Theorem fovcdmd 6091

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 6089	. 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 2176   X. cxp 4673   -->wf 5267  (class class class)co 5944

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 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-14 2179  ax-ext 2187  ax-sep 4162  ax-pow 4218  ax-pr 4253

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-rex 2490  df-v 2774  df-sbc 2999  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-br 4045  df-opab 4106  df-id 4340  df-xp 4681  df-rel 4682  df-cnv 4683  df-co 4684  df-dm 4685  df-rn 4686  df-iota 5232  df-fun 5273  df-fn 5274  df-f 5275  df-fv 5279  df-ov 5947

This theorem is referenced by:  eroveu  6713  isxmet2d  14820  ismet2  14826  comet  14971  bdmetval  14972  txmetcnp  14990  limccnp2lem  15148  limccnp2cntop  15149