Theorem ovrspc2v 5904

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

Step	Hyp	Ref	Expression
1		oveq1 5885	. . 3 |- ( x = X -> ( x F y ) = ( X F y ) )
2	1	eleq1d 2246	. 2 |- ( x = X -> ( ( x F y ) e. C <-> ( X F y ) e. C ) )
3		oveq2 5886	. . 3 |- ( y = Y -> ( X F y ) = ( X F Y ) )
4	3	eleq1d 2246	. 2 |- ( y = Y -> ( ( X F y ) e. C <-> ( X F Y ) e. C ) )
5	2, 4	rspc2va 2857	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 )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1353   e. wcel 2148   A.wral 2455  (class class class)co 5878

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 2741  df-un 3135  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-br 4006  df-iota 5180  df-fv 5226  df-ov 5881

This theorem is referenced by:  ercpbl  12756  mgmcl  12784  mndpropd  12847  issubmnd  12849  submcl  12876  issubg2m  13055  lmodprop2d  13444  lsspropdg  13523