Theorem ixpfn 7546

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 5674	. 2 |- ( f = F -> ( f Fn A <-> F Fn A ) )
2		elixp2 7544	. . 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 1046	. 2 |- ( f e. X_ x e. A B -> f Fn A )
4	1, 3	vtoclga 3099	1 |- ( F e. X_ x e. A B -> F Fn A )

Syntax hints:   -> wi 4   e. wcel 1904   A.wral 2756   _Vcvv 3031   Fn wfn 5584   ` cfv 5589   X_cixp 7540

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451

This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-br 4396  df-opab 4455  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-iota 5553  df-fun 5591  df-fn 5592  df-fv 5597  df-ixp 7541

This theorem is referenced by:  ixpprc  7561  undifixp  7576  resixpfo  7578  boxcutc  7583  ixpiunwdom  8124  prdsbasfn  15447  xpsff1o  15552  sscfn1  15800  funcfn2  15852  natfn  15937  pthaus  20730  ptuncnv  20899  ptunhmeo  20900  ptcmplem2  21146  prdsbl  21584  finixpnum  31994  upixp  32120  prdstotbnd  32190  hoidmvlelem3  38537  hspdifhsp  38556  hspmbllem2  38567