Theorem restidsing 5377

Description: Restriction of the identity to a singleton. (Contributed by FL, 2-Aug-2009.) (Proof shortened by JJ, 25-Aug-2021.)

Assertion

Ref	Expression
restidsing	⊢ ( I ↾ {𝐴}) = ({𝐴} × {𝐴})

Proof of Theorem restidsing

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		relres 5346	. 2 ⊢ Rel ( I ↾ {𝐴})
2		relxp 5150	. 2 ⊢ Rel ({𝐴} × {𝐴})
3		velsn 4141	. . . . 5 ⊢ (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴)
4		velsn 4141	. . . . 5 ⊢ (𝑦 ∈ {𝐴} ↔ 𝑦 = 𝐴)
5	3, 4	anbi12i 729	. . . 4 ⊢ ((𝑥 ∈ {𝐴} ∧ 𝑦 ∈ {𝐴}) ↔ (𝑥 = 𝐴 ∧ 𝑦 = 𝐴))
6		vex 3176	. . . . . . 7 ⊢ 𝑦 ∈ V
7	6	ideq 5196	. . . . . 6 ⊢ (𝑥 I 𝑦 ↔ 𝑥 = 𝑦)
8	7, 3	anbi12i 729	. . . . 5 ⊢ ((𝑥 I 𝑦 ∧ 𝑥 ∈ {𝐴}) ↔ (𝑥 = 𝑦 ∧ 𝑥 = 𝐴))
9		ancom 465	. . . . 5 ⊢ ((𝑥 = 𝑦 ∧ 𝑥 = 𝐴) ↔ (𝑥 = 𝐴 ∧ 𝑥 = 𝑦))
10		eqeq1 2614	. . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝑥 = 𝑦 ↔ 𝐴 = 𝑦))
11		eqcom 2617	. . . . . . 7 ⊢ (𝐴 = 𝑦 ↔ 𝑦 = 𝐴)
12	10, 11	syl6bb 275	. . . . . 6 ⊢ (𝑥 = 𝐴 → (𝑥 = 𝑦 ↔ 𝑦 = 𝐴))
13	12	pm5.32i 667	. . . . 5 ⊢ ((𝑥 = 𝐴 ∧ 𝑥 = 𝑦) ↔ (𝑥 = 𝐴 ∧ 𝑦 = 𝐴))
14	8, 9, 13	3bitri 285	. . . 4 ⊢ ((𝑥 I 𝑦 ∧ 𝑥 ∈ {𝐴}) ↔ (𝑥 = 𝐴 ∧ 𝑦 = 𝐴))
15		df-br 4584	. . . . 5 ⊢ (𝑥 I 𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ I )
16	15	anbi1i 727	. . . 4 ⊢ ((𝑥 I 𝑦 ∧ 𝑥 ∈ {𝐴}) ↔ (⟨𝑥, 𝑦⟩ ∈ I ∧ 𝑥 ∈ {𝐴}))
17	5, 14, 16	3bitr2ri 288	. . 3 ⊢ ((⟨𝑥, 𝑦⟩ ∈ I ∧ 𝑥 ∈ {𝐴}) ↔ (𝑥 ∈ {𝐴} ∧ 𝑦 ∈ {𝐴}))
18	6	opelres 5322	. . 3 ⊢ (⟨𝑥, 𝑦⟩ ∈ ( I ↾ {𝐴}) ↔ (⟨𝑥, 𝑦⟩ ∈ I ∧ 𝑥 ∈ {𝐴}))
19		opelxp 5070	. . 3 ⊢ (⟨𝑥, 𝑦⟩ ∈ ({𝐴} × {𝐴}) ↔ (𝑥 ∈ {𝐴} ∧ 𝑦 ∈ {𝐴}))
20	17, 18, 19	3bitr4i 291	. 2 ⊢ (⟨𝑥, 𝑦⟩ ∈ ( I ↾ {𝐴}) ↔ ⟨𝑥, 𝑦⟩ ∈ ({𝐴} × {𝐴}))
21	1, 2, 20	eqrelriiv 5137	1 ⊢ ( I ↾ {𝐴}) = ({𝐴} × {𝐴})

Syntax hints:   ∧ wa 383   = wceq 1475   ∈ wcel 1977  {csn 4125  ⟨cop 4131   class class class wbr 4583   I cid 4948   × cxp 5036   ↾ 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-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  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-br 4584  df-opab 4644  df-id 4953  df-xp 5044  df-rel 5045  df-res 5050

This theorem is referenced by:  residpr  6315  grp1inv  17346  psgnsn  17763  m1detdiag  20222