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Theorem restidsing 6061
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 6015 . 2 Rel ( I ↾ {𝐴})
2 relxp 5700 . 2 Rel ({𝐴} × {𝐴})
3 velsn 4648 . . . . 5 (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴)
4 velsn 4648 . . . . 5 (𝑦 ∈ {𝐴} ↔ 𝑦 = 𝐴)
53, 4anbi12i 626 . . . 4 ((𝑥 ∈ {𝐴} ∧ 𝑦 ∈ {𝐴}) ↔ (𝑥 = 𝐴𝑦 = 𝐴))
6 vex 3477 . . . . . . 7 𝑦 ∈ V
76ideq 5859 . . . . . 6 (𝑥 I 𝑦𝑥 = 𝑦)
83, 7anbi12i 626 . . . . 5 ((𝑥 ∈ {𝐴} ∧ 𝑥 I 𝑦) ↔ (𝑥 = 𝐴𝑥 = 𝑦))
9 eqeq1 2732 . . . . . . 7 (𝑥 = 𝐴 → (𝑥 = 𝑦𝐴 = 𝑦))
10 eqcom 2735 . . . . . . 7 (𝐴 = 𝑦𝑦 = 𝐴)
119, 10bitrdi 286 . . . . . 6 (𝑥 = 𝐴 → (𝑥 = 𝑦𝑦 = 𝐴))
1211pm5.32i 573 . . . . 5 ((𝑥 = 𝐴𝑥 = 𝑦) ↔ (𝑥 = 𝐴𝑦 = 𝐴))
138, 12bitri 274 . . . 4 ((𝑥 ∈ {𝐴} ∧ 𝑥 I 𝑦) ↔ (𝑥 = 𝐴𝑦 = 𝐴))
14 df-br 5153 . . . . 5 (𝑥 I 𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ I )
1514anbi2i 621 . . . 4 ((𝑥 ∈ {𝐴} ∧ 𝑥 I 𝑦) ↔ (𝑥 ∈ {𝐴} ∧ ⟨𝑥, 𝑦⟩ ∈ I ))
165, 13, 153bitr2ri 299 . . 3 ((𝑥 ∈ {𝐴} ∧ ⟨𝑥, 𝑦⟩ ∈ I ) ↔ (𝑥 ∈ {𝐴} ∧ 𝑦 ∈ {𝐴}))
176opelresi 5997 . . 3 (⟨𝑥, 𝑦⟩ ∈ ( I ↾ {𝐴}) ↔ (𝑥 ∈ {𝐴} ∧ ⟨𝑥, 𝑦⟩ ∈ I ))
18 opelxp 5718 . . 3 (⟨𝑥, 𝑦⟩ ∈ ({𝐴} × {𝐴}) ↔ (𝑥 ∈ {𝐴} ∧ 𝑦 ∈ {𝐴}))
1916, 17, 183bitr4i 302 . 2 (⟨𝑥, 𝑦⟩ ∈ ( I ↾ {𝐴}) ↔ ⟨𝑥, 𝑦⟩ ∈ ({𝐴} × {𝐴}))
201, 2, 19eqrelriiv 5796 1 ( I ↾ {𝐴}) = ({𝐴} × {𝐴})
Colors of variables: wff setvar class
Syntax hints:  wa 394   = wceq 1533  wcel 2098  {csn 4632  cop 4638   class class class wbr 5152   I cid 5579   × cxp 5680  cres 5684
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pr 5433
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2706  df-cleq 2720  df-clel 2806  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-br 5153  df-opab 5215  df-id 5580  df-xp 5688  df-rel 5689  df-res 5694
This theorem is referenced by:  residpr  7158  grp1inv  19011  psgnsn  19482  m1detdiag  22519
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