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Theorem dmopabelb 5870
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 → (𝜑𝜓))
Assertion
Ref Expression
dmopabelb (𝑋𝑉 → (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
StepHypRef Expression
1 dmopab 5869 . . 3 dom {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {𝑥 ∣ ∃𝑦𝜑}
21eleq2i 2829 . 2 (𝑋 ∈ dom {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ 𝑋 ∈ {𝑥 ∣ ∃𝑦𝜑})
3 dmopabel.d . . . 4 (𝑥 = 𝑋 → (𝜑𝜓))
43exbidv 1924 . . 3 (𝑥 = 𝑋 → (∃𝑦𝜑 ↔ ∃𝑦𝜓))
5 eqid 2736 . . 3 {𝑥 ∣ ∃𝑦𝜑} = {𝑥 ∣ ∃𝑦𝜑}
64, 5elab2g 3630 . 2 (𝑋𝑉 → (𝑋 ∈ {𝑥 ∣ ∃𝑦𝜑} ↔ ∃𝑦𝜓))
72, 6bitrid 282 1 (𝑋𝑉 → (𝑋 ∈ dom {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ ∃𝑦𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205   = wceq 1541  wex 1781  wcel 2106  {cab 2713  {copab 5165  dom cdm 5631
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 2707  ax-sep 5254  ax-nul 5261  ax-pr 5382
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 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2887  df-rab 3406  df-v 3445  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4281  df-if 4485  df-sn 4585  df-pr 4587  df-op 4591  df-br 5104  df-opab 5166  df-dm 5641
This theorem is referenced by:  dmopab2rex  5871  dmopab3rexdif  33868
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