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Theorem fovrnda 6117
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 6116 . . 3  |-  ( ( F : ( R  X.  S ) --> C  /\  A  e.  R  /\  B  e.  S
)  ->  ( A F B )  e.  C
)
31, 2syl3an1 1216 . 2  |-  ( (
ph  /\  A  e.  R  /\  B  e.  S
)  ->  ( A F B )  e.  C
)
433expb 1153 1  |-  ( (
ph  /\  ( A  e.  R  /\  B  e.  S ) )  -> 
( A F B )  e.  C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    e. wcel 1715    X. cxp 4790   -->wf 5354  (class class class)co 5981
This theorem is referenced by:  yonedalem3  14264  yonedainv  14265
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-sep 4243  ax-nul 4251  ax-pr 4316
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-ral 2633  df-rex 2634  df-rab 2637  df-v 2875  df-sbc 3078  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-nul 3544  df-if 3655  df-sn 3735  df-pr 3736  df-op 3738  df-uni 3930  df-br 4126  df-opab 4180  df-id 4412  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-rn 4803  df-iota 5322  df-fun 5360  df-fn 5361  df-f 5362  df-fv 5366  df-ov 5984
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