Theorem rnopab3 5895

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 3048	. 2 ⊢ (∀𝑦 ∈ 𝐴 ∃𝑥𝜑 ↔ ∀𝑦(𝑦 ∈ 𝐴 → ∃𝑥𝜑))
2		pm4.71 557	. . 3 ⊢ ((𝑦 ∈ 𝐴 → ∃𝑥𝜑) ↔ (𝑦 ∈ 𝐴 ↔ (𝑦 ∈ 𝐴 ∧ ∃𝑥𝜑)))
3	2	albii 1820	. 2 ⊢ (∀𝑦(𝑦 ∈ 𝐴 → ∃𝑥𝜑) ↔ ∀𝑦(𝑦 ∈ 𝐴 ↔ (𝑦 ∈ 𝐴 ∧ ∃𝑥𝜑)))
4		rnopab 5893	. . . . 5 ⊢ ran {⟨𝑥, 𝑦⟩ ∣ (𝑦 ∈ 𝐴 ∧ 𝜑)} = {𝑦 ∣ ∃𝑥(𝑦 ∈ 𝐴 ∧ 𝜑)}
5		19.42v 1954	. . . . . 6 ⊢ (∃𝑥(𝑦 ∈ 𝐴 ∧ 𝜑) ↔ (𝑦 ∈ 𝐴 ∧ ∃𝑥𝜑))
6	5	abbii 2798	. . . . 5 ⊢ {𝑦 ∣ ∃𝑥(𝑦 ∈ 𝐴 ∧ 𝜑)} = {𝑦 ∣ (𝑦 ∈ 𝐴 ∧ ∃𝑥𝜑)}
7	4, 6	eqtri 2754	. . . 4 ⊢ ran {⟨𝑥, 𝑦⟩ ∣ (𝑦 ∈ 𝐴 ∧ 𝜑)} = {𝑦 ∣ (𝑦 ∈ 𝐴 ∧ ∃𝑥𝜑)}
8	7	eqeq1i 2736	. . 3 ⊢ (ran {⟨𝑥, 𝑦⟩ ∣ (𝑦 ∈ 𝐴 ∧ 𝜑)} = 𝐴 ↔ {𝑦 ∣ (𝑦 ∈ 𝐴 ∧ ∃𝑥𝜑)} = 𝐴)
9		eqcom 2738	. . 3 ⊢ (𝐴 = {𝑦 ∣ (𝑦 ∈ 𝐴 ∧ ∃𝑥𝜑)} ↔ {𝑦 ∣ (𝑦 ∈ 𝐴 ∧ ∃𝑥𝜑)} = 𝐴)
10		eqabb 2870	. . 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 1539   = wceq 1541  ∃wex 1780   ∈ wcel 2111  {cab 2709  ∀wral 3047  {copab 5151  ran crn 5615

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5232  ax-nul 5242  ax-pr 5368

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ral 3048  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-ss 3914  df-nul 4281  df-if 4473  df-sn 4574  df-pr 4576  df-op 4580  df-br 5090  df-opab 5152  df-cnv 5622  df-dm 5624  df-rn 5625

This theorem is referenced by: (None)