
inject_wf, 
fpf_wf, 
deq_wf, 
safe-assert-deq, 
l_member_wf, 
assert-deq-member, 
equal_wf, 
squash_wf, 
true_wf, 
subtype_rel-equal, 
iff_weakening_equal
Rules used in proof : 
sqequalSubstitution, 
sqequalTransitivity, 
computationStep, 
sqequalReflexivity, 
sqequalHypSubstitution, 
productElimination, 
thin, 
sqequalRule, 
cut, 
introduction, 
extract_by_obid, 
isectElimination, 
hypothesisEquality, 
dependent_functionElimination, 
lambdaEquality, 
applyEquality, 
setElimination, 
rename, 
hypothesis, 
functionExtensionality, 
cumulativity, 
independent_functionElimination, 
independent_isectElimination, 
lambdaFormation, 
isect_memberEquality, 
voidElimination, 
voidEquality, 
because_Cache, 
functionEquality, 
universeEquality, 
isect_memberFormation, 
axiomEquality, 
equalityTransitivity, 
equalitySymmetry, 
dependent_pairFormation, 
independent_pairFormation, 
productEquality, 
dependent_set_memberEquality, 
imageElimination, 
instantiate, 
setEquality, 
natural_numberEquality, 
imageMemberEquality, 
baseClosed
Latex:
\mforall{}[A,C:Type].  \mforall{}[B:A  {}\mrightarrow{}  Type].  \mforall{}[eqa:EqDecider(A)].  \mforall{}[eqc:EqDecider(C)].  \mforall{}[r:A  {}\mrightarrow{}  C].
\mforall{}[f:a:A  fp->  B[a]].  \mforall{}[a:A].
    (rename(r;f)(r  a)  =  f(a))  supposing  ((\muparrow{}a  \mmember{}  dom(f))  and  Inj(A;C;r))
Date html generated:
2018_05_21-PM-09_26_56
Last ObjectModification:
2018_02_09-AM-10_22_14
Theory : finite!partial!functions
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