Theorem mpteq12f 5194

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 2191	. . . 4 ⊢ Ⅎ𝑥∀𝑥 𝐴 = 𝐶
2		nfra1 3290	. . . 4 ⊢ Ⅎ𝑥∀𝑥 ∈ 𝐴 𝐵 = 𝐷
3	1, 2	nfan 1934	. . 3 ⊢ Ⅎ𝑥(∀𝑥 𝐴 = 𝐶 ∧ ∀𝑥 ∈ 𝐴 𝐵 = 𝐷)
4		nfv 1949	. . 3 ⊢ Ⅎ𝑦(∀𝑥 𝐴 = 𝐶 ∧ ∀𝑥 ∈ 𝐴 𝐵 = 𝐷)
5		rspa 3255	. . . . . 6 ⊢ ((∀𝑥 ∈ 𝐴 𝐵 = 𝐷 ∧ 𝑥 ∈ 𝐴) → 𝐵 = 𝐷)
6	5	eqeq2d 2774	. . . . 5 ⊢ ((∀𝑥 ∈ 𝐴 𝐵 = 𝐷 ∧ 𝑥 ∈ 𝐴) → (𝑦 = 𝐵 ↔ 𝑦 = 𝐷))
7	6	pm5.32da 590	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 𝐵 = 𝐷 → ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷)))
8		sp 2223	. . . . . 6 ⊢ (∀𝑥 𝐴 = 𝐶 → 𝐴 = 𝐶)
9	8	eleq2d 2850	. . . . 5 ⊢ (∀𝑥 𝐴 = 𝐶 → (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐶))
10	9	anbi1d 643	. . . 4 ⊢ (∀𝑥 𝐴 = 𝐶 → ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷) ↔ (𝑥 ∈ 𝐶 ∧ 𝑦 = 𝐷)))
11	7, 10	sylan9bbr 520	. . 3 ⊢ ((∀𝑥 𝐴 = 𝐶 ∧ ∀𝑥 ∈ 𝐴 𝐵 = 𝐷) → ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵) ↔ (𝑥 ∈ 𝐶 ∧ 𝑦 = 𝐷)))
12	3, 4, 11	opabbid 5174	. 2 ⊢ ((∀𝑥 𝐴 = 𝐶 ∧ ∀𝑥 ∈ 𝐴 𝐵 = 𝐷) → {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)} = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐶 ∧ 𝑦 = 𝐷)})
13		df-mpt 5191	. 2 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)}
14		df-mpt 5191	. 2 ⊢ (𝑥 ∈ 𝐶 ↦ 𝐷) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐶 ∧ 𝑦 = 𝐷)}
15	12, 13, 14	3eqtr4g 2823	1 ⊢ ((∀𝑥 𝐴 = 𝐶 ∧ ∀𝑥 ∈ 𝐴 𝐵 = 𝐷) → (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐶 ↦ 𝐷))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401  ∀wal 1568   = wceq 1570   ∈ wcel 2147  ∀wral 3080  {copab 5171   ↦ cmpt 5190

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-12 2216  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-ex 1813  df-nf 1817  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ral 3081  df-opab 5172  df-mpt 5191

This theorem is used by:  mpteq12  5197  esumeq12dvaf  34746  refsum2cnlem1  46270  smfinflem  48042