Theorem fovcdmda 6054

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

Hypothesis

Ref	Expression
fovcdmd.1	|- ( ph -> F : ( R X. S ) --> C )

Assertion

Ref	Expression
fovcdmda	|- ( ( ph /\ ( A e. R /\ B e. S ) ) -> ( A F B ) e. C )

Proof of Theorem fovcdmda

Step	Hyp	Ref	Expression
1		fovcdmd.1	. . 3 |- ( ph -> F : ( R X. S ) --> C )
2		fovcdm 6053	. . 3 |- ( ( F : ( R X. S ) --> C /\ A e. R /\ B e. S ) -> ( A F B ) e. C )
3	1, 2	syl3an1 1282	. 2 |- ( ( ph /\ A e. R /\ B e. S ) -> ( A F B ) e. C )
4	3	3expb 1206	1 |- ( ( ph /\ ( A e. R /\ B e. S ) ) -> ( A F B ) e. C )

Syntax hints:   -> wi 4   /\ wa 104   e. wcel 2164   X. cxp 4653   -->wf 5242  (class class class)co 5910

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 4322  df-xp 4661  df-rel 4662  df-cnv 4663  df-co 4664  df-dm 4665  df-rn 4666  df-iota 5207  df-fun 5248  df-fn 5249  df-f 5250  df-fv 5254  df-ov 5913

This theorem is referenced by:  eroprf  6673  isxmet2d  14493