Theorem rnopab3 5937

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 3077	. 2 ⊢ (∀𝑦 ∈ 𝐴 ∃𝑥𝜑 ↔ ∀𝑦(𝑦 ∈ 𝐴 → ∃𝑥𝜑))
2		pm4.71 567	. . 3 ⊢ ((𝑦 ∈ 𝐴 → ∃𝑥𝜑) ↔ (𝑦 ∈ 𝐴 ↔ (𝑦 ∈ 𝐴 ∧ ∃𝑥𝜑)))
3	2	albii 1851	. 2 ⊢ (∀𝑦(𝑦 ∈ 𝐴 → ∃𝑥𝜑) ↔ ∀𝑦(𝑦 ∈ 𝐴 ↔ (𝑦 ∈ 𝐴 ∧ ∃𝑥𝜑)))
4		rnopab 5935	. . . . 5 ⊢ ran {⟨𝑥, 𝑦⟩ ∣ (𝑦 ∈ 𝐴 ∧ 𝜑)} = {𝑦 ∣ ∃𝑥(𝑦 ∈ 𝐴 ∧ 𝜑)}
5		19.42v 1985	. . . . . 6 ⊢ (∃𝑥(𝑦 ∈ 𝐴 ∧ 𝜑) ↔ (𝑦 ∈ 𝐴 ∧ ∃𝑥𝜑))
6	5	abbii 2827	. . . . 5 ⊢ {𝑦 ∣ ∃𝑥(𝑦 ∈ 𝐴 ∧ 𝜑)} = {𝑦 ∣ (𝑦 ∈ 𝐴 ∧ ∃𝑥𝜑)}
7	4, 6	eqtri 2783	. . . 4 ⊢ ran {⟨𝑥, 𝑦⟩ ∣ (𝑦 ∈ 𝐴 ∧ 𝜑)} = {𝑦 ∣ (𝑦 ∈ 𝐴 ∧ ∃𝑥𝜑)}
8	7	eqeq1i 2765	. . 3 ⊢ (ran {⟨𝑥, 𝑦⟩ ∣ (𝑦 ∈ 𝐴 ∧ 𝜑)} = 𝐴 ↔ {𝑦 ∣ (𝑦 ∈ 𝐴 ∧ ∃𝑥𝜑)} = 𝐴)
9		eqcom 2767	. . 3 ⊢ (𝐴 = {𝑦 ∣ (𝑦 ∈ 𝐴 ∧ ∃𝑥𝜑)} ↔ {𝑦 ∣ (𝑦 ∈ 𝐴 ∧ ∃𝑥𝜑)} = 𝐴)
10		eqabb 2899	. . 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 401  ∀wal 1567   = wceq 1569  ∃wex 1811   ∈ wcel 2144  {cab 2738  ∀wral 3076  {copab 5166  ran crn 5651

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ral 3077  df-rab 3413  df-v 3452  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-sn 4584  df-pr 4586  df-op 4590  df-br 5103  df-opab 5167  df-cnv 5658  df-dm 5660  df-rn 5661

This theorem is used by: (None)