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 ∈ A, y ∈ B ↦ E) = (x ∈ C, y ∈ 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 ⊢ ∀y ∈ B E = E
3	2	jctr 526	. . 3 ⊢ (B = D → (B = D ∧ ∀y ∈ B E = E))
4	3	ralrimivw 2699	. 2 ⊢ (B = D → ∀x ∈ A (B = D ∧ ∀y ∈ B E = E))
5		mpt2eq123 5662	. 2 ⊢ ((A = C ∧ ∀x ∈ A (B = D ∧ ∀y ∈ B E = E)) → (x ∈ A, y ∈ B ↦ E) = (x ∈ C, y ∈ D ↦ E))
6	4, 5	sylan2 460	1 ⊢ ((A = C ∧ B = D) → (x ∈ A, y ∈ B ↦ E) = (x ∈ C, y ∈ D ↦ E))

Syntax hints:   → wi 4   ∧ wa 358   = wceq 1642  ∀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)