Theorem fovrnda 5985

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 5984	. . 3 |- ( ( F : ( R X. S ) --> C /\ A e. R /\ B e. S ) -> ( A F B ) e. C )
3	1, 2	syl3an1 1261	. 2 |- ( ( ph /\ A e. R /\ B e. S ) -> ( A F B ) e. C )
4	3	3expb 1194	1 |- ( ( ph /\ ( A e. R /\ B e. S ) ) -> ( A F B ) e. C )

Syntax hints:   -> wi 4   /\ wa 103   e. wcel 2136   X. cxp 4602   -->wf 5184  (class class class)co 5842

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 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-sbc 2952  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-br 3983  df-opab 4044  df-id 4271  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-fv 5196  df-ov 5845

This theorem is referenced by:  eroprf  6594  isxmet2d  12988