Theorem ovrspc2v 5868

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 5849	. . 3 |- ( x = X -> ( x F y ) = ( X F y ) )
2	1	eleq1d 2235	. 2 |- ( x = X -> ( ( x F y ) e. C <-> ( X F y ) e. C ) )
3		oveq2 5850	. . 3 |- ( y = Y -> ( X F y ) = ( X F Y ) )
4	3	eleq1d 2235	. 2 |- ( y = Y -> ( ( X F y ) e. C <-> ( X F Y ) e. C ) )
5	2, 4	rspc2va 2844	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 103   = wceq 1343   e. wcel 2136   A.wral 2444  (class class class)co 5842

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-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-un 3120  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-br 3983  df-iota 5153  df-fv 5196  df-ov 5845

This theorem is referenced by:  mgmcl  12590