ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  opabbi2dv GIF version

Theorem opabbi2dv 4777
Description: Deduce equality of a relation and an ordered-pair class builder. Compare abbi2dv 2296. (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 4759 . . 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 4070 . 2 (𝜑 → {⟨𝑥, 𝑦⟩ ∣ ⟨𝑥, 𝑦⟩ ∈ 𝐴} = {⟨𝑥, 𝑦⟩ ∣ 𝜓})
63, 5eqtr3id 2224 1 (𝜑𝐴 = {⟨𝑥, 𝑦⟩ ∣ 𝜓})
Colors of variables: wff set class
Syntax hints:  wi 4  wb 105   = wceq 1353  wcel 2148  cop 3596  {copab 4064  Rel wrel 4632
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4122  ax-pow 4175  ax-pr 4210
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2740  df-un 3134  df-in 3136  df-ss 3143  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-opab 4066  df-xp 4633  df-rel 4634
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator