Theorem mpteq12f 5164

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 2162	. . . 4 ⊢ Ⅎ𝑥∀𝑥 𝐴 = 𝐶
2		nfra1 3264	. . . 4 ⊢ Ⅎ𝑥∀𝑥 ∈ 𝐴 𝐵 = 𝐷
3	1, 2	nfan 1906	. . 3 ⊢ Ⅎ𝑥(∀𝑥 𝐴 = 𝐶 ∧ ∀𝑥 ∈ 𝐴 𝐵 = 𝐷)
4		nfv 1921	. . 3 ⊢ Ⅎ𝑦(∀𝑥 𝐴 = 𝐶 ∧ ∀𝑥 ∈ 𝐴 𝐵 = 𝐷)
5		rspa 3229	. . . . . 6 ⊢ ((∀𝑥 ∈ 𝐴 𝐵 = 𝐷 ∧ 𝑥 ∈ 𝐴) → 𝐵 = 𝐷)
6	5	eqeq2d 2751	. . . . 5 ⊢ ((∀𝑥 ∈ 𝐴 𝐵 = 𝐷 ∧ 𝑥 ∈ 𝐴) → (𝑦 = 𝐵 ↔ 𝑦 = 𝐷))
7	6	pm5.32da 584	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 𝐵 = 𝐷 → ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷)))
8		sp 2195	. . . . . 6 ⊢ (∀𝑥 𝐴 = 𝐶 → 𝐴 = 𝐶)
9	8	eleq2d 2826	. . . . 5 ⊢ (∀𝑥 𝐴 = 𝐶 → (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐶))
10	9	anbi1d 637	. . . 4 ⊢ (∀𝑥 𝐴 = 𝐶 → ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷) ↔ (𝑥 ∈ 𝐶 ∧ 𝑦 = 𝐷)))
11	7, 10	sylan9bbr 515	. . 3 ⊢ ((∀𝑥 𝐴 = 𝐶 ∧ ∀𝑥 ∈ 𝐴 𝐵 = 𝐷) → ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵) ↔ (𝑥 ∈ 𝐶 ∧ 𝑦 = 𝐷)))
12	3, 4, 11	opabbid 5144	. 2 ⊢ ((∀𝑥 𝐴 = 𝐶 ∧ ∀𝑥 ∈ 𝐴 𝐵 = 𝐷) → {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)} = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐶 ∧ 𝑦 = 𝐷)})
13		df-mpt 5161	. 2 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)}
14		df-mpt 5161	. 2 ⊢ (𝑥 ∈ 𝐶 ↦ 𝐷) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐶 ∧ 𝑦 = 𝐷)}
15	12, 13, 14	3eqtr4g 2800	1 ⊢ ((∀𝑥 𝐴 = 𝐶 ∧ ∀𝑥 ∈ 𝐴 𝐵 = 𝐷) → (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐶 ↦ 𝐷))

Syntax hints:   → wi 4   ∧ wa 396  ∀wal 1545   = wceq 1547   ∈ wcel 2119  ∀wral 3054  {copab 5141   ↦ cmpt 5160

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-12 2189  ax-ext 2712

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-tru 1550  df-ex 1787  df-nf 1791  df-sb 2074  df-clab 2719  df-cleq 2732  df-clel 2815  df-ral 3055  df-opab 5142  df-mpt 5161

This theorem is referenced by:  mpteq12  5167  esumeq12dvaf  34222  refsum2cnlem1  45492  smfinflem  47267