Theorem rnopabss 5904

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 5903	. 2 ⊢ ran {⟨𝑥, 𝑦⟩ ∣ (𝑦 ∈ 𝐴 ∧ 𝜑)} = {𝑦 ∣ ∃𝑥(𝑦 ∈ 𝐴 ∧ 𝜑)}
2		19.42v 1955	. . . 4 ⊢ (∃𝑥(𝑦 ∈ 𝐴 ∧ 𝜑) ↔ (𝑦 ∈ 𝐴 ∧ ∃𝑥𝜑))
3	2	abbii 2804	. . 3 ⊢ {𝑦 ∣ ∃𝑥(𝑦 ∈ 𝐴 ∧ 𝜑)} = {𝑦 ∣ (𝑦 ∈ 𝐴 ∧ ∃𝑥𝜑)}
4		ssab2 4020	. . 3 ⊢ {𝑦 ∣ (𝑦 ∈ 𝐴 ∧ ∃𝑥𝜑)} ⊆ 𝐴
5	3, 4	eqsstri 3969	. 2 ⊢ {𝑦 ∣ ∃𝑥(𝑦 ∈ 𝐴 ∧ 𝜑)} ⊆ 𝐴
6	1, 5	eqsstri 3969	1 ⊢ ran {⟨𝑥, 𝑦⟩ ∣ (𝑦 ∈ 𝐴 ∧ 𝜑)} ⊆ 𝐴

Syntax hints:   ∧ wa 395  ∃wex 1781   ∈ wcel 2114  {cab 2715   ⊆ wss 3890  {copab 5148  ran crn 5625

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5231  ax-pr 5370

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-br 5087  df-opab 5149  df-cnv 5632  df-dm 5634  df-rn 5635

This theorem is referenced by:  modelaxreplem2  45424