Theorem fovcdmd 6177

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

Hypotheses

Ref	Expression
fovcdmd.1	|- ( ph -> F : ( R X. S ) --> C )
fovcdmd.2	|- ( ph -> A e. R )
fovcdmd.3	|- ( ph -> B e. S )

Assertion

Ref	Expression
fovcdmd	|- ( ph -> ( A F B ) e. C )

Proof of Theorem fovcdmd

Step	Hyp	Ref	Expression
1		fovcdmd.1	. 2 |- ( ph -> F : ( R X. S ) --> C )
2		fovcdmd.2	. 2 |- ( ph -> A e. R )
3		fovcdmd.3	. 2 |- ( ph -> B e. S )
4		fovcdm 6175	. 2 |- ( ( F : ( R X. S ) --> C /\ A e. R /\ B e. S ) -> ( A F B ) e. C )
5	1, 2, 3, 4	syl3anc 1274	1 |- ( ph -> ( A F B ) e. C )

Syntax hints:   -> wi 4   e. wcel 2202   X. cxp 4729   -->wf 5329  (class class class)co 6028

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-v 2805  df-sbc 3033  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-br 4094  df-opab 4156  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-fv 5341  df-ov 6031

This theorem is referenced by:  eroveu  6838  isxmet2d  15142  ismet2  15148  comet  15293  bdmetval  15294  txmetcnp  15312  limccnp2lem  15470  limccnp2cntop  15471