Theorem fovcdmda 6168

Description: An operation's value belongs to its codomain. (Contributed by Mario Carneiro, 29-Dec-2016.)

Hypothesis

Ref	Expression
fovcdmd.1	|- ( ph -> F : ( R X. S ) --> C )

Assertion

Ref	Expression
fovcdmda	|- ( ( ph /\ ( A e. R /\ B e. S ) ) -> ( A F B ) e. C )

Proof of Theorem fovcdmda

Step	Hyp	Ref	Expression
1		fovcdmd.1	. . 3 |- ( ph -> F : ( R X. S ) --> C )
2		fovcdm 6167	. . 3 |- ( ( F : ( R X. S ) --> C /\ A e. R /\ B e. S ) -> ( A F B ) e. C )
3	1, 2	syl3an1 1306	. 2 |- ( ( ph /\ A e. R /\ B e. S ) -> ( A F B ) e. C )
4	3	3expb 1230	1 |- ( ( ph /\ ( A e. R /\ B e. S ) ) -> ( A F B ) e. C )

Syntax hints:   -> wi 4   /\ wa 104   e. wcel 2201   X. cxp 4722   -->wf 5321  (class class class)co 6020

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2204  ax-ext 2212  ax-sep 4206  ax-pow 4263  ax-pr 4298

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1810  df-eu 2081  df-mo 2082  df-clab 2217  df-cleq 2223  df-clel 2226  df-nfc 2362  df-ral 2514  df-rex 2515  df-v 2803  df-sbc 3031  df-un 3203  df-in 3205  df-ss 3212  df-pw 3653  df-sn 3674  df-pr 3675  df-op 3677  df-uni 3893  df-br 4088  df-opab 4150  df-id 4389  df-xp 4730  df-rel 4731  df-cnv 4732  df-co 4733  df-dm 4734  df-rn 4735  df-iota 5285  df-fun 5327  df-fn 5328  df-f 5329  df-fv 5333  df-ov 6023

This theorem is referenced by:  eroprf  6799  isxmet2d  15098