Theorem opabresid 4944

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 4935	. 2 ⊢ ({⟨𝑥, 𝑦⟩ ∣ 𝑦 = 𝑥} ↾ 𝐴) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝑥)}
2		equcom 1699	. . . . 5 ⊢ (𝑦 = 𝑥 ↔ 𝑥 = 𝑦)
3	2	opabbii 4056	. . . 4 ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝑦 = 𝑥} = {⟨𝑥, 𝑦⟩ ∣ 𝑥 = 𝑦}
4		df-id 4278	. . . 4 ⊢ I = {⟨𝑥, 𝑦⟩ ∣ 𝑥 = 𝑦}
5	3, 4	eqtr4i 2194	. . 3 ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝑦 = 𝑥} = I
6	5	reseq1i 4887	. 2 ⊢ ({⟨𝑥, 𝑦⟩ ∣ 𝑦 = 𝑥} ↾ 𝐴) = ( I ↾ 𝐴)
7	1, 6	eqtr3i 2193	1 ⊢ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝑥)} = ( I ↾ 𝐴)

Syntax hints:   ∧ wa 103   = wceq 1348   ∈ wcel 2141  {copab 4049   I cid 4273   ↾ cres 4613

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-opab 4051  df-id 4278  df-xp 4617  df-rel 4618  df-res 4623

This theorem is referenced by:  mptresid  4945