Theorem ixpfn 7914

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 5979	. 2 |- ( f = F -> ( f Fn A <-> F Fn A ) )
2		elixp2 7912	. . 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 1077	. 2 |- ( f e. X_ x e. A B -> f Fn A )
4	1, 3	vtoclga 3272	1 |- ( F e. X_ x e. A B -> F Fn A )

Syntax hints:   -> wi 4   e. wcel 1990   A.wral 2912   _Vcvv 3200   Fn wfn 5883   ` cfv 5888   X_cixp 7908

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fn 5891  df-fv 5896  df-ixp 7909

This theorem is referenced by:  ixpprc  7929  undifixp  7944  resixpfo  7946  boxcutc  7951  ixpiunwdom  8496  prdsbasfn  16131  xpsff1o  16228  sscfn1  16477  funcfn2  16529  natfn  16614  pthaus  21441  ptuncnv  21610  ptunhmeo  21611  ptcmplem2  21857  prdsbl  22296  finixpnum  33394  upixp  33524  prdstotbnd  33593  rrxsnicc  40520  ioorrnopnxrlem  40526  hoidmvlelem3  40811  hspdifhsp  40830  hspmbllem2  40841