Theorem ixpfn 6670

Description: A nuple is a function. (Contributed by FL, 6-Jun-2011.) (Revised by Mario Carneiro, 31-May-2014.)

Assertion

Ref	Expression
ixpfn	|- ( F e. X_ x e. A B -> F Fn A )

Distinct variable group:   x, A

Allowed substitution hints:   B( x)   F( x)

Proof of Theorem ixpfn

Dummy variable f is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fneq1 5276	. 2 |- ( f = F -> ( f Fn A <-> F Fn A ) )
2		elixp2 6668	. . 3 |- ( f e. X_ x e. A B <-> ( f e. _V /\ f Fn A /\ A. x e. A ( f ` x ) e. B ) )
3	2	simp2bi 1003	. 2 |- ( f e. X_ x e. A B -> f Fn A )
4	1, 3	vtoclga 2792	1 |- ( F e. X_ x e. A B -> F Fn A )

Syntax hints:   -> wi 4   e. wcel 2136   A.wral 2444   _Vcvv 2726   Fn wfn 5183   ` cfv 5188   X_cixp 6664

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-un 3120  df-in 3122  df-ss 3129  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-br 3983  df-opab 4044  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-iota 5153  df-fun 5190  df-fn 5191  df-fv 5196  df-ixp 6665

This theorem is referenced by:  ixpprc  6685  ixpssmap2g  6693  ixpssmapg  6694