Theorem fovrnda 5993

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

Hypothesis

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

Assertion

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

Proof of Theorem fovrnda

Step	Hyp	Ref	Expression
1		fovrnd.1	. . 3 |- ( ph -> F : ( R X. S ) --> C )
2		fovrn 5992	. . 3 |- ( ( F : ( R X. S ) --> C /\ A e. R /\ B e. S ) -> ( A F B ) e. C )
3	1, 2	syl3an1 1266	. 2 |- ( ( ph /\ A e. R /\ B e. S ) -> ( A F B ) e. C )
4	3	3expb 1199	1 |- ( ( ph /\ ( A e. R /\ B e. S ) ) -> ( A F B ) e. C )

Syntax hints:   -> wi 4   /\ wa 103   e. wcel 2141   X. cxp 4607   -->wf 5192  (class class class)co 5850

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4105  ax-pow 4158  ax-pr 4192

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-sbc 2956  df-un 3125  df-in 3127  df-ss 3134  df-pw 3566  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3795  df-br 3988  df-opab 4049  df-id 4276  df-xp 4615  df-rel 4616  df-cnv 4617  df-co 4618  df-dm 4619  df-rn 4620  df-iota 5158  df-fun 5198  df-fn 5199  df-f 5200  df-fv 5204  df-ov 5853

This theorem is referenced by:  eroprf  6602  isxmet2d  13101