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Theorem restidsing 5608
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.
StepHypRef Expression
1 relres 5576 . 2 Rel ( I ↾ {𝐴})
2 relxp 5275 . 2 Rel ({𝐴} × {𝐴})
3 velsn 4329 . . . . 5 (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴)
4 velsn 4329 . . . . 5 (𝑦 ∈ {𝐴} ↔ 𝑦 = 𝐴)
53, 4anbi12i 735 . . . 4 ((𝑥 ∈ {𝐴} ∧ 𝑦 ∈ {𝐴}) ↔ (𝑥 = 𝐴𝑦 = 𝐴))
6 vex 3335 . . . . . . 7 𝑦 ∈ V
76ideq 5422 . . . . . 6 (𝑥 I 𝑦𝑥 = 𝑦)
87, 3anbi12i 735 . . . . 5 ((𝑥 I 𝑦𝑥 ∈ {𝐴}) ↔ (𝑥 = 𝑦𝑥 = 𝐴))
9 ancom 465 . . . . 5 ((𝑥 = 𝑦𝑥 = 𝐴) ↔ (𝑥 = 𝐴𝑥 = 𝑦))
10 eqeq1 2756 . . . . . . 7 (𝑥 = 𝐴 → (𝑥 = 𝑦𝐴 = 𝑦))
11 eqcom 2759 . . . . . . 7 (𝐴 = 𝑦𝑦 = 𝐴)
1210, 11syl6bb 276 . . . . . 6 (𝑥 = 𝐴 → (𝑥 = 𝑦𝑦 = 𝐴))
1312pm5.32i 672 . . . . 5 ((𝑥 = 𝐴𝑥 = 𝑦) ↔ (𝑥 = 𝐴𝑦 = 𝐴))
148, 9, 133bitri 286 . . . 4 ((𝑥 I 𝑦𝑥 ∈ {𝐴}) ↔ (𝑥 = 𝐴𝑦 = 𝐴))
15 df-br 4797 . . . . 5 (𝑥 I 𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ I )
1615anbi1i 733 . . . 4 ((𝑥 I 𝑦𝑥 ∈ {𝐴}) ↔ (⟨𝑥, 𝑦⟩ ∈ I ∧ 𝑥 ∈ {𝐴}))
175, 14, 163bitr2ri 289 . . 3 ((⟨𝑥, 𝑦⟩ ∈ I ∧ 𝑥 ∈ {𝐴}) ↔ (𝑥 ∈ {𝐴} ∧ 𝑦 ∈ {𝐴}))
186opelres 5551 . . 3 (⟨𝑥, 𝑦⟩ ∈ ( I ↾ {𝐴}) ↔ (⟨𝑥, 𝑦⟩ ∈ I ∧ 𝑥 ∈ {𝐴}))
19 opelxp 5295 . . 3 (⟨𝑥, 𝑦⟩ ∈ ({𝐴} × {𝐴}) ↔ (𝑥 ∈ {𝐴} ∧ 𝑦 ∈ {𝐴}))
2017, 18, 193bitr4i 292 . 2 (⟨𝑥, 𝑦⟩ ∈ ( I ↾ {𝐴}) ↔ ⟨𝑥, 𝑦⟩ ∈ ({𝐴} × {𝐴}))
211, 2, 20eqrelriiv 5363 1 ( I ↾ {𝐴}) = ({𝐴} × {𝐴})
Colors of variables: wff setvar class
Syntax hints:  wa 383   = wceq 1624  wcel 2131  {csn 4313  cop 4319   class class class wbr 4796   I cid 5165   × cxp 5256  cres 5260
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732  ax-sep 4925  ax-nul 4933  ax-pr 5047
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1627  df-ex 1846  df-nf 1851  df-sb 2039  df-eu 2603  df-mo 2604  df-clab 2739  df-cleq 2745  df-clel 2748  df-nfc 2883  df-ral 3047  df-rex 3048  df-rab 3051  df-v 3334  df-dif 3710  df-un 3712  df-in 3714  df-ss 3721  df-nul 4051  df-if 4223  df-sn 4314  df-pr 4316  df-op 4320  df-br 4797  df-opab 4857  df-id 5166  df-xp 5264  df-rel 5265  df-res 5270
This theorem is referenced by:  residpr  6564  grp1inv  17716  psgnsn  18132  m1detdiag  20597
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