Theorem fovcdmd 6063

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 6061	. 2 |- ( ( F : ( R X. S ) --> C /\ A e. R /\ B e. S ) -> ( A F B ) e. C )
5	1, 2, 3, 4	syl3anc 1249	1 |- ( ph -> ( A F B ) e. C )

Syntax hints:   -> wi 4   e. wcel 2164   X. cxp 4657   -->wf 5250  (class class class)co 5918

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-sbc 2986  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-br 4030  df-opab 4091  df-id 4324  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-fv 5262  df-ov 5921

This theorem is referenced by:  eroveu  6680  isxmet2d  14516  ismet2  14522  comet  14667  bdmetval  14668  txmetcnp  14686  limccnp2lem  14830  limccnp2cntop  14831