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Theorem fovrnda 6337
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 6336 . . 3  |-  ( ( F : ( R  X.  S ) --> C  /\  A  e.  R  /\  B  e.  S
)  ->  ( A F B )  e.  C
)
31, 2syl3an1 1252 . 2  |-  ( (
ph  /\  A  e.  R  /\  B  e.  S
)  ->  ( A F B )  e.  C
)
433expb 1189 1  |-  ( (
ph  /\  ( A  e.  R  /\  B  e.  S ) )  -> 
( A F B )  e.  C )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    e. wcel 1758    X. cxp 4939   -->wf 5515  (class class class)co 6193
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4514  ax-nul 4522  ax-pr 4632
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3073  df-sbc 3288  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-nul 3739  df-if 3893  df-sn 3979  df-pr 3981  df-op 3985  df-uni 4193  df-br 4394  df-opab 4452  df-id 4737  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-rn 4952  df-iota 5482  df-fun 5521  df-fn 5522  df-f 5523  df-fv 5527  df-ov 6196
This theorem is referenced by:  eroprf  7301  yonedalem3  15201  yonedainv  15202  gass  15930  mamulid  18422  mamurid  18423  maducoeval2  18571  madutpos  18573  madugsum  18574  madurid  18575  isxmet2d  20027  prdsxmetlem  20068  rrxds  21022  metideq  26458  sibfof  26863  sseqfv2  26914
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