Theorem rnopabss 5944

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 5943	. 2 ⊢ ran {⟨𝑥, 𝑦⟩ ∣ (𝑦 ∈ 𝐴 ∧ 𝜑)} = {𝑦 ∣ ∃𝑥(𝑦 ∈ 𝐴 ∧ 𝜑)}
2		19.42v 1982	. . . 4 ⊢ (∃𝑥(𝑦 ∈ 𝐴 ∧ 𝜑) ↔ (𝑦 ∈ 𝐴 ∧ ∃𝑥𝜑))
3	2	abbii 2829	. . 3 ⊢ {𝑦 ∣ ∃𝑥(𝑦 ∈ 𝐴 ∧ 𝜑)} = {𝑦 ∣ (𝑦 ∈ 𝐴 ∧ ∃𝑥𝜑)}
4		ssab2 4032	. . 3 ⊢ {𝑦 ∣ (𝑦 ∈ 𝐴 ∧ ∃𝑥𝜑)} ⊆ 𝐴
5	3, 4	eqsstri 3982	. 2 ⊢ {𝑦 ∣ ∃𝑥(𝑦 ∈ 𝐴 ∧ 𝜑)} ⊆ 𝐴
6	1, 5	eqsstri 3982	1 ⊢ ran {⟨𝑥, 𝑦⟩ ∣ (𝑦 ∈ 𝐴 ∧ 𝜑)} ⊆ 𝐴

This proof depends on syntax axioms:   ∧ wa 400  ∃wex 1808   ∈ wcel 2142  {cab 2740   ⊆ wss 3904  {copab 5172  ran crn 5661

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-sep 5256  ax-pr 5403

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-nf 1813  df-sb 2096  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-rab 3416  df-v 3456  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4487  df-sn 4589  df-pr 4591  df-op 4595  df-br 5109  df-opab 5173  df-cnv 5668  df-dm 5670  df-rn 5671

This theorem is used by:  modelaxreplem2  45716