Theorem mpoeq12 6091

Description: An equality theorem for the maps-to notation. (Contributed by Mario Carneiro, 16-Dec-2013.)

Assertion

Ref	Expression
mpoeq12	|- ( ( A = C /\ B = D ) -> ( x e. A , y e. B |-> E ) = ( x e. C , y e. D |-> E ) )

Distinct variable groups:   x, y, A   x, B, y   x, C, y   x, D, y

Allowed substitution hints:   E( x, y)

Proof of Theorem mpoeq12

Step	Hyp	Ref	Expression
1		eqid 2231	. . . . 5 |- E = E
2	1	rgenw 2588	. . . 4 |- A. y e. B E = E
3	2	jctr 315	. . 3 |- ( B = D -> ( B = D /\ A. y e. B E = E ) )
4	3	ralrimivw 2607	. 2 |- ( B = D -> A. x e. A ( B = D /\ A. y e. B E = E ) )
5		mpoeq123 6090	. 2 |- ( ( A = C /\ A. x e. A ( B = D /\ A. y e. B E = E ) ) -> ( x e. A , y e. B |-> E ) = ( x e. C , y e. D |-> E ) )
6	4, 5	sylan2 286	1 |- ( ( A = C /\ B = D ) -> ( x e. A , y e. B |-> E ) = ( x e. C , y e. D |-> E ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1398   A.wral 2511   e. cmpo 6030

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-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-11 1555  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-oprab 6032  df-mpo 6033

This theorem is referenced by:  seqeq1  10775  xpsval  13515  grpsubpropd2  13768  txvalex  15065  txval  15066