Theorem ovrspc2v 6084

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 6065	. . 3 |- ( x = X -> ( x F y ) = ( X F y ) )
2	1	eleq1d 2303	. 2 |- ( x = X -> ( ( x F y ) e. C <-> ( X F y ) e. C ) )
3		oveq2 6066	. . 3 |- ( y = Y -> ( X F y ) = ( X F Y ) )
4	3	eleq1d 2303	. 2 |- ( y = Y -> ( ( X F y ) e. C <-> ( X F Y ) e. C ) )
5	2, 4	rspc2va 2938	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 1398   e. wcel 2205   A.wral 2522  (class class class)co 6058

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2216

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-v 2817  df-un 3218  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-br 4115  df-iota 5317  df-fv 5365  df-ov 6061

This theorem is referenced by:  ercpbl  13595  mgmcl  13622  sgrppropd  13676  mndpropd  13701  issubmnd  13703  submcl  13734  issubg2m  13942  lmodprop2d  14622  lsspropdg  14705