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Theorem ovmpt2g 5663
Description: Value of an operation given by a maps-to rule. Special case. (Contributed by NM, 14-Sep-1999.) (Revised by David Abernethy, 19-Jun-2012.)
Hypotheses
Ref Expression
ovmpt2g.1  |-  ( x  =  A  ->  R  =  G )
ovmpt2g.2  |-  ( y  =  B  ->  G  =  S )
ovmpt2g.3  |-  F  =  ( x  e.  C ,  y  e.  D  |->  R )
Assertion
Ref Expression
ovmpt2g  |-  ( ( A  e.  C  /\  B  e.  D  /\  S  e.  H )  ->  ( A F B )  =  S )
Distinct variable groups:    x, y, A   
x, B, y    x, C, y    x, D, y   
x, S, y
Allowed substitution hints:    R( x, y)    F( x, y)    G( x, y)    H( x, y)

Proof of Theorem ovmpt2g
StepHypRef Expression
1 ovmpt2g.1 . . 3  |-  ( x  =  A  ->  R  =  G )
2 ovmpt2g.2 . . 3  |-  ( y  =  B  ->  G  =  S )
31, 2sylan9eq 2108 . 2  |-  ( ( x  =  A  /\  y  =  B )  ->  R  =  S )
4 ovmpt2g.3 . 2  |-  F  =  ( x  e.  C ,  y  e.  D  |->  R )
53, 4ovmpt2ga 5658 1  |-  ( ( A  e.  C  /\  B  e.  D  /\  S  e.  H )  ->  ( A F B )  =  S )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 896    = wceq 1259    e. wcel 1409  (class class class)co 5540    |-> cmpt2 5542
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-pow 3955  ax-pr 3972  ax-setind 4290
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-fal 1265  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ne 2221  df-ral 2328  df-rex 2329  df-v 2576  df-sbc 2788  df-dif 2948  df-un 2950  df-in 2952  df-ss 2959  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-br 3793  df-opab 3847  df-id 4058  df-xp 4379  df-rel 4380  df-cnv 4381  df-co 4382  df-dm 4383  df-iota 4895  df-fun 4932  df-fv 4938  df-ov 5543  df-oprab 5544  df-mpt2 5545
This theorem is referenced by:  ovmpt2  5664  oav  6065  omv  6066  oeiv  6067  mulpipq2  6527  genipv  6665  genpelxp  6667  subval  7266  divvalap  7727  cnref1o  8680  modqval  9274  frecuzrdgrrn  9358  frec2uzrdg  9359  frecuzrdgsuc  9365  iseqval  9384  iseqp1  9389  expival  9422  bcval  9617  shftfvalg  9647  shftfval  9650  cnrecnv  9738
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