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

Theorem elopab 5144
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 3365 . 2 (𝐴 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} → 𝐴 ∈ V)
2 opex 5088 . . . . 5 𝑥, 𝑦⟩ ∈ V
3 eleq1 2832 . . . . 5 (𝐴 = ⟨𝑥, 𝑦⟩ → (𝐴 ∈ V ↔ ⟨𝑥, 𝑦⟩ ∈ V))
42, 3mpbiri 249 . . . 4 (𝐴 = ⟨𝑥, 𝑦⟩ → 𝐴 ∈ V)
54adantr 472 . . 3 ((𝐴 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) → 𝐴 ∈ V)
65exlimivv 2027 . 2 (∃𝑥𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) → 𝐴 ∈ V)
7 eqeq1 2769 . . . . 5 (𝑧 = 𝐴 → (𝑧 = ⟨𝑥, 𝑦⟩ ↔ 𝐴 = ⟨𝑥, 𝑦⟩))
87anbi1d 623 . . . 4 (𝑧 = 𝐴 → ((𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ (𝐴 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)))
982exbidv 2019 . . 3 (𝑧 = 𝐴 → (∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ ∃𝑥𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)))
10 df-opab 4872 . . 3 {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {𝑧 ∣ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)}
119, 10elab2g 3508 . 2 (𝐴 ∈ V → (𝐴 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ ∃𝑥𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)))
121, 6, 11pm5.21nii 369 1 (𝐴 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ ∃𝑥𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ 𝜑))
Colors of variables: wff setvar class
Syntax hints:  wb 197  wa 384   = wceq 1652  wex 1874  wcel 2155  Vcvv 3350  cop 4340  {copab 4871
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2069  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4941  ax-nul 4949  ax-pr 5062
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2062  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-v 3352  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-nul 4080  df-if 4244  df-sn 4335  df-pr 4337  df-op 4341  df-opab 4872
This theorem is referenced by:  opelopabsbALT  5145  opelopabsb  5146  opelopabt  5148  opelopabga  5149  opabn0  5167  iunopab  5173  elopabr  5174  epelg  5191  elxp  5300  elopaelxp  5361  elopaba  5400  elcnv  5467  dfmpt3  6192  fmptsng  6627  fmptsnd  6628  0neqopab  6896  opabex3d  7343  opabex3  7344  fsplit  7484  rtrclreclem3  14087  isfunc  16791  griedg0ssusgr  26436  rgrusgrprc  26776  brabgaf  29803  qqhval2  30408  eulerpartlemgvv  30820  poimirlem26  33791  ecxrn  34510  dicelval3  37068  pellexlem5  38007  pellex  38009  opelopab4  39361  sprsymrelfvlem  42341  uspgrsprf  42355  uspgrsprf1  42356
  Copyright terms: Public domain W3C validator