Theorem mpoeq12 6080

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 2231	. . . . 5 ⊢ 𝐸 = 𝐸
2	1	rgenw 2587	. . . 4 ⊢ ∀𝑦 ∈ 𝐵 𝐸 = 𝐸
3	2	jctr 315	. . 3 ⊢ (𝐵 = 𝐷 → (𝐵 = 𝐷 ∧ ∀𝑦 ∈ 𝐵 𝐸 = 𝐸))
4	3	ralrimivw 2606	. 2 ⊢ (𝐵 = 𝐷 → ∀𝑥 ∈ 𝐴 (𝐵 = 𝐷 ∧ ∀𝑦 ∈ 𝐵 𝐸 = 𝐸))
5		mpoeq123 6079	. 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 1397  ∀wral 2510   ∈ cmpo 6019

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 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-11 1554  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-oprab 6021  df-mpo 6022

This theorem is referenced by:  seqeq1  10711  xpsval  13434  grpsubpropd2  13687  txvalex  14977  txval  14978