Theorem dmopabelb 5916

Description: A set is an element of the domain of a ordered pair class abstraction iff there is a second set so that both sets fulfil the wff of the class abstraction. (Contributed by AV, 19-Oct-2023.)

Hypothesis

Ref	Expression
dmopabel.d	⊢ (𝑥 = 𝑋 → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
dmopabelb	⊢ (𝑋 ∈ 𝑉 → (𝑋 ∈ dom {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ ∃𝑦𝜓))

Distinct variable groups:   𝑥,𝑋,𝑦   𝜓,𝑥

Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑦)   𝑉(𝑥,𝑦)

Proof of Theorem dmopabelb

Step	Hyp	Ref	Expression
1		dmopab 5915	. . 3 ⊢ dom {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {𝑥 ∣ ∃𝑦𝜑}
2	1	eleq2i 2825	. 2 ⊢ (𝑋 ∈ dom {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ 𝑋 ∈ {𝑥 ∣ ∃𝑦𝜑})
3		dmopabel.d	. . . 4 ⊢ (𝑥 = 𝑋 → (𝜑 ↔ 𝜓))
4	3	exbidv 1924	. . 3 ⊢ (𝑥 = 𝑋 → (∃𝑦𝜑 ↔ ∃𝑦𝜓))
5		eqid 2732	. . 3 ⊢ {𝑥 ∣ ∃𝑦𝜑} = {𝑥 ∣ ∃𝑦𝜑}
6	4, 5	elab2g 3670	. 2 ⊢ (𝑋 ∈ 𝑉 → (𝑋 ∈ {𝑥 ∣ ∃𝑦𝜑} ↔ ∃𝑦𝜓))
7	2, 6	bitrid 282	1 ⊢ (𝑋 ∈ 𝑉 → (𝑋 ∈ dom {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ ∃𝑦𝜓))

Syntax hints:   → wi 4   ↔ wb 205   = wceq 1541  ∃wex 1781   ∈ wcel 2106  {cab 2709  {copab 5210  dom cdm 5676

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-br 5149  df-opab 5211  df-dm 5686

This theorem is referenced by:  dmopab2rex  5917  dmopab3rexdif  34391