Theorem mpteq12f 5162

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 2148	. . . 4 ⊢ Ⅎ𝑥∀𝑥 𝐴 = 𝐶
2		nfra1 3144	. . . 4 ⊢ Ⅎ𝑥∀𝑥 ∈ 𝐴 𝐵 = 𝐷
3	1, 2	nfan 1902	. . 3 ⊢ Ⅎ𝑥(∀𝑥 𝐴 = 𝐶 ∧ ∀𝑥 ∈ 𝐴 𝐵 = 𝐷)
4		nfv 1917	. . 3 ⊢ Ⅎ𝑦(∀𝑥 𝐴 = 𝐶 ∧ ∀𝑥 ∈ 𝐴 𝐵 = 𝐷)
5		rspa 3132	. . . . . 6 ⊢ ((∀𝑥 ∈ 𝐴 𝐵 = 𝐷 ∧ 𝑥 ∈ 𝐴) → 𝐵 = 𝐷)
6	5	eqeq2d 2749	. . . . 5 ⊢ ((∀𝑥 ∈ 𝐴 𝐵 = 𝐷 ∧ 𝑥 ∈ 𝐴) → (𝑦 = 𝐵 ↔ 𝑦 = 𝐷))
7	6	pm5.32da 579	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 𝐵 = 𝐷 → ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷)))
8		sp 2176	. . . . . 6 ⊢ (∀𝑥 𝐴 = 𝐶 → 𝐴 = 𝐶)
9	8	eleq2d 2824	. . . . 5 ⊢ (∀𝑥 𝐴 = 𝐶 → (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐶))
10	9	anbi1d 630	. . . 4 ⊢ (∀𝑥 𝐴 = 𝐶 → ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷) ↔ (𝑥 ∈ 𝐶 ∧ 𝑦 = 𝐷)))
11	7, 10	sylan9bbr 511	. . 3 ⊢ ((∀𝑥 𝐴 = 𝐶 ∧ ∀𝑥 ∈ 𝐴 𝐵 = 𝐷) → ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵) ↔ (𝑥 ∈ 𝐶 ∧ 𝑦 = 𝐷)))
12	3, 4, 11	opabbid 5139	. 2 ⊢ ((∀𝑥 𝐴 = 𝐶 ∧ ∀𝑥 ∈ 𝐴 𝐵 = 𝐷) → {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)} = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐶 ∧ 𝑦 = 𝐷)})
13		df-mpt 5158	. 2 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)}
14		df-mpt 5158	. 2 ⊢ (𝑥 ∈ 𝐶 ↦ 𝐷) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐶 ∧ 𝑦 = 𝐷)}
15	12, 13, 14	3eqtr4g 2803	1 ⊢ ((∀𝑥 𝐴 = 𝐶 ∧ ∀𝑥 ∈ 𝐴 𝐵 = 𝐷) → (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐶 ↦ 𝐷))

Syntax hints:   → wi 4   ∧ wa 396  ∀wal 1537   = wceq 1539   ∈ wcel 2106  ∀wral 3064  {copab 5136   ↦ cmpt 5157

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-12 2171  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-tru 1542  df-ex 1783  df-nf 1787  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3069  df-opab 5137  df-mpt 5158

This theorem is referenced by:  mpteq12dvaOLD  5164  mpteq12  5166  mpteq2daOLD  5173  mpteq2iaOLD  5178  esumeq12dvaf  31999  refsum2cnlem1  42580  mpteq1dfOLD  42780  mpteq12daOLD  42787  smfinflem  44350  smfinfmpt  44352