Theorem opabresid 5374

Description: The restricted identity expressed with the class builder. (Contributed by FL, 25-Apr-2012.)

Assertion

Ref	Expression
opabresid	⊢ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝑥)} = ( I ↾ 𝐴)

Distinct variable group:   𝑥,𝐴,𝑦

Proof of Theorem opabresid

Step	Hyp	Ref	Expression
1		resopab 5366	. 2 ⊢ ({⟨𝑥, 𝑦⟩ ∣ 𝑦 = 𝑥} ↾ 𝐴) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝑥)}
2		equcom 1932	. . . . 5 ⊢ (𝑦 = 𝑥 ↔ 𝑥 = 𝑦)
3	2	opabbii 4649	. . . 4 ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝑦 = 𝑥} = {⟨𝑥, 𝑦⟩ ∣ 𝑥 = 𝑦}
4		dfid3 4954	. . . 4 ⊢ I = {⟨𝑥, 𝑦⟩ ∣ 𝑥 = 𝑦}
5	3, 4	eqtr4i 2635	. . 3 ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝑦 = 𝑥} = I
6	5	reseq1i 5313	. 2 ⊢ ({⟨𝑥, 𝑦⟩ ∣ 𝑦 = 𝑥} ↾ 𝐴) = ( I ↾ 𝐴)
7	1, 6	eqtr3i 2634	1 ⊢ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝑥)} = ( I ↾ 𝐴)

Syntax hints:   ∧ wa 383   = wceq 1475   ∈ wcel 1977  {copab 4642   I cid 4948   ↾ cres 5040

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-opab 4644  df-id 4953  df-xp 5044  df-rel 5045  df-res 5050

This theorem is referenced by:  mptresid  5375  pospo  16796  opsrtoslem1  19305  tgphaus  21730  relexp0eq  37012