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Theorem opabbi2dv 5475
Description: Deduce equality of a relation and an ordered-pair class builder. Compare abbi2dv 2919. (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 5455 . . 3 (Rel 𝐴 → {⟨𝑥, 𝑦⟩ ∣ ⟨𝑥, 𝑦⟩ ∈ 𝐴} = 𝐴)
31, 2ax-mp 5 . 2 {⟨𝑥, 𝑦⟩ ∣ ⟨𝑥, 𝑦⟩ ∈ 𝐴} = 𝐴
4 opabbi2dv.3 . . 3 (𝜑 → (⟨𝑥, 𝑦⟩ ∈ 𝐴𝜓))
54opabbidv 4909 . 2 (𝜑 → {⟨𝑥, 𝑦⟩ ∣ ⟨𝑥, 𝑦⟩ ∈ 𝐴} = {⟨𝑥, 𝑦⟩ ∣ 𝜓})
63, 5syl5eqr 2847 1 (𝜑𝐴 = {⟨𝑥, 𝑦⟩ ∣ 𝜓})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198   = wceq 1653  wcel 2157  cop 4374  {copab 4905  Rel wrel 5317
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-sep 4975  ax-nul 4983  ax-pr 5097
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-rab 3098  df-v 3387  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-sn 4369  df-pr 4371  df-op 4375  df-opab 4906  df-xp 5318  df-rel 5319
This theorem is referenced by:  recmulnq  10074  dmscut  32431  dib1dim  37186
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