Theorem ixpfn 7536

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 5682	. 2 |- ( f = F -> ( f Fn A <-> F Fn A ) )
2		elixp2 7534	. . 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 1021	. 2 |- ( f e. X_ x e. A B -> f Fn A )
4	1, 3	vtoclga 3151	1 |- ( F e. X_ x e. A B -> F Fn A )

Syntax hints:   -> wi 4   e. wcel 1870   A.wral 2782   _Vcvv 3087   Fn wfn 5596   ` cfv 5601   X_cixp 7530

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407

This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ral 2787  df-rex 2788  df-rab 2791  df-v 3089  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-br 4427  df-opab 4485  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-iota 5565  df-fun 5603  df-fn 5604  df-fv 5609  df-ixp 7531

This theorem is referenced by:  ixpprc  7551  undifixp  7566  resixpfo  7568  boxcutc  7573  ixpiunwdom  8106  prdsbasfn  15328  xpsff1o  15425  sscfn1  15673  funcfn2  15725  natfn  15810  pthaus  20584  ptuncnv  20753  ptunhmeo  20754  ptcmplem2  20999  prdsbl  21437  finixpnum  31634  upixp  31760  prdstotbnd  31830