MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  dmopabelb Structured version   Visualization version   GIF version

Theorem dmopabelb 5927
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 5926 . . 3 dom {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {𝑥 ∣ ∃𝑦𝜑}
21eleq2i 2833 . 2 (𝑋 ∈ dom {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ 𝑋 ∈ {𝑥 ∣ ∃𝑦𝜑})
3 dmopabel.d . . . 4 (𝑥 = 𝑋 → (𝜑𝜓))
43exbidv 1921 . . 3 (𝑥 = 𝑋 → (∃𝑦𝜑 ↔ ∃𝑦𝜓))
5 eqid 2737 . . 3 {𝑥 ∣ ∃𝑦𝜑} = {𝑥 ∣ ∃𝑦𝜑}
64, 5elab2g 3680 . 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 1540  wex 1779  wcel 2108  {cab 2714  {copab 5205  dom cdm 5685
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-dm 5695
This theorem is referenced by:  dmopab2rex  5928  dmopab3rexdif  35410
  Copyright terms: Public domain W3C validator