Theorem restidsing 5093

Description: Restriction of the identity to a singleton. (Contributed by FL, 2-Aug-2009.) (Proof shortened by JJ, 25-Aug-2021.) (Proof shortened by Peter Mazsa, 6-Oct-2022.)

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 5065	. 2 ⊢ Rel ( I ↾ {𝐴})
2		relxp 4858	. 2 ⊢ Rel ({𝐴} × {𝐴})
3		velsn 3705	. . . . 5 ⊢ (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴)
4		velsn 3705	. . . . 5 ⊢ (𝑦 ∈ {𝐴} ↔ 𝑦 = 𝐴)
5	3, 4	anbi12i 460	. . . 4 ⊢ ((𝑥 ∈ {𝐴} ∧ 𝑦 ∈ {𝐴}) ↔ (𝑥 = 𝐴 ∧ 𝑦 = 𝐴))
6		vex 2815	. . . . . . 7 ⊢ 𝑦 ∈ V
7	6	ideq 4906	. . . . . 6 ⊢ (𝑥 I 𝑦 ↔ 𝑥 = 𝑦)
8	3, 7	anbi12i 460	. . . . 5 ⊢ ((𝑥 ∈ {𝐴} ∧ 𝑥 I 𝑦) ↔ (𝑥 = 𝐴 ∧ 𝑥 = 𝑦))
9		eqeq1 2239	. . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝑥 = 𝑦 ↔ 𝐴 = 𝑦))
10		eqcom 2234	. . . . . . 7 ⊢ (𝐴 = 𝑦 ↔ 𝑦 = 𝐴)
11	9, 10	bitrdi 196	. . . . . 6 ⊢ (𝑥 = 𝐴 → (𝑥 = 𝑦 ↔ 𝑦 = 𝐴))
12	11	pm5.32i 454	. . . . 5 ⊢ ((𝑥 = 𝐴 ∧ 𝑥 = 𝑦) ↔ (𝑥 = 𝐴 ∧ 𝑦 = 𝐴))
13	8, 12	bitri 184	. . . 4 ⊢ ((𝑥 ∈ {𝐴} ∧ 𝑥 I 𝑦) ↔ (𝑥 = 𝐴 ∧ 𝑦 = 𝐴))
14		df-br 4109	. . . . 5 ⊢ (𝑥 I 𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ I )
15	14	anbi2i 457	. . . 4 ⊢ ((𝑥 ∈ {𝐴} ∧ 𝑥 I 𝑦) ↔ (𝑥 ∈ {𝐴} ∧ ⟨𝑥, 𝑦⟩ ∈ I ))
16	5, 13, 15	3bitr2ri 209	. . 3 ⊢ ((𝑥 ∈ {𝐴} ∧ ⟨𝑥, 𝑦⟩ ∈ I ) ↔ (𝑥 ∈ {𝐴} ∧ 𝑦 ∈ {𝐴}))
17	6	opelres 5042	. . . 4 ⊢ (⟨𝑥, 𝑦⟩ ∈ ( I ↾ {𝐴}) ↔ (⟨𝑥, 𝑦⟩ ∈ I ∧ 𝑥 ∈ {𝐴}))
18	17	biancomi 270	. . 3 ⊢ (⟨𝑥, 𝑦⟩ ∈ ( I ↾ {𝐴}) ↔ (𝑥 ∈ {𝐴} ∧ ⟨𝑥, 𝑦⟩ ∈ I ))
19		opelxp 4778	. . 3 ⊢ (⟨𝑥, 𝑦⟩ ∈ ({𝐴} × {𝐴}) ↔ (𝑥 ∈ {𝐴} ∧ 𝑦 ∈ {𝐴}))
20	16, 18, 19	3bitr4i 212	. 2 ⊢ (⟨𝑥, 𝑦⟩ ∈ ( I ↾ {𝐴}) ↔ ⟨𝑥, 𝑦⟩ ∈ ({𝐴} × {𝐴}))
21	1, 2, 20	eqrelriiv 4843	1 ⊢ ( I ↾ {𝐴}) = ({𝐴} × {𝐴})

Syntax hints:   ∧ wa 104   = wceq 1398   ∈ wcel 2203  {csn 3688  ⟨cop 3691   class class class wbr 4108   I cid 4408   × cxp 4746   ↾ cres 4750

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 2206  ax-ext 2214  ax-sep 4227  ax-pow 4286  ax-pr 4321

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-v 2814  df-un 3214  df-in 3216  df-ss 3223  df-pw 3670  df-sn 3694  df-pr 3695  df-op 3697  df-br 4109  df-opab 4171  df-id 4413  df-xp 4754  df-rel 4755  df-res 4760

This theorem is referenced by:  grp1inv  13809