Theorem opabresid 6052

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 5556	. . . 4 ⊢ I = {⟨𝑥, 𝑦⟩ ∣ 𝑥 = 𝑦}
2		equcom 2048	. . . . 5 ⊢ (𝑥 = 𝑦 ↔ 𝑦 = 𝑥)
3	2	opabbii 5178	. . . 4 ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝑥 = 𝑦} = {⟨𝑥, 𝑦⟩ ∣ 𝑦 = 𝑥}
4	1, 3	eqtri 2786	. . 3 ⊢ I = {⟨𝑥, 𝑦⟩ ∣ 𝑦 = 𝑥}
5	4	reseq1i 5974	. 2 ⊢ ( I ↾ 𝐴) = ({⟨𝑥, 𝑦⟩ ∣ 𝑦 = 𝑥} ↾ 𝐴)
6		resopab 6036	. 2 ⊢ ({⟨𝑥, 𝑦⟩ ∣ 𝑦 = 𝑥} ↾ 𝐴) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝑥)}
7	5, 6	eqtri 2786	1 ⊢ ( I ↾ 𝐴) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝑥)}

Syntax hints:   ∧ wa 400   = wceq 1570   ∈ wcel 2143  {copab 5173   I cid 5555   ↾ cres 5663

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-12 2213  ax-ext 2735  ax-sep 5257  ax-pr 5404

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-rab 3417  df-v 3457  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-sn 4590  df-pr 4592  df-op 4596  df-opab 5174  df-id 5556  df-xp 5667  df-rel 5668  df-res 5673

This theorem is referenced by:  mptresid  6053  iresn0n0  6056  pospo  18394  eqg0subg  19262  opsrtoslem1  22206  tgphaus  24274  relexp0eq  44447