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Theorem ov4gc 25124
Description: Value of a composition. ovmpt4g 5970 adapted to the special case of a composite. (Contributed by FL, 14-Jul-2012.)
Hypothesis
Ref Expression
ov4gc.1  |-  O  =  ( x  e.  C ,  y  e.  D  |->  ( x  o.  y
) )
Assertion
Ref Expression
ov4gc  |-  ( ( x  e.  C  /\  y  e.  D )  ->  ( x O y )  =  ( x  o.  y ) )
Distinct variable groups:    x, C, y    x, D, y
Allowed substitution hints:    O( x, y)

Proof of Theorem ov4gc
StepHypRef Expression
1 coexg 5215 . 2  |-  ( ( x  e.  C  /\  y  e.  D )  ->  ( x  o.  y
)  e.  _V )
2 ov4gc.1 . . 3  |-  O  =  ( x  e.  C ,  y  e.  D  |->  ( x  o.  y
) )
32ovmpt4g 5970 . 2  |-  ( ( x  e.  C  /\  y  e.  D  /\  ( x  o.  y
)  e.  _V )  ->  ( x O y )  =  ( x  o.  y ) )
41, 3mpd3an3 1278 1  |-  ( ( x  e.  C  /\  y  e.  D )  ->  ( x O y )  =  ( x  o.  y ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   _Vcvv 2788    o. ccom 4693  (class class class)co 5858    e. cmpt2 5860
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-13 1686  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-pow 4188  ax-pr 4214  ax-un 4512
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-pw 3627  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-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863
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