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Theorem fovrnd 5968
Description: An operation's value belongs to its codomain. (Contributed by Mario Carneiro, 29-Dec-2016.)
Hypotheses
Ref Expression
fovrnd.1  |-  ( ph  ->  F : ( R  X.  S ) --> C )
fovrnd.2  |-  ( ph  ->  A  e.  R )
fovrnd.3  |-  ( ph  ->  B  e.  S )
Assertion
Ref Expression
fovrnd  |-  ( ph  ->  ( A F B )  e.  C )

Proof of Theorem fovrnd
StepHypRef Expression
1 fovrnd.1 . 2  |-  ( ph  ->  F : ( R  X.  S ) --> C )
2 fovrnd.2 . 2  |-  ( ph  ->  A  e.  R )
3 fovrnd.3 . 2  |-  ( ph  ->  B  e.  S )
4 fovrn 5966 . 2  |-  ( ( F : ( R  X.  S ) --> C  /\  A  e.  R  /\  B  e.  S
)  ->  ( A F B )  e.  C
)
51, 2, 3, 4syl3anc 1220 1  |-  ( ph  ->  ( A F B )  e.  C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 2128    X. cxp 4587   -->wf 5169  (class class class)co 5827
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 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-14 2131  ax-ext 2139  ax-sep 4085  ax-pow 4138  ax-pr 4172
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-v 2714  df-sbc 2938  df-un 3106  df-in 3108  df-ss 3115  df-pw 3546  df-sn 3567  df-pr 3568  df-op 3570  df-uni 3775  df-br 3968  df-opab 4029  df-id 4256  df-xp 4595  df-rel 4596  df-cnv 4597  df-co 4598  df-dm 4599  df-rn 4600  df-iota 5138  df-fun 5175  df-fn 5176  df-f 5177  df-fv 5181  df-ov 5830
This theorem is referenced by:  eroveu  6574  isxmet2d  12844  ismet2  12850  comet  12995  bdmetval  12996  txmetcnp  13014  limccnp2lem  13141  limccnp2cntop  13142
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