Theorem ixpfn 7380

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 5608	. 2 |- ( f = F -> ( f Fn A <-> F Fn A ) )
2		elixp2 7378	. . 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 1004	. 2 |- ( f e. X_ x e. A B -> f Fn A )
4	1, 3	vtoclga 3142	1 |- ( F e. X_ x e. A B -> F Fn A )

Syntax hints:   -> wi 4   e. wcel 1758   A.wral 2799   _Vcvv 3078   Fn wfn 5522   ` cfv 5527   X_cixp 7374

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3747  df-if 3901  df-sn 3987  df-pr 3989  df-op 3993  df-uni 4201  df-br 4402  df-opab 4460  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-iota 5490  df-fun 5529  df-fn 5530  df-fv 5535  df-ixp 7375

This theorem is referenced by:  ixpprc  7395  undifixp  7410  resixpfo  7412  boxcutc  7417  ixpiunwdom  7918  prdsbasfn  14529  xpsff1o  14626  sscfn1  14850  funcfn2  14899  natfn  14984  dprdvalOLD  16610  pthaus  19344  ptuncnv  19513  ptunhmeo  19514  ptcmplem2  19758  prdsbl  20199  finixpnum  28563  upixp  28772  prdstotbnd  28842