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Theorem ovrspc2v 5895
Description: If an operation value is element of a class for all operands of two classes, then the operation value is an element of the class for specific operands of the two classes. (Contributed by Mario Carneiro, 6-Dec-2014.)
Assertion
Ref Expression
ovrspc2v  |-  ( ( ( X  e.  A  /\  Y  e.  B
)  /\  A. x  e.  A  A. y  e.  B  ( x F y )  e.  C )  ->  ( X F Y )  e.  C )
Distinct variable groups:    x, y, A   
x, B, y    x, C, y    x, F, y   
y, Y    x, X, y
Allowed substitution hint:    Y( x)

Proof of Theorem ovrspc2v
StepHypRef Expression
1 oveq1 5876 . . 3  |-  ( x  =  X  ->  (
x F y )  =  ( X F y ) )
21eleq1d 2246 . 2  |-  ( x  =  X  ->  (
( x F y )  e.  C  <->  ( X F y )  e.  C ) )
3 oveq2 5877 . . 3  |-  ( y  =  Y  ->  ( X F y )  =  ( X F Y ) )
43eleq1d 2246 . 2  |-  ( y  =  Y  ->  (
( X F y )  e.  C  <->  ( X F Y )  e.  C
) )
52, 4rspc2va 2855 1  |-  ( ( ( X  e.  A  /\  Y  e.  B
)  /\  A. x  e.  A  A. y  e.  B  ( x F y )  e.  C )  ->  ( X F Y )  e.  C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    = wceq 1353    e. wcel 2148   A.wral 2455  (class class class)co 5869
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-un 3133  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-br 4001  df-iota 5174  df-fv 5220  df-ov 5872
This theorem is referenced by:  mgmcl  12670  mndpropd  12733  issubmnd  12735  submcl  12760
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