Theorem fex2 5551

Description: A function with bounded domain and codomain is a set. This version is proven without the Axiom of Replacement. (Contributed by Mario Carneiro, 24-Jun-2015.)

Assertion

Ref	Expression
fex2	|- ( ( F : A --> B /\ A e. V /\ B e. W ) -> F e. _V )

Proof of Theorem fex2

Step	Hyp	Ref	Expression
1		xpexg 4884	. . 3 |- ( ( A e. V /\ B e. W ) -> ( A X. B ) e. _V )
2	1	3adant1 1046	. 2 |- ( ( F : A --> B /\ A e. V /\ B e. W ) -> ( A X. B ) e. _V )
3		fssxp 5550	. . 3 |- ( F : A --> B -> F C_ ( A X. B ) )
4	3	3ad2ant1 1049	. 2 |- ( ( F : A --> B /\ A e. V /\ B e. W ) -> F C_ ( A X. B ) )
5	2, 4	ssexd 4268	1 |- ( ( F : A --> B /\ A e. V /\ B e. W ) -> F e. _V )

Syntax hints:   -> wi 4   /\ w3a 1009   e. wcel 2209   _Vcvv 2821   C_ wss 3220   X. cxp 4767   -->wf 5368

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-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341  ax-un 4573

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  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-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-br 4126  df-opab 4188  df-xp 4775  df-rel 4776  df-cnv 4777  df-dm 4779  df-rn 4780  df-fun 5374  df-fn 5375  df-f 5376

This theorem is referenced by:  elmapg  6925  f1oen2g  7031  f1dom2g  7032  dom3d  7050  mapxpen  7138  addex  10031  mulex  10032  climrecvg1n  12092  cnpfval  15219  txcn  15299  blfvalps  15409