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Theorem ovmpt2dx 5990
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 1609 . 2  |-  F/ x ph
8 nfv 1609 . 2  |-  F/ y
ph
9 nfcv 2432 . 2  |-  F/_ y A
10 nfcv 2432 . 2  |-  F/_ x B
11 nfcv 2432 . 2  |-  F/_ x S
12 nfcv 2432 . 2  |-  F/_ y S
131, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12ovmpt2dxf 5989 1  |-  ( ph  ->  ( A F B )  =  S )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696  (class class class)co 5874    e. cmpt2 5876
This theorem is referenced by:  ovmpt2d  5991  ovmpt2x  5992  dpjfval  15306  fgval  17581  om1val  18544  pi1val  18551  dvfval  19263  dvnfval  19287  taylfval  19754
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-iota 5235  df-fun 5273  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879
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