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Theorem fovrnda 6220
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
StepHypRef Expression
1 fovrnd.1 . . 3  |-  ( ph  ->  F : ( R  X.  S ) --> C )
2 fovrn 6219 . . 3  |-  ( ( F : ( R  X.  S ) --> C  /\  A  e.  R  /\  B  e.  S
)  ->  ( A F B )  e.  C
)
31, 2syl3an1 1218 . 2  |-  ( (
ph  /\  A  e.  R  /\  B  e.  S
)  ->  ( A F B )  e.  C
)
433expb 1155 1  |-  ( (
ph  /\  ( A  e.  R  /\  B  e.  S ) )  -> 
( A F B )  e.  C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    e. wcel 1726    X. cxp 4879   -->wf 5453  (class class class)co 6084
This theorem is referenced by:  eroprf  7005  yonedalem3  14382  yonedainv  14383  gass  15083  isxmet2d  18362  prdsxmetlem  18403  metideq  24293  sibfof  24659  mamulid  27449  mamurid  27450
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pr 4406
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4216  df-opab 4270  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-fv 5465  df-ov 6087
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