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Theorem ovmpt2x 6063
Description: The value of an operation class abstraction. Variant of ovmpt2ga 6064 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 2872 . 2  |-  ( S  e.  H  ->  S  e.  _V )
2 ovmpt2x.3 . . . 4  |-  F  =  ( x  e.  C ,  y  e.  D  |->  R )
32a1i 10 . . 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 452 . . 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 452 . . 3  |-  ( ( ( A  e.  C  /\  B  e.  L  /\  S  e.  _V )  /\  x  =  A )  ->  D  =  L )
8 simp1 955 . . 3  |-  ( ( A  e.  C  /\  B  e.  L  /\  S  e.  _V )  ->  A  e.  C )
9 simp2 956 . . 3  |-  ( ( A  e.  C  /\  B  e.  L  /\  S  e.  _V )  ->  B  e.  L )
10 simp3 957 . . 3  |-  ( ( A  e.  C  /\  B  e.  L  /\  S  e.  _V )  ->  S  e.  _V )
113, 5, 7, 8, 9, 10ovmpt2dx 6061 . 2  |-  ( ( A  e.  C  /\  B  e.  L  /\  S  e.  _V )  ->  ( A F B )  =  S )
121, 11syl3an3 1217 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 358    /\ w3a 934    = wceq 1642    e. wcel 1710   _Vcvv 2864  (class class class)co 5945    e. cmpt2 5947
This theorem is referenced by:  ptbasfi  17382  igenval  26009
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4222  ax-nul 4230  ax-pr 4295
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-rab 2628  df-v 2866  df-sbc 3068  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3909  df-br 4105  df-opab 4159  df-id 4391  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-iota 5301  df-fun 5339  df-fv 5345  df-ov 5948  df-oprab 5949  df-mpt2 5950
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