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Theorem restidsing 5422
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 5390 . 2 Rel ( I ↾ {𝐴})
2 relxp 5193 . 2 Rel ({𝐴} × {𝐴})
3 velsn 4169 . . . . 5 (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴)
4 velsn 4169 . . . . 5 (𝑦 ∈ {𝐴} ↔ 𝑦 = 𝐴)
53, 4anbi12i 732 . . . 4 ((𝑥 ∈ {𝐴} ∧ 𝑦 ∈ {𝐴}) ↔ (𝑥 = 𝐴𝑦 = 𝐴))
6 vex 3192 . . . . . . 7 𝑦 ∈ V
76ideq 5239 . . . . . 6 (𝑥 I 𝑦𝑥 = 𝑦)
87, 3anbi12i 732 . . . . 5 ((𝑥 I 𝑦𝑥 ∈ {𝐴}) ↔ (𝑥 = 𝑦𝑥 = 𝐴))
9 ancom 466 . . . . 5 ((𝑥 = 𝑦𝑥 = 𝐴) ↔ (𝑥 = 𝐴𝑥 = 𝑦))
10 eqeq1 2625 . . . . . . 7 (𝑥 = 𝐴 → (𝑥 = 𝑦𝐴 = 𝑦))
11 eqcom 2628 . . . . . . 7 (𝐴 = 𝑦𝑦 = 𝐴)
1210, 11syl6bb 276 . . . . . 6 (𝑥 = 𝐴 → (𝑥 = 𝑦𝑦 = 𝐴))
1312pm5.32i 668 . . . . 5 ((𝑥 = 𝐴𝑥 = 𝑦) ↔ (𝑥 = 𝐴𝑦 = 𝐴))
148, 9, 133bitri 286 . . . 4 ((𝑥 I 𝑦𝑥 ∈ {𝐴}) ↔ (𝑥 = 𝐴𝑦 = 𝐴))
15 df-br 4619 . . . . 5 (𝑥 I 𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ I )
1615anbi1i 730 . . . 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 5366 . . 3 (⟨𝑥, 𝑦⟩ ∈ ( I ↾ {𝐴}) ↔ (⟨𝑥, 𝑦⟩ ∈ I ∧ 𝑥 ∈ {𝐴}))
19 opelxp 5111 . . 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 5180 1 ( I ↾ {𝐴}) = ({𝐴} × {𝐴})
Colors of variables: wff setvar class
Syntax hints:  wa 384   = wceq 1480  wcel 1987  {csn 4153  cop 4159   class class class wbr 4618   I cid 4989   × cxp 5077  cres 5081
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4746  ax-nul 4754  ax-pr 4872
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3191  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-br 4619  df-opab 4679  df-id 4994  df-xp 5085  df-rel 5086  df-res 5091
This theorem is referenced by:  residpr  6369  grp1inv  17451  psgnsn  17868  m1detdiag  20331
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