Theorem fovcdmd 6199

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 6197	. 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 2203   X. cxp 4747   -->wf 5348  (class class class)co 6050

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 2206  ax-ext 2214  ax-sep 4228  ax-pow 4287  ax-pr 4322

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-v 2815  df-sbc 3043  df-un 3215  df-in 3217  df-ss 3224  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-br 4110  df-opab 4172  df-id 4414  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-fv 5360  df-ov 6053

This theorem is referenced by:  eroveu  6860  isxmet2d  15213  ismet2  15219  comet  15364  bdmetval  15365  txmetcnp  15383  limccnp2lem  15541  limccnp2cntop  15542