Theorem opabresid 5072

Description: The restricted identity relation expressed as an ordered-pair class abstraction. (Contributed by FL, 25-Apr-2012.)

Assertion

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

Distinct variable group:   𝑥,𝐴,𝑦

Proof of Theorem opabresid

Step	Hyp	Ref	Expression
1		df-id 4396	. . . 4 ⊢ I = {⟨𝑥, 𝑦⟩ ∣ 𝑥 = 𝑦}
2		equcom 1754	. . . . 5 ⊢ (𝑥 = 𝑦 ↔ 𝑦 = 𝑥)
3	2	opabbii 4161	. . . 4 ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝑥 = 𝑦} = {⟨𝑥, 𝑦⟩ ∣ 𝑦 = 𝑥}
4	1, 3	eqtri 2252	. . 3 ⊢ I = {⟨𝑥, 𝑦⟩ ∣ 𝑦 = 𝑥}
5	4	reseq1i 5015	. 2 ⊢ ( I ↾ 𝐴) = ({⟨𝑥, 𝑦⟩ ∣ 𝑦 = 𝑥} ↾ 𝐴)
6		resopab 5063	. 2 ⊢ ({⟨𝑥, 𝑦⟩ ∣ 𝑦 = 𝑥} ↾ 𝐴) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝑥)}
7	5, 6	eqtri 2252	1 ⊢ ( I ↾ 𝐴) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝑥)}

Syntax hints:   ∧ wa 104   = wceq 1398   ∈ wcel 2202  {copab 4154   I cid 4391   ↾ cres 4733

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-v 2805  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-opab 4156  df-id 4396  df-xp 4737  df-rel 4738  df-res 4743

This theorem is referenced by:  mptresid  5073