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Theorem opabbi2dv 4585
Description: Deduce equality of a relation and an ordered-pair class builder. Compare abbi2dv 2206. (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 4567 . . 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 3904 . 2 (𝜑 → {⟨𝑥, 𝑦⟩ ∣ ⟨𝑥, 𝑦⟩ ∈ 𝐴} = {⟨𝑥, 𝑦⟩ ∣ 𝜓})
63, 5syl5eqr 2134 1 (𝜑𝐴 = {⟨𝑥, 𝑦⟩ ∣ 𝜓})
Colors of variables: wff set class
Syntax hints:  wi 4  wb 103   = wceq 1289  wcel 1438  cop 3449  {copab 3898  Rel wrel 4443
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pow 4009  ax-pr 4036
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-un 3003  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-opab 3900  df-xp 4444  df-rel 4445
This theorem is referenced by: (None)
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