Theorem dmopabelb 5892

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 5891	. . 3 ⊢ dom {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {𝑥 ∣ ∃𝑦𝜑}
2	1	eleq2i 2854	. 2 ⊢ (𝑋 ∈ dom {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ 𝑋 ∈ {𝑥 ∣ ∃𝑦𝜑})
3		dmopabel.d	. . . 4 ⊢ (𝑥 = 𝑋 → (𝜑 ↔ 𝜓))
4	3	exbidv 1941	. . 3 ⊢ (𝑥 = 𝑋 → (∃𝑦𝜑 ↔ ∃𝑦𝜓))
5		eqid 2762	. . 3 ⊢ {𝑥 ∣ ∃𝑦𝜑} = {𝑥 ∣ ∃𝑦𝜑}
6	4, 5	elab2g 3639	. 2 ⊢ (𝑋 ∈ 𝑉 → (𝑋 ∈ {𝑥 ∣ ∃𝑦𝜑} ↔ ∃𝑦𝜓))
7	2, 6	bitrid 285	1 ⊢ (𝑋 ∈ 𝑉 → (𝑋 ∈ dom {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ ∃𝑦𝜓))

Syntax hints:   → wi 4   ↔ wb 208   = wceq 1560  ∃wex 1799   ∈ wcel 2142  {cab 2740  {copab 5162  dom cdm 5647

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-sep 5246  ax-pr 5390

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-nf 1804  df-sb 2091  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-rab 3415  df-v 3456  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4481  df-sn 4583  df-pr 4585  df-op 4589  df-br 5101  df-opab 5163  df-dm 5657

This theorem is referenced by:  dmopab2rex  5893  dmopab3rexdif  35755