MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  restidsing Structured version   Visualization version   GIF version

Theorem restidsing 6073
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.
StepHypRef Expression
1 relres 6026 . 2 Rel ( I ↾ {𝐴})
2 relxp 5707 . 2 Rel ({𝐴} × {𝐴})
3 velsn 4647 . . . . 5 (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴)
4 velsn 4647 . . . . 5 (𝑦 ∈ {𝐴} ↔ 𝑦 = 𝐴)
53, 4anbi12i 628 . . . 4 ((𝑥 ∈ {𝐴} ∧ 𝑦 ∈ {𝐴}) ↔ (𝑥 = 𝐴𝑦 = 𝐴))
6 vex 3482 . . . . . . 7 𝑦 ∈ V
76ideq 5866 . . . . . 6 (𝑥 I 𝑦𝑥 = 𝑦)
83, 7anbi12i 628 . . . . 5 ((𝑥 ∈ {𝐴} ∧ 𝑥 I 𝑦) ↔ (𝑥 = 𝐴𝑥 = 𝑦))
9 eqeq1 2739 . . . . . . 7 (𝑥 = 𝐴 → (𝑥 = 𝑦𝐴 = 𝑦))
10 eqcom 2742 . . . . . . 7 (𝐴 = 𝑦𝑦 = 𝐴)
119, 10bitrdi 287 . . . . . 6 (𝑥 = 𝐴 → (𝑥 = 𝑦𝑦 = 𝐴))
1211pm5.32i 574 . . . . 5 ((𝑥 = 𝐴𝑥 = 𝑦) ↔ (𝑥 = 𝐴𝑦 = 𝐴))
138, 12bitri 275 . . . 4 ((𝑥 ∈ {𝐴} ∧ 𝑥 I 𝑦) ↔ (𝑥 = 𝐴𝑦 = 𝐴))
14 df-br 5149 . . . . 5 (𝑥 I 𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ I )
1514anbi2i 623 . . . 4 ((𝑥 ∈ {𝐴} ∧ 𝑥 I 𝑦) ↔ (𝑥 ∈ {𝐴} ∧ ⟨𝑥, 𝑦⟩ ∈ I ))
165, 13, 153bitr2ri 300 . . 3 ((𝑥 ∈ {𝐴} ∧ ⟨𝑥, 𝑦⟩ ∈ I ) ↔ (𝑥 ∈ {𝐴} ∧ 𝑦 ∈ {𝐴}))
176opelresi 6008 . . 3 (⟨𝑥, 𝑦⟩ ∈ ( I ↾ {𝐴}) ↔ (𝑥 ∈ {𝐴} ∧ ⟨𝑥, 𝑦⟩ ∈ I ))
18 opelxp 5725 . . 3 (⟨𝑥, 𝑦⟩ ∈ ({𝐴} × {𝐴}) ↔ (𝑥 ∈ {𝐴} ∧ 𝑦 ∈ {𝐴}))
1916, 17, 183bitr4i 303 . 2 (⟨𝑥, 𝑦⟩ ∈ ( I ↾ {𝐴}) ↔ ⟨𝑥, 𝑦⟩ ∈ ({𝐴} × {𝐴}))
201, 2, 19eqrelriiv 5803 1 ( I ↾ {𝐴}) = ({𝐴} × {𝐴})
Colors of variables: wff setvar class
Syntax hints:  wa 395   = wceq 1537  wcel 2106  {csn 4631  cop 4637   class class class wbr 5148   I cid 5582   × cxp 5687  cres 5691
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  df-id 5583  df-xp 5695  df-rel 5696  df-res 5701
This theorem is referenced by:  residpr  7163  grp1inv  19079  psgnsn  19553  m1detdiag  22619
  Copyright terms: Public domain W3C validator