Theorem mpt2eq12 5663

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 2353	. . . . 5 |- E = E
2	1	rgenw 2682	. . . 4 |- A.y e. B E = E
3	2	jctr 526	. . 3 |- (B = D -> (B = D /\ A.y e. B E = E))
4	3	ralrimivw 2699	. 2 |- (B = D -> A.x e. A (B = D /\ A.y e. B E = E))
5		mpt2eq123 5662	. 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 460	1 |- ((A = C /\ B = D) -> (x e. A, y e. B |-> E) = (x e. C, y e. D |-> E))

Syntax hints:   -> wi 4   /\ wa 358   = wceq 1642  A.wral 2615   |-> cmpt2 5654

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ral 2620  df-oprab 5529  df-mpt2 5655

This theorem is referenced by: (None)