Theorem opabbi2dv 5832

Description: Deduce equality of a relation and an ordered-pair class abstraction. Compare eqabdv 2897. (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 5811	. . 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 5174	. 2 ⊢ (𝜑 → {⟨𝑥, 𝑦⟩ ∣ ⟨𝑥, 𝑦⟩ ∈ 𝐴} = {⟨𝑥, 𝑦⟩ ∣ 𝜓})
6	3, 5	eqtr3id 2812	1 ⊢ (𝜑 → 𝐴 = {⟨𝑥, 𝑦⟩ ∣ 𝜓})

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   = wceq 1570   ∈ wcel 2147  ⟨cop 4838  {copab 5170  Rel wrel 5660

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735  ax-sep 5252  ax-pr 5395

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-opab 5171  df-xp 5661  df-rel 5662

This theorem is used by:  recmulnq  11073  dmcuts  28204  dib1dim  42413