Nuprl Lemma : fpf-rename_wf
∀[A,C:Type]. ∀[B:A ⟶ Type]. ∀[D:C ⟶ Type]. ∀[eq:EqDecider(C)]. ∀[r:A ⟶ C]. ∀[f:a:A fp-> B[a]].
  rename(r;f) ∈ c:C fp-> D[c] supposing ∀a:A. (D[r a] = B[a] ∈ Type)
Proof
Definitions occuring in Statement : 
fpf-rename: rename(r;f)
, 
fpf: a:A fp-> B[a]
, 
deq: EqDecider(T)
, 
uimplies: b supposing a
, 
uall: ∀[x:A]. B[x]
, 
so_apply: x[s]
, 
all: ∀x:A. B[x]
, 
member: t ∈ T
, 
apply: f a
, 
function: x:A ⟶ B[x]
, 
universe: Type
, 
equal: s = t ∈ T
Definitions unfolded in proof : 
uall: ∀[x:A]. B[x]
, 
uimplies: b supposing a
, 
member: t ∈ T
, 
fpf: a:A fp-> B[a]
, 
fpf-rename: rename(r;f)
, 
pi1: fst(t)
, 
pi2: snd(t)
, 
all: ∀x:A. B[x]
, 
prop: ℙ
, 
so_lambda: λ2x.t[x]
, 
so_apply: x[s]
, 
guard: {T}
, 
implies: P 
⇒ Q
, 
subtype_rel: A ⊆r B
, 
squash: ↓T
, 
true: True
, 
iff: P 
⇐⇒ Q
, 
and: P ∧ Q
, 
rev_implies: P 
⇐ Q
, 
deq: EqDecider(T)
, 
exists: ∃x:A. B[x]
, 
cand: A c∧ B
, 
eqof: eqof(d)
, 
uiff: uiff(P;Q)
, 
rev_uimplies: rev_uimplies(P;Q)
Lemmas referenced : 
set_wf, 
l_member_wf, 
map_wf, 
all_wf, 
equal_wf, 
fpf_wf, 
deq_wf, 
subtype_rel-equal, 
squash_wf, 
true_wf, 
subtype_rel_self, 
iff_weakening_equal, 
hd-filter, 
member_map, 
safe-assert-deq, 
assert_wf
Rules used in proof : 
sqequalSubstitution, 
sqequalTransitivity, 
computationStep, 
sqequalReflexivity, 
isect_memberFormation, 
cut, 
sqequalHypSubstitution, 
productElimination, 
thin, 
sqequalRule, 
lambdaFormation, 
introduction, 
extract_by_obid, 
isectElimination, 
hypothesisEquality, 
lambdaEquality, 
hypothesis, 
dependent_pairEquality, 
dependent_functionElimination, 
setEquality, 
functionEquality, 
applyEquality, 
setElimination, 
rename, 
instantiate, 
cumulativity, 
universeEquality, 
dependent_set_memberEquality, 
independent_isectElimination, 
imageElimination, 
equalityTransitivity, 
equalitySymmetry, 
natural_numberEquality, 
imageMemberEquality, 
baseClosed, 
independent_functionElimination, 
because_Cache, 
dependent_pairFormation, 
independent_pairFormation, 
productEquality
Latex:
\mforall{}[A,C:Type].  \mforall{}[B:A  {}\mrightarrow{}  Type].  \mforall{}[D:C  {}\mrightarrow{}  Type].  \mforall{}[eq:EqDecider(C)].  \mforall{}[r:A  {}\mrightarrow{}  C].  \mforall{}[f:a:A  fp->  B[a]].
    rename(r;f)  \mmember{}  c:C  fp->  D[c]  supposing  \mforall{}a:A.  (D[r  a]  =  B[a])
Date html generated:
2018_05_21-PM-09_26_42
Last ObjectModification:
2018_05_19-PM-04_37_56
Theory : finite!partial!functions
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