Theorem mpoeq12 5913

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 2170	. . . . 5 |- E = E
2	1	rgenw 2525	. . . 4 |- A. y e. B E = E
3	2	jctr 313	. . 3 |- ( B = D -> ( B = D /\ A. y e. B E = E ) )
4	3	ralrimivw 2544	. 2 |- ( B = D -> A. x e. A ( B = D /\ A. y e. B E = E ) )
5		mpoeq123 5912	. 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 284	1 |- ( ( A = C /\ B = D ) -> ( x e. A , y e. B |-> E ) = ( x e. C , y e. D |-> E ) )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1348   A.wral 2448   e. cmpo 5855

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-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-11 1499  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-oprab 5857  df-mpo 5858

This theorem is referenced by:  seqeq1  10404  txvalex  13048  txval  13049