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Theorem ovmpod 5905
Description: Value of an operation given by a maps-to rule, deduction form. (Contributed by Mario Carneiro, 7-Dec-2014.)
Hypotheses
Ref Expression
ovmpod.1  |-  ( ph  ->  F  =  ( x  e.  C ,  y  e.  D  |->  R ) )
ovmpod.2  |-  ( (
ph  /\  ( x  =  A  /\  y  =  B ) )  ->  R  =  S )
ovmpod.3  |-  ( ph  ->  A  e.  C )
ovmpod.4  |-  ( ph  ->  B  e.  D )
ovmpod.5  |-  ( ph  ->  S  e.  X )
Assertion
Ref Expression
ovmpod  |-  ( ph  ->  ( A F B )  =  S )
Distinct variable groups:    x, y, A   
x, B, y    x, S, y    ph, x, y
Allowed substitution hints:    C( x, y)    D( x, y)    R( x, y)    F( x, y)    X( x, y)

Proof of Theorem ovmpod
StepHypRef Expression
1 ovmpod.1 . 2  |-  ( ph  ->  F  =  ( x  e.  C ,  y  e.  D  |->  R ) )
2 ovmpod.2 . 2  |-  ( (
ph  /\  ( x  =  A  /\  y  =  B ) )  ->  R  =  S )
3 eqidd 2141 . 2  |-  ( (
ph  /\  x  =  A )  ->  D  =  D )
4 ovmpod.3 . 2  |-  ( ph  ->  A  e.  C )
5 ovmpod.4 . 2  |-  ( ph  ->  B  e.  D )
6 ovmpod.5 . 2  |-  ( ph  ->  S  e.  X )
71, 2, 3, 4, 5, 6ovmpodx 5904 1  |-  ( ph  ->  ( A F B )  =  S )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1332    e. wcel 1481  (class class class)co 5781    e. cmpo 5783
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4053  ax-pow 4105  ax-pr 4138  ax-setind 4459
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-ral 2422  df-rex 2423  df-v 2691  df-sbc 2913  df-dif 3077  df-un 3079  df-in 3081  df-ss 3088  df-pw 3516  df-sn 3537  df-pr 3538  df-op 3540  df-uni 3744  df-br 3937  df-opab 3997  df-id 4222  df-xp 4552  df-rel 4553  df-cnv 4554  df-co 4555  df-dm 4556  df-iota 5095  df-fun 5132  df-fv 5138  df-ov 5784  df-oprab 5785  df-mpo 5786
This theorem is referenced by:  ovmpoga  5907  iseqovex  10259  seqvalcd  10262  resqrexlemp1rp  10809  resqrexlemfp1  10812  lcmval  11778  ennnfonelemg  11950  cnfval  12400  cnpfval  12401  blvalps  12594  blval  12595
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