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Theorem ovmpox 5978
Description: The value of an operation class abstraction. Variant of ovmpoga 5979 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
ovmpox.1  |-  ( ( x  =  A  /\  y  =  B )  ->  R  =  S )
ovmpox.2  |-  ( x  =  A  ->  D  =  L )
ovmpox.3  |-  F  =  ( x  e.  C ,  y  e.  D  |->  R )
Assertion
Ref Expression
ovmpox  |-  ( ( 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 ovmpox
StepHypRef Expression
1 elex 2741 . 2  |-  ( S  e.  H  ->  S  e.  _V )
2 ovmpox.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 ovmpox.1 . . . 4  |-  ( ( x  =  A  /\  y  =  B )  ->  R  =  S )
54adantl 275 . . 3  |-  ( ( ( A  e.  C  /\  B  e.  L  /\  S  e.  _V )  /\  ( x  =  A  /\  y  =  B ) )  ->  R  =  S )
6 ovmpox.2 . . . 4  |-  ( x  =  A  ->  D  =  L )
76adantl 275 . . 3  |-  ( ( ( A  e.  C  /\  B  e.  L  /\  S  e.  _V )  /\  x  =  A )  ->  D  =  L )
8 simp1 992 . . 3  |-  ( ( A  e.  C  /\  B  e.  L  /\  S  e.  _V )  ->  A  e.  C )
9 simp2 993 . . 3  |-  ( ( A  e.  C  /\  B  e.  L  /\  S  e.  _V )  ->  B  e.  L )
10 simp3 994 . . 3  |-  ( ( A  e.  C  /\  B  e.  L  /\  S  e.  _V )  ->  S  e.  _V )
113, 5, 7, 8, 9, 10ovmpodx 5976 . 2  |-  ( ( A  e.  C  /\  B  e.  L  /\  S  e.  _V )  ->  ( A F B )  =  S )
121, 11syl3an3 1268 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 103    /\ w3a 973    = wceq 1348    e. wcel 2141   _Vcvv 2730  (class class class)co 5850    e. cmpo 5852
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4105  ax-pow 4158  ax-pr 4192  ax-setind 4519
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-v 2732  df-sbc 2956  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-pw 3566  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3795  df-br 3988  df-opab 4049  df-id 4276  df-xp 4615  df-rel 4616  df-cnv 4617  df-co 4618  df-dm 4619  df-iota 5158  df-fun 5198  df-fv 5204  df-ov 5853  df-oprab 5854  df-mpo 5855
This theorem is referenced by:  reldvg  13401
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