Theorem rnopab3 5934

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		pm4.71 565	. . 3 ⊢ ((𝑦 ∈ 𝐴 → ∃𝑥𝜑) ↔ (𝑦 ∈ 𝐴 ↔ (𝑦 ∈ 𝐴 ∧ ∃𝑥𝜑)))
3	2	albii 1841	. 2 ⊢ (∀𝑦(𝑦 ∈ 𝐴 → ∃𝑥𝜑) ↔ ∀𝑦(𝑦 ∈ 𝐴 ↔ (𝑦 ∈ 𝐴 ∧ ∃𝑥𝜑)))
4		rnopab 5932	. . . . 5 ⊢ ran {⟨𝑥, 𝑦⟩ ∣ (𝑦 ∈ 𝐴 ∧ 𝜑)} = {𝑦 ∣ ∃𝑥(𝑦 ∈ 𝐴 ∧ 𝜑)}
5		19.42v 1975	. . . . . 6 ⊢ (∃𝑥(𝑦 ∈ 𝐴 ∧ 𝜑) ↔ (𝑦 ∈ 𝐴 ∧ ∃𝑥𝜑))
6	5	abbii 2831	. . . . 5 ⊢ {𝑦 ∣ ∃𝑥(𝑦 ∈ 𝐴 ∧ 𝜑)} = {𝑦 ∣ (𝑦 ∈ 𝐴 ∧ ∃𝑥𝜑)}
7	4, 6	eqtri 2787	. . . 4 ⊢ ran {⟨𝑥, 𝑦⟩ ∣ (𝑦 ∈ 𝐴 ∧ 𝜑)} = {𝑦 ∣ (𝑦 ∈ 𝐴 ∧ ∃𝑥𝜑)}
8	7	eqeq1i 2769	. . 3 ⊢ (ran {⟨𝑥, 𝑦⟩ ∣ (𝑦 ∈ 𝐴 ∧ 𝜑)} = 𝐴 ↔ {𝑦 ∣ (𝑦 ∈ 𝐴 ∧ ∃𝑥𝜑)} = 𝐴)
9		eqcom 2771	. . 3 ⊢ (𝐴 = {𝑦 ∣ (𝑦 ∈ 𝐴 ∧ ∃𝑥𝜑)} ↔ {𝑦 ∣ (𝑦 ∈ 𝐴 ∧ ∃𝑥𝜑)} = 𝐴)
10		eqabb 2903	. . 3 ⊢ (𝐴 = {𝑦 ∣ (𝑦 ∈ 𝐴 ∧ ∃𝑥𝜑)} ↔ ∀𝑦(𝑦 ∈ 𝐴 ↔ (𝑦 ∈ 𝐴 ∧ ∃𝑥𝜑)))
11	8, 9, 10	3bitr2ri 302	. 2 ⊢ (∀𝑦(𝑦 ∈ 𝐴 ↔ (𝑦 ∈ 𝐴 ∧ ∃𝑥𝜑)) ↔ ran {⟨𝑥, 𝑦⟩ ∣ (𝑦 ∈ 𝐴 ∧ 𝜑)} = 𝐴)
12	1, 3, 11	3bitri 299	1 ⊢ (∀𝑦 ∈ 𝐴 ∃𝑥𝜑 ↔ ran {⟨𝑥, 𝑦⟩ ∣ (𝑦 ∈ 𝐴 ∧ 𝜑)} = 𝐴)

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399  ∀wal 1560   = wceq 1562  ∃wex 1801   ∈ wcel 2144  {cab 2742  ∀wral 3078  {copab 5164  ran crn 5650

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2214  ax-ext 2736  ax-sep 5248  ax-pr 5392

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1101  df-tru 1565  df-fal 1575  df-ex 1802  df-nf 1806  df-sb 2093  df-mo 2568  df-eu 2598  df-clab 2743  df-cleq 2756  df-clel 2839  df-nfc 2913  df-ral 3079  df-rab 3417  df-v 3458  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-br 5103  df-opab 5165  df-cnv 5657  df-dm 5659  df-rn 5660

This theorem is referenced by: (None)