Theorem mpoeq123dva 6064

Description: An equality deduction for the maps-to notation. (Contributed by Mario Carneiro, 26-Jan-2017.)

Hypotheses

Ref	Expression
mpoeq123dv.1	⊢ (𝜑 → 𝐴 = 𝐷)
mpoeq123dva.2	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 = 𝐸)
mpoeq123dva.3	⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → 𝐶 = 𝐹)

Assertion

Ref	Expression
mpoeq123dva	⊢ (𝜑 → (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = (𝑥 ∈ 𝐷, 𝑦 ∈ 𝐸 ↦ 𝐹))

Distinct variable groups:   𝜑,𝑥   𝜑,𝑦

Allowed substitution hints:   𝐴(𝑥,𝑦)   𝐵(𝑥,𝑦)   𝐶(𝑥,𝑦)   𝐷(𝑥,𝑦)   𝐸(𝑥,𝑦)   𝐹(𝑥,𝑦)

Proof of Theorem mpoeq123dva

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		mpoeq123dva.3	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → 𝐶 = 𝐹)
2	1	eqeq2d 2241	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → (𝑧 = 𝐶 ↔ 𝑧 = 𝐹))
3	2	pm5.32da 452	. . . 4 ⊢ (𝜑 → (((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐹)))
4		mpoeq123dva.2	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 = 𝐸)
5	4	eleq2d 2299	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑦 ∈ 𝐵 ↔ 𝑦 ∈ 𝐸))
6	5	pm5.32da 452	. . . . . 6 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐸)))
7		mpoeq123dv.1	. . . . . . . 8 ⊢ (𝜑 → 𝐴 = 𝐷)
8	7	eleq2d 2299	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐷))
9	8	anbi1d 465	. . . . . 6 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐸) ↔ (𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐸)))
10	6, 9	bitrd 188	. . . . 5 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ↔ (𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐸)))
11	10	anbi1d 465	. . . 4 ⊢ (𝜑 → (((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐹) ↔ ((𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐸) ∧ 𝑧 = 𝐹)))
12	3, 11	bitrd 188	. . 3 ⊢ (𝜑 → (((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶) ↔ ((𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐸) ∧ 𝑧 = 𝐹)))
13	12	oprabbidv 6057	. 2 ⊢ (𝜑 → {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶)} = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐸) ∧ 𝑧 = 𝐹)})
14		df-mpo 6005	. 2 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶)}
15		df-mpo 6005	. 2 ⊢ (𝑥 ∈ 𝐷, 𝑦 ∈ 𝐸 ↦ 𝐹) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐸) ∧ 𝑧 = 𝐹)}
16	13, 14, 15	3eqtr4g 2287	1 ⊢ (𝜑 → (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = (𝑥 ∈ 𝐷, 𝑦 ∈ 𝐸 ↦ 𝐹))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1395   ∈ wcel 2200  {coprab 6001   ∈ cmpo 6002

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 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-11 1552  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-oprab 6004  df-mpo 6005

This theorem is referenced by:  mpoeq123dv  6065