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Theorem opabbi2dv 4485
Description: Deduce equality of a relation and an ordered-pair class builder. Compare abbi2dv 2156. (Contributed by NM, 24-Feb-2014.)
Hypotheses
Ref Expression
opabbi2dv.1 Rel 𝐴
opabbi2dv.3 (𝜑 → (⟨𝑥, 𝑦⟩ ∈ 𝐴𝜓))
Assertion
Ref Expression
opabbi2dv (𝜑𝐴 = {⟨𝑥, 𝑦⟩ ∣ 𝜓})
Distinct variable groups:   𝑥,𝑦,𝐴   𝜑,𝑥,𝑦
Allowed substitution hints:   𝜓(𝑥,𝑦)

Proof of Theorem opabbi2dv
StepHypRef Expression
1 opabbi2dv.1 . . 3 Rel 𝐴
2 opabid2 4467 . . 3 (Rel 𝐴 → {⟨𝑥, 𝑦⟩ ∣ ⟨𝑥, 𝑦⟩ ∈ 𝐴} = 𝐴)
31, 2ax-mp 7 . 2 {⟨𝑥, 𝑦⟩ ∣ ⟨𝑥, 𝑦⟩ ∈ 𝐴} = 𝐴
4 opabbi2dv.3 . . 3 (𝜑 → (⟨𝑥, 𝑦⟩ ∈ 𝐴𝜓))
54opabbidv 3823 . 2 (𝜑 → {⟨𝑥, 𝑦⟩ ∣ ⟨𝑥, 𝑦⟩ ∈ 𝐴} = {⟨𝑥, 𝑦⟩ ∣ 𝜓})
63, 5syl5eqr 2086 1 (𝜑𝐴 = {⟨𝑥, 𝑦⟩ ∣ 𝜓})
Colors of variables: wff set class
Syntax hints:  wi 4  wb 98   = wceq 1243  wcel 1393  cop 3378  {copab 3817  Rel wrel 4350
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944
This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-opab 3819  df-xp 4351  df-rel 4352
This theorem is referenced by: (None)
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