Theorem elopab 4908

Description: Membership in a class abstraction of pairs. (Contributed by NM, 24-Mar-1998.)

Assertion

Ref	Expression
elopab	⊢ (𝐴 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ ∃𝑥∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ 𝜑))

Distinct variable groups:   𝑥,𝐴   𝑦,𝐴

Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem elopab

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex 3185	. 2 ⊢ (𝐴 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} → 𝐴 ∈ V)
2		opex 4859	. . . . 5 ⊢ ⟨𝑥, 𝑦⟩ ∈ V
3		eleq1 2676	. . . . 5 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → (𝐴 ∈ V ↔ ⟨𝑥, 𝑦⟩ ∈ V))
4	2, 3	mpbiri 247	. . . 4 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → 𝐴 ∈ V)
5	4	adantr 480	. . 3 ⊢ ((𝐴 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) → 𝐴 ∈ V)
6	5	exlimivv 1847	. 2 ⊢ (∃𝑥∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) → 𝐴 ∈ V)
7		eqeq1 2614	. . . . 5 ⊢ (𝑧 = 𝐴 → (𝑧 = ⟨𝑥, 𝑦⟩ ↔ 𝐴 = ⟨𝑥, 𝑦⟩))
8	7	anbi1d 737	. . . 4 ⊢ (𝑧 = 𝐴 → ((𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ (𝐴 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)))
9	8	2exbidv 1839	. . 3 ⊢ (𝑧 = 𝐴 → (∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ ∃𝑥∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)))
10		df-opab 4644	. . 3 ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {𝑧 ∣ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)}
11	9, 10	elab2g 3322	. 2 ⊢ (𝐴 ∈ V → (𝐴 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ ∃𝑥∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)))
12	1, 6, 11	pm5.21nii 367	1 ⊢ (𝐴 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ ∃𝑥∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ 𝜑))

Syntax hints:   ↔ wb 195   ∧ wa 383   = wceq 1475  ∃wex 1695   ∈ wcel 1977  Vcvv 3173  ⟨cop 4131  {copab 4642

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-opab 4644

This theorem is referenced by:  opelopabsbALT  4909  opelopabsb  4910  opelopabt  4912  opelopabga  4913  opabn0  4931  iunopab  4936  elopabr  4937  epelg  4950  elxp  5055  elopaelxp  5114  elopaba  5155  elcnv  5221  dfmpt3  5927  fmptsng  6339  fmptsnd  6340  0neqopab  6596  opabex3d  7037  opabex3  7038  fsplit  7169  rtrclreclem3  13648  isfunc  16347  usgraop  25879  clwlkswlks  26286  brabgaf  28800  qqhval2  29354  eulerpartlemgvv  29765  poimirlem26  32605  dicelval3  35487  pellexlem5  36415  pellex  36417  opelopab4  37788  griedg0ssusgr  40489  rgrusgrprc  40789