Theorem rnopabss 5946

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

Assertion

Ref	Expression
rnopabss	⊢ ran {⟨𝑥, 𝑦⟩ ∣ (𝑦 ∈ 𝐴 ∧ 𝜑)} ⊆ 𝐴

Distinct variable group:   𝑥,𝑦,𝐴

Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem rnopabss

Step	Hyp	Ref	Expression
1		rnopab 5945	. 2 ⊢ ran {⟨𝑥, 𝑦⟩ ∣ (𝑦 ∈ 𝐴 ∧ 𝜑)} = {𝑦 ∣ ∃𝑥(𝑦 ∈ 𝐴 ∧ 𝜑)}
2		19.42v 1952	. . . 4 ⊢ (∃𝑥(𝑦 ∈ 𝐴 ∧ 𝜑) ↔ (𝑦 ∈ 𝐴 ∧ ∃𝑥𝜑))
3	2	abbii 2801	. . 3 ⊢ {𝑦 ∣ ∃𝑥(𝑦 ∈ 𝐴 ∧ 𝜑)} = {𝑦 ∣ (𝑦 ∈ 𝐴 ∧ ∃𝑥𝜑)}
4		ssab2 4059	. . 3 ⊢ {𝑦 ∣ (𝑦 ∈ 𝐴 ∧ ∃𝑥𝜑)} ⊆ 𝐴
5	3, 4	eqsstri 4010	. 2 ⊢ {𝑦 ∣ ∃𝑥(𝑦 ∈ 𝐴 ∧ 𝜑)} ⊆ 𝐴
6	1, 5	eqsstri 4010	1 ⊢ ran {⟨𝑥, 𝑦⟩ ∣ (𝑦 ∈ 𝐴 ∧ 𝜑)} ⊆ 𝐴

Syntax hints:   ∧ wa 395  ∃wex 1778   ∈ wcel 2107  {cab 2712   ⊆ wss 3931  {copab 5185  ran crn 5666

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2706  ax-sep 5276  ax-nul 5286  ax-pr 5412

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2808  df-nfc 2884  df-rab 3420  df-v 3465  df-dif 3934  df-un 3936  df-ss 3948  df-nul 4314  df-if 4506  df-sn 4607  df-pr 4609  df-op 4613  df-br 5124  df-opab 5186  df-cnv 5673  df-dm 5675  df-rn 5676

This theorem is referenced by:  modelaxreplem2  44953