Theorem dmopab3 5893

Description: The domain of a restricted class of ordered pairs. (Contributed by NM, 31-Jan-2004.)

Assertion

Ref	Expression
dmopab3	⊢ (∀𝑥 ∈ 𝐴 ∃𝑦𝜑 ↔ dom {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} = 𝐴)

Distinct variable group:   𝑥,𝑦,𝐴

Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem dmopab3

Step	Hyp	Ref	Expression
1		df-ral 3076	. 2 ⊢ (∀𝑥 ∈ 𝐴 ∃𝑦𝜑 ↔ ∀𝑥(𝑥 ∈ 𝐴 → ∃𝑦𝜑))
2		pm4.71 565	. . 3 ⊢ ((𝑥 ∈ 𝐴 → ∃𝑦𝜑) ↔ (𝑥 ∈ 𝐴 ↔ (𝑥 ∈ 𝐴 ∧ ∃𝑦𝜑)))
3	2	albii 1838	. 2 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → ∃𝑦𝜑) ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ (𝑥 ∈ 𝐴 ∧ ∃𝑦𝜑)))
4		dmopab 5889	. . . . 5 ⊢ dom {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} = {𝑥 ∣ ∃𝑦(𝑥 ∈ 𝐴 ∧ 𝜑)}
5		19.42v 1972	. . . . . 6 ⊢ (∃𝑦(𝑥 ∈ 𝐴 ∧ 𝜑) ↔ (𝑥 ∈ 𝐴 ∧ ∃𝑦𝜑))
6	5	abbii 2828	. . . . 5 ⊢ {𝑥 ∣ ∃𝑦(𝑥 ∈ 𝐴 ∧ 𝜑)} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ ∃𝑦𝜑)}
7	4, 6	eqtri 2784	. . . 4 ⊢ dom {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ ∃𝑦𝜑)}
8	7	eqeq1i 2766	. . 3 ⊢ (dom {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} = 𝐴 ↔ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ ∃𝑦𝜑)} = 𝐴)
9		eqcom 2768	. . 3 ⊢ (𝐴 = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ ∃𝑦𝜑)} ↔ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ ∃𝑦𝜑)} = 𝐴)
10		eqabb 2900	. . 3 ⊢ (𝐴 = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ ∃𝑦𝜑)} ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ (𝑥 ∈ 𝐴 ∧ ∃𝑦𝜑)))
11	8, 9, 10	3bitr2ri 302	. 2 ⊢ (∀𝑥(𝑥 ∈ 𝐴 ↔ (𝑥 ∈ 𝐴 ∧ ∃𝑦𝜑)) ↔ dom {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} = 𝐴)
12	1, 3, 11	3bitri 299	1 ⊢ (∀𝑥 ∈ 𝐴 ∃𝑦𝜑 ↔ dom {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} = 𝐴)

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399  ∀wal 1557   = wceq 1559  ∃wex 1798   ∈ wcel 2141  {cab 2739  ∀wral 3075  {copab 5161  dom cdm 5645

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-sep 5245  ax-pr 5389

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ral 3076  df-rab 3414  df-v 3455  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4480  df-sn 4582  df-pr 4584  df-op 4588  df-br 5100  df-opab 5162  df-dm 5655

This theorem is referenced by:  fnopabg  6654  opabn1stprc  8035  n0el2  38798