Theorem mpoeq12 5937

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 ∈ 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 mpoeq12

Step	Hyp	Ref	Expression
1		eqid 2177	. . . . 5 ⊢ 𝐸 = 𝐸
2	1	rgenw 2532	. . . 4 ⊢ ∀𝑦 ∈ 𝐵 𝐸 = 𝐸
3	2	jctr 315	. . 3 ⊢ (𝐵 = 𝐷 → (𝐵 = 𝐷 ∧ ∀𝑦 ∈ 𝐵 𝐸 = 𝐸))
4	3	ralrimivw 2551	. 2 ⊢ (𝐵 = 𝐷 → ∀𝑥 ∈ 𝐴 (𝐵 = 𝐷 ∧ ∀𝑦 ∈ 𝐵 𝐸 = 𝐸))
5		mpoeq123 5936	. 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 286	1 ⊢ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) → (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐸) = (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝐸))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1353  ∀wral 2455   ∈ cmpo 5879

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 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-11 1506  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-oprab 5881  df-mpo 5882

This theorem is referenced by:  seqeq1  10450  xpsval  12776  grpsubpropd2  12980  txvalex  13839  txval  13840