Theorem opabbi2dv 4929

Description: Deduce equality of a relation and an ordered-pair class builder. Compare abbi2dv 2359. (Contributed by NM, 24-Feb-2014.)

Hypotheses

Ref	Expression
opabbi2dv.1	⊢ Rel 𝐴
opabbi2dv.3	⊢ (𝜑 → (⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ 𝜓))

Assertion

Ref	Expression
opabbi2dv	⊢ (𝜑 → 𝐴 = {⟨𝑥, 𝑦⟩ ∣ 𝜓})

Distinct variable groups:   𝑥,𝑦,𝐴   𝜑,𝑥,𝑦

Allowed substitution hints:   𝜓(𝑥, 𝑦)

Proof of Theorem opabbi2dv

Step	Hyp	Ref	Expression
1		opabbi2dv.1	. . 3 ⊢ Rel 𝐴
2		opabid2 4911	. . 3 ⊢ (Rel 𝐴 → {⟨𝑥, 𝑦⟩ ∣ ⟨𝑥, 𝑦⟩ ∈ 𝐴} = 𝐴)
3	1, 2	ax-mp 5	. 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ ⟨𝑥, 𝑦⟩ ∈ 𝐴} = 𝐴
4		opabbi2dv.3	. . 3 ⊢ (𝜑 → (⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ 𝜓))
5	4	opabbidv 4197	. 2 ⊢ (𝜑 → {⟨𝑥, 𝑦⟩ ∣ ⟨𝑥, 𝑦⟩ ∈ 𝐴} = {⟨𝑥, 𝑦⟩ ∣ 𝜓})
6	3, 5	eqtr3id 2285	1 ⊢ (𝜑 → 𝐴 = {⟨𝑥, 𝑦⟩ ∣ 𝜓})

This proof depends on syntax axioms:   → wi 4   ↔ wb 105   = wceq 1402   ∈ wcel 2209  ⟨cop 3712  {copab 4191  Rel wrel 4779

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-pow 4311  ax-pr 4346

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-opab 4193  df-xp 4780  df-rel 4781

This theorem is used by: (None)