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Theorem fovrnda 5675
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 5674 . . 3  |-  ( ( F : ( R  X.  S ) --> C  /\  A  e.  R  /\  B  e.  S
)  ->  ( A F B )  e.  C
)
31, 2syl3an1 1203 . 2  |-  ( (
ph  /\  A  e.  R  /\  B  e.  S
)  ->  ( A F B )  e.  C
)
433expb 1140 1  |-  ( (
ph  /\  ( A  e.  R  /\  B  e.  S ) )  -> 
( A F B )  e.  C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    e. wcel 1434    X. cxp 4369   -->wf 4928  (class class class)co 5543
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3904  ax-pow 3956  ax-pr 3972
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rex 2355  df-v 2604  df-sbc 2817  df-un 2978  df-in 2980  df-ss 2987  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-uni 3610  df-br 3794  df-opab 3848  df-id 4056  df-xp 4377  df-rel 4378  df-cnv 4379  df-co 4380  df-dm 4381  df-rn 4382  df-iota 4897  df-fun 4934  df-fn 4935  df-f 4936  df-fv 4940  df-ov 5546
This theorem is referenced by:  eroprf  6265
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