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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.
StepHypRef 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)
53, 4anbi12i 729 . . . 4 ((𝑥 ∈ {𝐴} ∧ 𝑦 ∈ {𝐴}) ↔ (𝑥 = 𝐴𝑦 = 𝐴))
6 vex 3176 . . . . . . 7 𝑦 ∈ V
76ideq 5196 . . . . . 6 (𝑥 I 𝑦𝑥 = 𝑦)
87, 3anbi12i 729 . . . . 5 ((𝑥 I 𝑦𝑥 ∈ {𝐴}) ↔ (𝑥 = 𝑦𝑥 = 𝐴))
9 ancom 465 . . . . 5 ((𝑥 = 𝑦𝑥 = 𝐴) ↔ (𝑥 = 𝐴𝑥 = 𝑦))
10 eqeq1 2614 . . . . . . 7 (𝑥 = 𝐴 → (𝑥 = 𝑦𝐴 = 𝑦))
11 eqcom 2617 . . . . . . 7 (𝐴 = 𝑦𝑦 = 𝐴)
1210, 11syl6bb 275 . . . . . 6 (𝑥 = 𝐴 → (𝑥 = 𝑦𝑦 = 𝐴))
1312pm5.32i 667 . . . . 5 ((𝑥 = 𝐴𝑥 = 𝑦) ↔ (𝑥 = 𝐴𝑦 = 𝐴))
148, 9, 133bitri 285 . . . 4 ((𝑥 I 𝑦𝑥 ∈ {𝐴}) ↔ (𝑥 = 𝐴𝑦 = 𝐴))
15 df-br 4584 . . . . 5 (𝑥 I 𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ I )
1615anbi1i 727 . . . 4 ((𝑥 I 𝑦𝑥 ∈ {𝐴}) ↔ (⟨𝑥, 𝑦⟩ ∈ I ∧ 𝑥 ∈ {𝐴}))
175, 14, 163bitr2ri 288 . . 3 ((⟨𝑥, 𝑦⟩ ∈ I ∧ 𝑥 ∈ {𝐴}) ↔ (𝑥 ∈ {𝐴} ∧ 𝑦 ∈ {𝐴}))
186opelres 5322 . . 3 (⟨𝑥, 𝑦⟩ ∈ ( I ↾ {𝐴}) ↔ (⟨𝑥, 𝑦⟩ ∈ I ∧ 𝑥 ∈ {𝐴}))
19 opelxp 5070 . . 3 (⟨𝑥, 𝑦⟩ ∈ ({𝐴} × {𝐴}) ↔ (𝑥 ∈ {𝐴} ∧ 𝑦 ∈ {𝐴}))
2017, 18, 193bitr4i 291 . 2 (⟨𝑥, 𝑦⟩ ∈ ( I ↾ {𝐴}) ↔ ⟨𝑥, 𝑦⟩ ∈ ({𝐴} × {𝐴}))
211, 2, 20eqrelriiv 5137 1 ( I ↾ {𝐴}) = ({𝐴} × {𝐴})
 Colors of variables: wff setvar class 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
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