Theorem restidsing 5458

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 5426	. 2 ⊢ Rel ( I ↾ {𝐴})
2		relxp 5227	. 2 ⊢ Rel ({𝐴} × {𝐴})
3		velsn 4193	. . . . 5 ⊢ (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴)
4		velsn 4193	. . . . 5 ⊢ (𝑦 ∈ {𝐴} ↔ 𝑦 = 𝐴)
5	3, 4	anbi12i 733	. . . 4 ⊢ ((𝑥 ∈ {𝐴} ∧ 𝑦 ∈ {𝐴}) ↔ (𝑥 = 𝐴 ∧ 𝑦 = 𝐴))
6		vex 3203	. . . . . . 7 ⊢ 𝑦 ∈ V
7	6	ideq 5274	. . . . . 6 ⊢ (𝑥 I 𝑦 ↔ 𝑥 = 𝑦)
8	7, 3	anbi12i 733	. . . . 5 ⊢ ((𝑥 I 𝑦 ∧ 𝑥 ∈ {𝐴}) ↔ (𝑥 = 𝑦 ∧ 𝑥 = 𝐴))
9		ancom 466	. . . . 5 ⊢ ((𝑥 = 𝑦 ∧ 𝑥 = 𝐴) ↔ (𝑥 = 𝐴 ∧ 𝑥 = 𝑦))
10		eqeq1 2626	. . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝑥 = 𝑦 ↔ 𝐴 = 𝑦))
11		eqcom 2629	. . . . . . 7 ⊢ (𝐴 = 𝑦 ↔ 𝑦 = 𝐴)
12	10, 11	syl6bb 276	. . . . . 6 ⊢ (𝑥 = 𝐴 → (𝑥 = 𝑦 ↔ 𝑦 = 𝐴))
13	12	pm5.32i 669	. . . . 5 ⊢ ((𝑥 = 𝐴 ∧ 𝑥 = 𝑦) ↔ (𝑥 = 𝐴 ∧ 𝑦 = 𝐴))
14	8, 9, 13	3bitri 286	. . . 4 ⊢ ((𝑥 I 𝑦 ∧ 𝑥 ∈ {𝐴}) ↔ (𝑥 = 𝐴 ∧ 𝑦 = 𝐴))
15		df-br 4654	. . . . 5 ⊢ (𝑥 I 𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ I )
16	15	anbi1i 731	. . . 4 ⊢ ((𝑥 I 𝑦 ∧ 𝑥 ∈ {𝐴}) ↔ (⟨𝑥, 𝑦⟩ ∈ I ∧ 𝑥 ∈ {𝐴}))
17	5, 14, 16	3bitr2ri 289	. . 3 ⊢ ((⟨𝑥, 𝑦⟩ ∈ I ∧ 𝑥 ∈ {𝐴}) ↔ (𝑥 ∈ {𝐴} ∧ 𝑦 ∈ {𝐴}))
18	6	opelres 5401	. . 3 ⊢ (⟨𝑥, 𝑦⟩ ∈ ( I ↾ {𝐴}) ↔ (⟨𝑥, 𝑦⟩ ∈ I ∧ 𝑥 ∈ {𝐴}))
19		opelxp 5146	. . 3 ⊢ (⟨𝑥, 𝑦⟩ ∈ ({𝐴} × {𝐴}) ↔ (𝑥 ∈ {𝐴} ∧ 𝑦 ∈ {𝐴}))
20	17, 18, 19	3bitr4i 292	. 2 ⊢ (⟨𝑥, 𝑦⟩ ∈ ( I ↾ {𝐴}) ↔ ⟨𝑥, 𝑦⟩ ∈ ({𝐴} × {𝐴}))
21	1, 2, 20	eqrelriiv 5214	1 ⊢ ( I ↾ {𝐴}) = ({𝐴} × {𝐴})

Syntax hints:   ∧ wa 384   = wceq 1483   ∈ wcel 1990  {csn 4177  ⟨cop 4183   class class class wbr 4653   I cid 5023   × cxp 5112   ↾ cres 5116

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-res 5126

This theorem is referenced by:  residpr  6409  grp1inv  17523  psgnsn  17940  m1detdiag  20403