Theorem fnovex 5538

Description: The result of an operation is a set. (Contributed by Jim Kingdon, 15-Jan-2019.)

Assertion

Ref	Expression
fnovex	|- ( ( F Fn ( C X. D ) /\ A e. C /\ B e. D ) -> ( A F B ) e. _V )

Proof of Theorem fnovex

Step	Hyp	Ref	Expression
1		df-ov 5515	. 2 |- ( A F B ) = ( F ` <. A , B >. )
2		opelxp 4374	. . . 4 |- ( <. A , B >. e. ( C X. D ) <-> ( A e. C /\ B e. D ) )
3		funfvex 5192	. . . . 5 |- ( ( Fun F /\ <. A , B >. e. dom F ) -> ( F ` <. A , B >. ) e. _V )
4	3	funfni 4999	. . . 4 |- ( ( F Fn ( C X. D ) /\ <. A , B >. e. ( C X. D ) ) -> ( F ` <. A , B >. ) e. _V )
5	2, 4	sylan2br 272	. . 3 |- ( ( F Fn ( C X. D ) /\ ( A e. C /\ B e. D ) ) -> ( F ` <. A , B >. ) e. _V )
6	5	3impb 1100	. 2 |- ( ( F Fn ( C X. D ) /\ A e. C /\ B e. D ) -> ( F ` <. A , B >. ) e. _V )
7	1, 6	syl5eqel 2124	1 |- ( ( F Fn ( C X. D ) /\ A e. C /\ B e. D ) -> ( A F B ) e. _V )

Syntax hints:   -> wi 4   /\ wa 97   /\ w3a 885   e. wcel 1393   _Vcvv 2557   <.cop 3378   X. cxp 4343   Fn wfn 4897   ` cfv 4902  (class class class)co 5512

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-sbc 2765  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-opab 3819  df-id 4030  df-xp 4351  df-cnv 4353  df-co 4354  df-dm 4355  df-iota 4867  df-fun 4904  df-fn 4905  df-fv 4910  df-ov 5515

This theorem is referenced by:  ovelrn  5649  fnofval  5721  fzen  8907