Theorem opabresid 5711

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 5696	. 2 ⊢ ({⟨𝑥, 𝑦⟩ ∣ 𝑦 = 𝑥} ↾ 𝐴) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝑥)}
2		equcom 2065	. . . . 5 ⊢ (𝑦 = 𝑥 ↔ 𝑥 = 𝑦)
3	2	opabbii 4953	. . . 4 ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝑦 = 𝑥} = {⟨𝑥, 𝑦⟩ ∣ 𝑥 = 𝑦}
4		df-id 5261	. . . 4 ⊢ I = {⟨𝑥, 𝑦⟩ ∣ 𝑥 = 𝑦}
5	3, 4	eqtr4i 2805	. . 3 ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝑦 = 𝑥} = I
6	5	reseq1i 5638	. 2 ⊢ ({⟨𝑥, 𝑦⟩ ∣ 𝑦 = 𝑥} ↾ 𝐴) = ( I ↾ 𝐴)
7	1, 6	eqtr3i 2804	1 ⊢ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝑥)} = ( I ↾ 𝐴)

Syntax hints:   ∧ wa 386   = wceq 1601   ∈ wcel 2107  {copab 4948   I cid 5260   ↾ cres 5357

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-sep 5017  ax-nul 5025  ax-pr 5138

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-rab 3099  df-v 3400  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-sn 4399  df-pr 4401  df-op 4405  df-opab 4949  df-id 5261  df-xp 5361  df-rel 5362  df-res 5367

This theorem is referenced by:  mptresid  5712  pospo  17359  opsrtoslem1  19880  tgphaus  22328  relexp0eq  38950