Theorem fovcdmd 6114

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 6112	. 2 |- ( ( F : ( R X. S ) --> C /\ A e. R /\ B e. S ) -> ( A F B ) e. C )
5	1, 2, 3, 4	syl3anc 1250	1 |- ( ph -> ( A F B ) e. C )

Syntax hints:   -> wi 4   e. wcel 2178   X. cxp 4691   -->wf 5286  (class class class)co 5967

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-14 2181  ax-ext 2189  ax-sep 4178  ax-pow 4234  ax-pr 4269

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ral 2491  df-rex 2492  df-v 2778  df-sbc 3006  df-un 3178  df-in 3180  df-ss 3187  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-br 4060  df-opab 4122  df-id 4358  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-iota 5251  df-fun 5292  df-fn 5293  df-f 5294  df-fv 5298  df-ov 5970

This theorem is referenced by:  eroveu  6736  isxmet2d  14935  ismet2  14941  comet  15086  bdmetval  15087  txmetcnp  15105  limccnp2lem  15263  limccnp2cntop  15264