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Theorem fovrnda 5991
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 5990 . . 3  |-  ( ( F : ( R  X.  S ) --> C  /\  A  e.  R  /\  B  e.  S
)  ->  ( A F B )  e.  C
)
31, 2syl3an1 1215 . 2  |-  ( (
ph  /\  A  e.  R  /\  B  e.  S
)  ->  ( A F B )  e.  C
)
433expb 1152 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 1684    X. cxp 4687   -->wf 5251  (class class class)co 5858
This theorem is referenced by:  yonedalem3  14054  yonedainv  14055
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-fv 5263  df-ov 5861
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