MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  opabbi2dv Structured version   Visualization version   GIF version

Theorem opabbi2dv 5693
Description: Deduce equality of a relation and an ordered-pair class builder. Compare abbi2dv 2949. (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 5673 . . 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 5105 . 2 (𝜑 → {⟨𝑥, 𝑦⟩ ∣ ⟨𝑥, 𝑦⟩ ∈ 𝐴} = {⟨𝑥, 𝑦⟩ ∣ 𝜓})
63, 5syl5eqr 2870 1 (𝜑𝐴 = {⟨𝑥, 𝑦⟩ ∣ 𝜓})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209   = wceq 1538  wcel 2115  cop 4546  {copab 5101  Rel wrel 5533
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793  ax-sep 5176  ax-nul 5183  ax-pr 5303
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2623  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-rab 3135  df-v 3473  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4267  df-if 4441  df-sn 4541  df-pr 4543  df-op 4547  df-opab 5102  df-xp 5534  df-rel 5535
This theorem is referenced by:  recmulnq  10363  dmscut  33279  dib1dim  38339
  Copyright terms: Public domain W3C validator