Theorem ixpfn 6976

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 5464	. 2 |- ( f = F -> ( f Fn A <-> F Fn A ) )
2		elixp2 6974	. . 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 1044	. 2 |- ( f e. X_ x e. A B -> f Fn A )
4	1, 3	vtoclga 2889	1 |- ( F e. X_ x e. A B -> F Fn A )

Syntax hints:   -> wi 4   e. wcel 2209   A.wral 2528   _Vcvv 2821   Fn wfn 5367   ` cfv 5372   X_cixp 6970

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-br 4126  df-opab 4188  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-iota 5332  df-fun 5374  df-fn 5375  df-fv 5380  df-ixp 6971

This theorem is referenced by:  ixpprc  6991  ixpssmap2g  6999  ixpssmapg  7000  xpsff1o  13647  prdsbasfn  14158  depindlem3  16663