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Theorem ovmpt2x 5657
Description: The value of an operation class abstraction. Variant of ovmpt2ga 5658 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 2583 . 2  |-  ( S  e.  H  ->  S  e.  _V )
2 ovmpt2x.3 . . . 4  |-  F  =  ( x  e.  C ,  y  e.  D  |->  R )
32a1i 9 . . 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 266 . . 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 266 . . 3  |-  ( ( ( A  e.  C  /\  B  e.  L  /\  S  e.  _V )  /\  x  =  A )  ->  D  =  L )
8 simp1 915 . . 3  |-  ( ( A  e.  C  /\  B  e.  L  /\  S  e.  _V )  ->  A  e.  C )
9 simp2 916 . . 3  |-  ( ( A  e.  C  /\  B  e.  L  /\  S  e.  _V )  ->  B  e.  L )
10 simp3 917 . . 3  |-  ( ( A  e.  C  /\  B  e.  L  /\  S  e.  _V )  ->  S  e.  _V )
113, 5, 7, 8, 9, 10ovmpt2dx 5655 . 2  |-  ( ( A  e.  C  /\  B  e.  L  /\  S  e.  _V )  ->  ( A F B )  =  S )
121, 11syl3an3 1181 1  |-  ( ( A  e.  C  /\  B  e.  L  /\  S  e.  H )  ->  ( A F B )  =  S )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 101    /\ w3a 896    = wceq 1259    e. wcel 1409   _Vcvv 2574  (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: (None)
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