Theorem opabbi2dv 5271

Description: Deduce equality of a relation and an ordered-pair class builder. Compare abbi2dv 2742. (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 5251	. . 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 4716	. 2 ⊢ (𝜑 → {⟨𝑥, 𝑦⟩ ∣ ⟨𝑥, 𝑦⟩ ∈ 𝐴} = {⟨𝑥, 𝑦⟩ ∣ 𝜓})
6	3, 5	syl5eqr 2670	1 ⊢ (𝜑 → 𝐴 = {⟨𝑥, 𝑦⟩ ∣ 𝜓})

Syntax hints:   → wi 4   ↔ wb 196   = wceq 1483   ∈ wcel 1990  ⟨cop 4183  {copab 4712  Rel wrel 5119

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-opab 4713  df-xp 5120  df-rel 5121

This theorem is referenced by:  recmulnq  9786  dmscut  31918  dib1dim  36454