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Theorem ovmpt2dx 6192
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 1629 . 2  |-  F/ x ph
8 nfv 1629 . 2  |-  F/ y
ph
9 nfcv 2571 . 2  |-  F/_ y A
10 nfcv 2571 . 2  |-  F/_ x B
11 nfcv 2571 . 2  |-  F/_ x S
12 nfcv 2571 . 2  |-  F/_ y S
131, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12ovmpt2dxf 6191 1  |-  ( ph  ->  ( A F B )  =  S )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725  (class class class)co 6073    e. cmpt2 6075
This theorem is referenced by:  ovmpt2d  6193  ovmpt2x  6194  dpjfval  15605  fgval  17894  om1val  19047  pi1val  19054  dvfval  19776  dvnfval  19800  taylfval  20267
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-iota 5410  df-fun 5448  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078
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