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Theorem ovmpt2dx 6140
Description: Value of an operation given by a maps-to rule, deduction form. (Contributed by Mario Carneiro, 29-Dec-2014.)
Hypotheses
Ref Expression
ovmpt2dx.1  |-  ( ph  ->  F  =  ( x  e.  C ,  y  e.  D  |->  R ) )
ovmpt2dx.2  |-  ( (
ph  /\  ( x  =  A  /\  y  =  B ) )  ->  R  =  S )
ovmpt2dx.3  |-  ( (
ph  /\  x  =  A )  ->  D  =  L )
ovmpt2dx.4  |-  ( ph  ->  A  e.  C )
ovmpt2dx.5  |-  ( ph  ->  B  e.  L )
ovmpt2dx.6  |-  ( ph  ->  S  e.  X )
Assertion
Ref Expression
ovmpt2dx  |-  ( ph  ->  ( A F B )  =  S )
Distinct variable groups:    x, y, A    y, B    y, A    x, B    x, S, y    ph, x, y
Allowed substitution hints:    C( x, y)    D( x, y)    R( x, y)    F( x, y)    L( x, y)    X( x, y)

Proof of Theorem ovmpt2dx
StepHypRef Expression
1 ovmpt2dx.1 . 2  |-  ( ph  ->  F  =  ( x  e.  C ,  y  e.  D  |->  R ) )
2 ovmpt2dx.2 . 2  |-  ( (
ph  /\  ( x  =  A  /\  y  =  B ) )  ->  R  =  S )
3 ovmpt2dx.3 . 2  |-  ( (
ph  /\  x  =  A )  ->  D  =  L )
4 ovmpt2dx.4 . 2  |-  ( ph  ->  A  e.  C )
5 ovmpt2dx.5 . 2  |-  ( ph  ->  B  e.  L )
6 ovmpt2dx.6 . 2  |-  ( ph  ->  S  e.  X )
7 nfv 1626 . 2  |-  F/ x ph
8 nfv 1626 . 2  |-  F/ y
ph
9 nfcv 2524 . 2  |-  F/_ y A
10 nfcv 2524 . 2  |-  F/_ x B
11 nfcv 2524 . 2  |-  F/_ x S
12 nfcv 2524 . 2  |-  F/_ y S
131, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12ovmpt2dxf 6139 1  |-  ( ph  ->  ( A F B )  =  S )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1717  (class class class)co 6021    e. cmpt2 6023
This theorem is referenced by:  ovmpt2d  6141  ovmpt2x  6142  dpjfval  15541  fgval  17824  om1val  18927  pi1val  18934  dvfval  19652  dvnfval  19676  taylfval  20143
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-sep 4272  ax-nul 4280  ax-pr 4345
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-ral 2655  df-rex 2656  df-rab 2659  df-v 2902  df-sbc 3106  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-nul 3573  df-if 3684  df-sn 3764  df-pr 3765  df-op 3767  df-uni 3959  df-br 4155  df-opab 4209  df-id 4440  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-iota 5359  df-fun 5397  df-fv 5403  df-ov 6024  df-oprab 6025  df-mpt2 6026
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