Theorem rnopab3 5923

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 3046	. 2 ⊢ (∀𝑦 ∈ 𝐴 ∃𝑥𝜑 ↔ ∀𝑦(𝑦 ∈ 𝐴 → ∃𝑥𝜑))
2		pm4.71 557	. . 3 ⊢ ((𝑦 ∈ 𝐴 → ∃𝑥𝜑) ↔ (𝑦 ∈ 𝐴 ↔ (𝑦 ∈ 𝐴 ∧ ∃𝑥𝜑)))
3	2	albii 1819	. 2 ⊢ (∀𝑦(𝑦 ∈ 𝐴 → ∃𝑥𝜑) ↔ ∀𝑦(𝑦 ∈ 𝐴 ↔ (𝑦 ∈ 𝐴 ∧ ∃𝑥𝜑)))
4		rnopab 5921	. . . . 5 ⊢ ran {⟨𝑥, 𝑦⟩ ∣ (𝑦 ∈ 𝐴 ∧ 𝜑)} = {𝑦 ∣ ∃𝑥(𝑦 ∈ 𝐴 ∧ 𝜑)}
5		19.42v 1953	. . . . . 6 ⊢ (∃𝑥(𝑦 ∈ 𝐴 ∧ 𝜑) ↔ (𝑦 ∈ 𝐴 ∧ ∃𝑥𝜑))
6	5	abbii 2797	. . . . 5 ⊢ {𝑦 ∣ ∃𝑥(𝑦 ∈ 𝐴 ∧ 𝜑)} = {𝑦 ∣ (𝑦 ∈ 𝐴 ∧ ∃𝑥𝜑)}
7	4, 6	eqtri 2753	. . . 4 ⊢ ran {⟨𝑥, 𝑦⟩ ∣ (𝑦 ∈ 𝐴 ∧ 𝜑)} = {𝑦 ∣ (𝑦 ∈ 𝐴 ∧ ∃𝑥𝜑)}
8	7	eqeq1i 2735	. . 3 ⊢ (ran {⟨𝑥, 𝑦⟩ ∣ (𝑦 ∈ 𝐴 ∧ 𝜑)} = 𝐴 ↔ {𝑦 ∣ (𝑦 ∈ 𝐴 ∧ ∃𝑥𝜑)} = 𝐴)
9		eqcom 2737	. . 3 ⊢ (𝐴 = {𝑦 ∣ (𝑦 ∈ 𝐴 ∧ ∃𝑥𝜑)} ↔ {𝑦 ∣ (𝑦 ∈ 𝐴 ∧ ∃𝑥𝜑)} = 𝐴)
10		eqabb 2868	. . 3 ⊢ (𝐴 = {𝑦 ∣ (𝑦 ∈ 𝐴 ∧ ∃𝑥𝜑)} ↔ ∀𝑦(𝑦 ∈ 𝐴 ↔ (𝑦 ∈ 𝐴 ∧ ∃𝑥𝜑)))
11	8, 9, 10	3bitr2ri 300	. 2 ⊢ (∀𝑦(𝑦 ∈ 𝐴 ↔ (𝑦 ∈ 𝐴 ∧ ∃𝑥𝜑)) ↔ ran {⟨𝑥, 𝑦⟩ ∣ (𝑦 ∈ 𝐴 ∧ 𝜑)} = 𝐴)
12	1, 3, 11	3bitri 297	1 ⊢ (∀𝑦 ∈ 𝐴 ∃𝑥𝜑 ↔ ran {⟨𝑥, 𝑦⟩ ∣ (𝑦 ∈ 𝐴 ∧ 𝜑)} = 𝐴)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395  ∀wal 1538   = wceq 1540  ∃wex 1779   ∈ wcel 2109  {cab 2708  ∀wral 3045  {copab 5172  ran crn 5642

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-sep 5254  ax-nul 5264  ax-pr 5390

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ral 3046  df-rab 3409  df-v 3452  df-dif 3920  df-un 3922  df-ss 3934  df-nul 4300  df-if 4492  df-sn 4593  df-pr 4595  df-op 4599  df-br 5111  df-opab 5173  df-cnv 5649  df-dm 5651  df-rn 5652

This theorem is referenced by: (None)