Theorem ixpfn 7476

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 5669	. 2 |- ( f = F -> ( f Fn A <-> F Fn A ) )
2		elixp2 7474	. . 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 1012	. 2 |- ( f e. X_ x e. A B -> f Fn A )
4	1, 3	vtoclga 3177	1 |- ( F e. X_ x e. A B -> F Fn A )

Syntax hints:   -> wi 4   e. wcel 1767   A.wral 2814   _Vcvv 3113   Fn wfn 5583   ` cfv 5588   X_cixp 7470

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-iota 5551  df-fun 5590  df-fn 5591  df-fv 5596  df-ixp 7471

This theorem is referenced by:  ixpprc  7491  undifixp  7506  resixpfo  7508  boxcutc  7513  ixpiunwdom  8018  prdsbasfn  14729  xpsff1o  14826  sscfn1  15050  funcfn2  15099  natfn  15184  dprdvalOLD  16851  pthaus  19966  ptuncnv  20135  ptunhmeo  20136  ptcmplem2  20380  prdsbl  20821  finixpnum  29891  upixp  30050  prdstotbnd  30120