Theorem mpt2eq12 5596

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

Assertion

Ref	Expression
mpt2eq12	|- ( ( 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 mpt2eq12

Step	Hyp	Ref	Expression
1		eqid 2082	. . . . 5 |- E = E
2	1	rgenw 2419	. . . 4 |- A. y e. B E = E
3	2	jctr 308	. . 3 |- ( B = D -> ( B = D /\ A. y e. B E = E ) )
4	3	ralrimivw 2436	. 2 |- ( B = D -> A. x e. A ( B = D /\ A. y e. B E = E ) )
5		mpt2eq123 5595	. 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 280	1 |- ( ( A = C /\ B = D ) -> ( x e. A , y e. B |-> E ) = ( x e. C , y e. D |-> E ) )

Syntax hints:   -> wi 4   /\ wa 102   = wceq 1285   A.wral 2349   |-> cmpt2 5545

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-11 1438  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064

This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-oprab 5547  df-mpt2 5548

This theorem is referenced by:  iseqeq1  9524  iseqeq4  9527