Theorem mpoeq12 6028

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 2207	. . . . 5 |- E = E
2	1	rgenw 2563	. . . 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 2582	. 2 |- ( B = D -> A. x e. A ( B = D /\ A. y e. B E = E ) )
5		mpoeq123 6027	. 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 1373   A.wral 2486   e. cmpo 5969

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 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-11 1530  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ral 2491  df-oprab 5971  df-mpo 5972

This theorem is referenced by:  seqeq1  10632  xpsval  13299  grpsubpropd2  13552  txvalex  14841  txval  14842