### Nuprl Lemma : fpf-rename-ap2

`∀[A,C:Type]. ∀[B:A ─→ Type]. ∀[eqa:EqDecider(A)]. ∀[eqc,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`
Lemmas :  safe-assert-deq l_member_wf assert_wf assert-deq-member subtype_rel-equal iff_weakening_equal
\mforall{}[A,C:Type].  \mforall{}[B:A  {}\mrightarrow{}  Type].  \mforall{}[eqa:EqDecider(A)].  \mforall{}[eqc,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: 2015_07_17-AM-11_10_56
Last ObjectModification: 2015_02_04-PM-05_06_32

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