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Theorem dmopabelb 5930
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 5929 . . 3 dom {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {𝑥 ∣ ∃𝑦𝜑}
21eleq2i 2831 . 2 (𝑋 ∈ dom {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ 𝑋 ∈ {𝑥 ∣ ∃𝑦𝜑})
3 dmopabel.d . . . 4 (𝑥 = 𝑋 → (𝜑𝜓))
43exbidv 1919 . . 3 (𝑥 = 𝑋 → (∃𝑦𝜑 ↔ ∃𝑦𝜓))
5 eqid 2735 . . 3 {𝑥 ∣ ∃𝑦𝜑} = {𝑥 ∣ ∃𝑦𝜑}
64, 5elab2g 3683 . 2 (𝑋𝑉 → (𝑋 ∈ {𝑥 ∣ ∃𝑦𝜑} ↔ ∃𝑦𝜓))
72, 6bitrid 283 1 (𝑋𝑉 → (𝑋 ∈ dom {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ ∃𝑦𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206   = wceq 1537  wex 1776  wcel 2106  {cab 2712  {copab 5210  dom cdm 5689
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  df-dm 5699
This theorem is referenced by:  dmopab2rex  5931  dmopab3rexdif  35390
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