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Theorem ovmpt2x 6165
Description: The value of an operation class abstraction. Variant of ovmpt2ga 6166 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 2928 . 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 453 . . 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 453 . . 3  |-  ( ( ( A  e.  C  /\  B  e.  L  /\  S  e.  _V )  /\  x  =  A )  ->  D  =  L )
8 simp1 957 . . 3  |-  ( ( A  e.  C  /\  B  e.  L  /\  S  e.  _V )  ->  A  e.  C )
9 simp2 958 . . 3  |-  ( ( A  e.  C  /\  B  e.  L  /\  S  e.  _V )  ->  B  e.  L )
10 simp3 959 . . 3  |-  ( ( A  e.  C  /\  B  e.  L  /\  S  e.  _V )  ->  S  e.  _V )
113, 5, 7, 8, 9, 10ovmpt2dx 6163 . 2  |-  ( ( A  e.  C  /\  B  e.  L  /\  S  e.  _V )  ->  ( A F B )  =  S )
121, 11syl3an3 1219 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 359    /\ w3a 936    = wceq 1649    e. wcel 1721   _Vcvv 2920  (class class class)co 6044    e. cmpt2 6046
This theorem is referenced by:  ptbasfi  17570  igenval  26565
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 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-sep 4294  ax-nul 4302  ax-pr 4367
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 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-ral 2675  df-rex 2676  df-rab 2679  df-v 2922  df-sbc 3126  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-nul 3593  df-if 3704  df-sn 3784  df-pr 3785  df-op 3787  df-uni 3980  df-br 4177  df-opab 4231  df-id 4462  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-iota 5381  df-fun 5419  df-fv 5425  df-ov 6047  df-oprab 6048  df-mpt2 6049
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