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Theorem elopab 4148
Description: Membership in a class abstraction of pairs. (Contributed by NM, 24-Mar-1998.)
Assertion
Ref Expression
elopab (𝐴 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ ∃𝑥𝑦(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.
StepHypRef Expression
1 elex 2669 . 2 (𝐴 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} → 𝐴 ∈ V)
2 vex 2661 . . . . . 6 𝑥 ∈ V
3 vex 2661 . . . . . 6 𝑦 ∈ V
42, 3opex 4119 . . . . 5 𝑥, 𝑦⟩ ∈ V
5 eleq1 2178 . . . . 5 (𝐴 = ⟨𝑥, 𝑦⟩ → (𝐴 ∈ V ↔ ⟨𝑥, 𝑦⟩ ∈ V))
64, 5mpbiri 167 . . . 4 (𝐴 = ⟨𝑥, 𝑦⟩ → 𝐴 ∈ V)
76adantr 272 . . 3 ((𝐴 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) → 𝐴 ∈ V)
87exlimivv 1850 . 2 (∃𝑥𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) → 𝐴 ∈ V)
9 eqeq1 2122 . . . . 5 (𝑧 = 𝐴 → (𝑧 = ⟨𝑥, 𝑦⟩ ↔ 𝐴 = ⟨𝑥, 𝑦⟩))
109anbi1d 458 . . . 4 (𝑧 = 𝐴 → ((𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ (𝐴 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)))
11102exbidv 1822 . . 3 (𝑧 = 𝐴 → (∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ ∃𝑥𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)))
12 df-opab 3958 . . 3 {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {𝑧 ∣ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)}
1311, 12elab2g 2802 . 2 (𝐴 ∈ V → (𝐴 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ ∃𝑥𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)))
141, 8, 13pm5.21nii 676 1 (𝐴 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ ∃𝑥𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ 𝜑))
Colors of variables: wff set class
Syntax hints:  wa 103  wb 104   = wceq 1314  wex 1451  wcel 1463  Vcvv 2658  cop 3498  {copab 3956
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-pow 4066  ax-pr 4099
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-v 2660  df-un 3043  df-in 3045  df-ss 3052  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-opab 3958
This theorem is referenced by:  opelopabsbALT  4149  opelopabsb  4150  opelopabt  4152  opelopabga  4153  opabm  4170  iunopab  4171  epelg  4180  elxp  4524  elco  4673  elcnv  4684  dfmpt3  5213  0neqopab  5782  brabvv  5783  opabex3d  5985  opabex3  5986
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