Theorem mpoeq123dva 7498

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 2771	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → (𝑧 = 𝐶 ↔ 𝑧 = 𝐹))
3	2	pm5.32da 590	. . . 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 2846	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑦 ∈ 𝐵 ↔ 𝑦 ∈ 𝐸))
6	5	pm5.32da 590	. . . . . 6 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐸)))
7		mpoeq123dv.1	. . . . . . . 8 ⊢ (𝜑 → 𝐴 = 𝐷)
8	7	eleq2d 2846	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐷))
9	8	anbi1d 643	. . . . . 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 643	. . . 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 7490	. 2 ⊢ (𝜑 → {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶)} = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐸) ∧ 𝑧 = 𝐹)})
14		df-mpo 7429	. 2 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶)}
15		df-mpo 7429	. 2 ⊢ (𝑥 ∈ 𝐷, 𝑦 ∈ 𝐸 ↦ 𝐹) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐸) ∧ 𝑧 = 𝐹)}
16	13, 14, 15	3eqtr4g 2820	1 ⊢ (𝜑 → (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = (𝑥 ∈ 𝐷, 𝑦 ∈ 𝐸 ↦ 𝐹))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1569   ∈ wcel 2144  {coprab 7425   ∈ cmpo 7426

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-oprab 7428  df-mpo 7429

This theorem is used by:  mpoeq123dv  7499  natpropd  18163  fucpropd  18164  curfpropd  18416  hofpropd  18450  rngcifuestrc  20900  funcrngcsetc  20901  funcrngcsetcALT  20902  funcringcsetc  20935  rrxdsfi  25741  eengv  29566  elntg  29571  submat1n  34487  rrxtopnfi  47405  iinfconstbas  50288  uppropd  50405  prcofpropd  50622  diag1f1olem  50776  lanpropd  50906  ranpropd  50907