Theorem mpteq12f 3858

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

Assertion

Ref	Expression
mpteq12f	⊢ ((∀𝑥 𝐴 = 𝐶 ∧ ∀𝑥 ∈ 𝐴 𝐵 = 𝐷) → (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐶 ↦ 𝐷))

Proof of Theorem mpteq12f

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfa1 1474	. . . 4 ⊢ Ⅎ𝑥∀𝑥 𝐴 = 𝐶
2		nfra1 2397	. . . 4 ⊢ Ⅎ𝑥∀𝑥 ∈ 𝐴 𝐵 = 𝐷
3	1, 2	nfan 1497	. . 3 ⊢ Ⅎ𝑥(∀𝑥 𝐴 = 𝐶 ∧ ∀𝑥 ∈ 𝐴 𝐵 = 𝐷)
4		nfv 1461	. . 3 ⊢ Ⅎ𝑦(∀𝑥 𝐴 = 𝐶 ∧ ∀𝑥 ∈ 𝐴 𝐵 = 𝐷)
5		rsp 2411	. . . . . . 7 ⊢ (∀𝑥 ∈ 𝐴 𝐵 = 𝐷 → (𝑥 ∈ 𝐴 → 𝐵 = 𝐷))
6	5	imp 122	. . . . . 6 ⊢ ((∀𝑥 ∈ 𝐴 𝐵 = 𝐷 ∧ 𝑥 ∈ 𝐴) → 𝐵 = 𝐷)
7	6	eqeq2d 2092	. . . . 5 ⊢ ((∀𝑥 ∈ 𝐴 𝐵 = 𝐷 ∧ 𝑥 ∈ 𝐴) → (𝑦 = 𝐵 ↔ 𝑦 = 𝐷))
8	7	pm5.32da 439	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 𝐵 = 𝐷 → ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷)))
9		sp 1441	. . . . . 6 ⊢ (∀𝑥 𝐴 = 𝐶 → 𝐴 = 𝐶)
10	9	eleq2d 2148	. . . . 5 ⊢ (∀𝑥 𝐴 = 𝐶 → (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐶))
11	10	anbi1d 452	. . . 4 ⊢ (∀𝑥 𝐴 = 𝐶 → ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷) ↔ (𝑥 ∈ 𝐶 ∧ 𝑦 = 𝐷)))
12	8, 11	sylan9bbr 450	. . 3 ⊢ ((∀𝑥 𝐴 = 𝐶 ∧ ∀𝑥 ∈ 𝐴 𝐵 = 𝐷) → ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵) ↔ (𝑥 ∈ 𝐶 ∧ 𝑦 = 𝐷)))
13	3, 4, 12	opabbid 3843	. 2 ⊢ ((∀𝑥 𝐴 = 𝐶 ∧ ∀𝑥 ∈ 𝐴 𝐵 = 𝐷) → {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)} = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐶 ∧ 𝑦 = 𝐷)})
14		df-mpt 3841	. 2 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)}
15		df-mpt 3841	. 2 ⊢ (𝑥 ∈ 𝐶 ↦ 𝐷) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐶 ∧ 𝑦 = 𝐷)}
16	13, 14, 15	3eqtr4g 2138	1 ⊢ ((∀𝑥 𝐴 = 𝐶 ∧ ∀𝑥 ∈ 𝐴 𝐵 = 𝐷) → (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐶 ↦ 𝐷))

Syntax hints:   → wi 4   ∧ wa 102  ∀wal 1282   = wceq 1284   ∈ wcel 1433  ∀wral 2348  {copab 3838   ↦ cmpt 3839

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 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-11 1437  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-ral 2353  df-opab 3840  df-mpt 3841

This theorem is referenced by:  mpteq12dva  3859  mpteq12  3861  mpteq2ia  3864  mpteq2da  3867