Theorem fex2 5500

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 4838	. . 3 |- ( ( A e. V /\ B e. W ) -> ( A X. B ) e. _V )
2	1	3adant1 1039	. 2 |- ( ( F : A --> B /\ A e. V /\ B e. W ) -> ( A X. B ) e. _V )
3		fssxp 5499	. . 3 |- ( F : A --> B -> F C_ ( A X. B ) )
4	3	3ad2ant1 1042	. 2 |- ( ( F : A --> B /\ A e. V /\ B e. W ) -> F C_ ( A X. B ) )
5	2, 4	ssexd 4227	1 |- ( ( F : A --> B /\ A e. V /\ B e. W ) -> F e. _V )

Syntax hints:   -> wi 4   /\ w3a 1002   e. wcel 2200   _Vcvv 2800   C_ wss 3198   X. cxp 4721   -->wf 5320

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4205  ax-pow 4262  ax-pr 4297  ax-un 4528

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2802  df-un 3202  df-in 3204  df-ss 3211  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-br 4087  df-opab 4149  df-xp 4729  df-rel 4730  df-cnv 4731  df-dm 4733  df-rn 4734  df-fun 5326  df-fn 5327  df-f 5328

This theorem is referenced by:  elmapg  6825  f1oen2g  6923  f1dom2g  6924  dom3d  6942  mapxpen  7029  addex  9876  mulex  9877  climrecvg1n  11899  cnpfval  14909  txcn  14989  blfvalps  15099