Theorem dmopabelb 5858

Description: A set is an element of the domain of an 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 5857	. . 3 ⊢ dom {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {𝑥 ∣ ∃𝑦𝜑}
2	1	eleq2i 2831	. 2 ⊢ (𝑋 ∈ dom {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ 𝑋 ∈ {𝑥 ∣ ∃𝑦𝜑})
3		dmopabel.d	. . . 4 ⊢ (𝑥 = 𝑋 → (𝜑 ↔ 𝜓))
4	3	exbidv 1928	. . 3 ⊢ (𝑥 = 𝑋 → (∃𝑦𝜑 ↔ ∃𝑦𝜓))
5		eqid 2739	. . 3 ⊢ {𝑥 ∣ ∃𝑦𝜑} = {𝑥 ∣ ∃𝑦𝜑}
6	4, 5	elab2g 3618	. 2 ⊢ (𝑋 ∈ 𝑉 → (𝑋 ∈ {𝑥 ∣ ∃𝑦𝜑} ↔ ∃𝑦𝜓))
7	2, 6	bitrid 284	1 ⊢ (𝑋 ∈ 𝑉 → (𝑋 ∈ dom {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ ∃𝑦𝜓))

Syntax hints:   → wi 4   ↔ wb 207   = wceq 1547  ∃wex 1786   ∈ wcel 2119  {cab 2717  {copab 5134  dom cdm 5618

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-sep 5218  ax-pr 5362

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-rab 3392  df-v 3433  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-sn 4556  df-pr 4558  df-op 4562  df-br 5073  df-opab 5135  df-dm 5628

This theorem is referenced by:  dmopab2rex  5859  dmopab3rexdif  35633