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Theorem ovmpt2x 6430
Description: The value of an operation class abstraction. Variant of ovmpt2ga 6431 which does not require  D and  x to be distinct. (Contributed by Jeff Madsen, 10-Jun-2010.) (Revised by Mario Carneiro, 20-Dec-2013.)
Hypotheses
Ref Expression
ovmpt2x.1  |-  ( ( x  =  A  /\  y  =  B )  ->  R  =  S )
ovmpt2x.2  |-  ( x  =  A  ->  D  =  L )
ovmpt2x.3  |-  F  =  ( x  e.  C ,  y  e.  D  |->  R )
Assertion
Ref Expression
ovmpt2x  |-  ( ( A  e.  C  /\  B  e.  L  /\  S  e.  H )  ->  ( A F B )  =  S )
Distinct variable groups:    x, y, A    x, B, y    x, C, y    x, L, y   
x, S, y
Allowed substitution hints:    D( x, y)    R( x, y)    F( x, y)    H( x, y)

Proof of Theorem ovmpt2x
StepHypRef Expression
1 elex 3118 . 2  |-  ( S  e.  H  ->  S  e.  _V )
2 ovmpt2x.3 . . . 4  |-  F  =  ( x  e.  C ,  y  e.  D  |->  R )
32a1i 11 . . 3  |-  ( ( A  e.  C  /\  B  e.  L  /\  S  e.  _V )  ->  F  =  ( x  e.  C ,  y  e.  D  |->  R ) )
4 ovmpt2x.1 . . . 4  |-  ( ( x  =  A  /\  y  =  B )  ->  R  =  S )
54adantl 466 . . 3  |-  ( ( ( A  e.  C  /\  B  e.  L  /\  S  e.  _V )  /\  ( x  =  A  /\  y  =  B ) )  ->  R  =  S )
6 ovmpt2x.2 . . . 4  |-  ( x  =  A  ->  D  =  L )
76adantl 466 . . 3  |-  ( ( ( A  e.  C  /\  B  e.  L  /\  S  e.  _V )  /\  x  =  A )  ->  D  =  L )
8 simp1 996 . . 3  |-  ( ( A  e.  C  /\  B  e.  L  /\  S  e.  _V )  ->  A  e.  C )
9 simp2 997 . . 3  |-  ( ( A  e.  C  /\  B  e.  L  /\  S  e.  _V )  ->  B  e.  L )
10 simp3 998 . . 3  |-  ( ( A  e.  C  /\  B  e.  L  /\  S  e.  _V )  ->  S  e.  _V )
113, 5, 7, 8, 9, 10ovmpt2dx 6428 . 2  |-  ( ( A  e.  C  /\  B  e.  L  /\  S  e.  _V )  ->  ( A F B )  =  S )
121, 11syl3an3 1263 1  |-  ( ( A  e.  C  /\  B  e.  L  /\  S  e.  H )  ->  ( A F B )  =  S )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 973    = wceq 1395    e. wcel 1819   _Vcvv 3109  (class class class)co 6296    |-> cmpt2 6298
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pr 4695
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-opab 4516  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-iota 5557  df-fun 5596  df-fv 5602  df-ov 6299  df-oprab 6300  df-mpt2 6301
This theorem is referenced by:  evls1fval  18482  ptbasfi  20207  tglngval  24063  igenval  30620  lcoop  33114
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