Theorem rnopab3 5945

Description: The range of a restricted class of ordered pairs. (Contributed by Eric Schmidt, 16-Sep-2025.)

Assertion

Ref	Expression
rnopab3	⊢ (∀𝑦 ∈ 𝐴 ∃𝑥𝜑 ↔ ran {⟨𝑥, 𝑦⟩ ∣ (𝑦 ∈ 𝐴 ∧ 𝜑)} = 𝐴)

Distinct variable group:   𝑥,𝑦,𝐴

Allowed substitution hints:   𝜑(𝑥, 𝑦)

Proof of Theorem rnopab3

Step	Hyp	Ref	Expression
1		df-ral 3079	. 2 ⊢ (∀𝑦 ∈ 𝐴 ∃𝑥𝜑 ↔ ∀𝑦(𝑦 ∈ 𝐴 → ∃𝑥𝜑))
2		pm4.71 566	. . 3 ⊢ ((𝑦 ∈ 𝐴 → ∃𝑥𝜑) ↔ (𝑦 ∈ 𝐴 ↔ (𝑦 ∈ 𝐴 ∧ ∃𝑥𝜑)))
3	2	albii 1848	. 2 ⊢ (∀𝑦(𝑦 ∈ 𝐴 → ∃𝑥𝜑) ↔ ∀𝑦(𝑦 ∈ 𝐴 ↔ (𝑦 ∈ 𝐴 ∧ ∃𝑥𝜑)))
4		rnopab 5943	. . . . 5 ⊢ ran {⟨𝑥, 𝑦⟩ ∣ (𝑦 ∈ 𝐴 ∧ 𝜑)} = {𝑦 ∣ ∃𝑥(𝑦 ∈ 𝐴 ∧ 𝜑)}
5		19.42v 1982	. . . . . 6 ⊢ (∃𝑥(𝑦 ∈ 𝐴 ∧ 𝜑) ↔ (𝑦 ∈ 𝐴 ∧ ∃𝑥𝜑))
6	5	abbii 2829	. . . . 5 ⊢ {𝑦 ∣ ∃𝑥(𝑦 ∈ 𝐴 ∧ 𝜑)} = {𝑦 ∣ (𝑦 ∈ 𝐴 ∧ ∃𝑥𝜑)}
7	4, 6	eqtri 2785	. . . 4 ⊢ ran {⟨𝑥, 𝑦⟩ ∣ (𝑦 ∈ 𝐴 ∧ 𝜑)} = {𝑦 ∣ (𝑦 ∈ 𝐴 ∧ ∃𝑥𝜑)}
8	7	eqeq1i 2767	. . 3 ⊢ (ran {⟨𝑥, 𝑦⟩ ∣ (𝑦 ∈ 𝐴 ∧ 𝜑)} = 𝐴 ↔ {𝑦 ∣ (𝑦 ∈ 𝐴 ∧ ∃𝑥𝜑)} = 𝐴)
9		eqcom 2769	. . 3 ⊢ (𝐴 = {𝑦 ∣ (𝑦 ∈ 𝐴 ∧ ∃𝑥𝜑)} ↔ {𝑦 ∣ (𝑦 ∈ 𝐴 ∧ ∃𝑥𝜑)} = 𝐴)
10		eqabb 2901	. . 3 ⊢ (𝐴 = {𝑦 ∣ (𝑦 ∈ 𝐴 ∧ ∃𝑥𝜑)} ↔ ∀𝑦(𝑦 ∈ 𝐴 ↔ (𝑦 ∈ 𝐴 ∧ ∃𝑥𝜑)))
11	8, 9, 10	3bitr2ri 303	. 2 ⊢ (∀𝑦(𝑦 ∈ 𝐴 ↔ (𝑦 ∈ 𝐴 ∧ ∃𝑥𝜑)) ↔ ran {⟨𝑥, 𝑦⟩ ∣ (𝑦 ∈ 𝐴 ∧ 𝜑)} = 𝐴)
12	1, 3, 11	3bitri 300	1 ⊢ (∀𝑦 ∈ 𝐴 ∃𝑥𝜑 ↔ ran {⟨𝑥, 𝑦⟩ ∣ (𝑦 ∈ 𝐴 ∧ 𝜑)} = 𝐴)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 400  ∀wal 1567   = wceq 1569  ∃wex 1808   ∈ wcel 2142  {cab 2740  ∀wral 3078  {copab 5172  ran crn 5661

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-sep 5256  ax-pr 5403

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-nf 1813  df-sb 2096  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ral 3079  df-rab 3416  df-v 3456  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4487  df-sn 4589  df-pr 4591  df-op 4595  df-br 5109  df-opab 5173  df-cnv 5668  df-dm 5670  df-rn 5671

This theorem is used by: (None)