Theorem mpoeq123dva 7285

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 2748	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → (𝑧 = 𝐶 ↔ 𝑧 = 𝐹))
3	2	pm5.32da 582	. . . 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 2823	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑦 ∈ 𝐵 ↔ 𝑦 ∈ 𝐸))
6	5	pm5.32da 582	. . . . . 6 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐸)))
7		mpoeq123dv.1	. . . . . . . 8 ⊢ (𝜑 → 𝐴 = 𝐷)
8	7	eleq2d 2823	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐷))
9	8	anbi1d 633	. . . . . 6 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐸) ↔ (𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐸)))
10	6, 9	bitrd 282	. . . . 5 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ↔ (𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐸)))
11	10	anbi1d 633	. . . 4 ⊢ (𝜑 → (((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐹) ↔ ((𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐸) ∧ 𝑧 = 𝐹)))
12	3, 11	bitrd 282	. . 3 ⊢ (𝜑 → (((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶) ↔ ((𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐸) ∧ 𝑧 = 𝐹)))
13	12	oprabbidv 7277	. 2 ⊢ (𝜑 → {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶)} = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐸) ∧ 𝑧 = 𝐹)})
14		df-mpo 7218	. 2 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶)}
15		df-mpo 7218	. 2 ⊢ (𝑥 ∈ 𝐷, 𝑦 ∈ 𝐸 ↦ 𝐹) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐸) ∧ 𝑧 = 𝐹)}
16	13, 14, 15	3eqtr4g 2803	1 ⊢ (𝜑 → (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = (𝑥 ∈ 𝐷, 𝑦 ∈ 𝐸 ↦ 𝐹))

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1543   ∈ wcel 2110  {coprab 7214   ∈ cmpo 7215

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-12 2175  ax-ext 2708

This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1788  df-nf 1792  df-sb 2071  df-clab 2715  df-cleq 2729  df-clel 2816  df-oprab 7217  df-mpo 7218

This theorem is referenced by:  mpoeq123dv  7286  natpropd  17485  fucpropd  17486  curfpropd  17741  hofpropd  17775  rrxdsfi  24308  istrkgl  26549  eengv  27070  elntg  27075  submat1n  31469  rrxtopnfi  43503  rngcifuestrc  45228  funcrngcsetc  45229  funcrngcsetcALT  45230  funcringcsetc  45266  eenglngeehlnm  45758