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Theorem ovmpod 6004
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 2178 . 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 6003 1  |-  ( ph  ->  ( A F B )  =  S )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    = wceq 1353    e. wcel 2148  (class class class)co 5877    e. cmpo 5879
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211  ax-setind 4538
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-v 2741  df-sbc 2965  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-br 4006  df-opab 4067  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-iota 5180  df-fun 5220  df-fv 5226  df-ov 5880  df-oprab 5881  df-mpo 5882
This theorem is referenced by:  ovmpoga  6006  iseqovex  10458  seqvalcd  10461  resqrexlemp1rp  11017  resqrexlemfp1  11020  lcmval  12065  ennnfonelemg  12406  imasival  12732  qusval  12749  plusfvalg  12787  grpsubval  12924  mulgval  12991  dvrvald  13308  scafvalg  13402  rmodislmodlem  13445  rmodislmod  13446  cnfval  13779  cnpfval  13780  blvalps  13973  blval  13974
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