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Theorem restidsingOLD 5378
Description: Obsolete proof of restidsing 5377 as of 25-Aug-2021. (Contributed by FL, 2-Aug-2009.) (New usage is discouraged.) (Proof modification is discouraged.)
Assertion
Ref Expression
restidsingOLD ( I ↾ {𝐴}) = ({𝐴} × {𝐴})

Proof of Theorem restidsingOLD
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 df-br 4584 . . . . . 6 (𝑥 I 𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ I )
43bicomi 213 . . . . 5 (⟨𝑥, 𝑦⟩ ∈ I ↔ 𝑥 I 𝑦)
54anbi1i 727 . . . 4 ((⟨𝑥, 𝑦⟩ ∈ I ∧ 𝑥 ∈ {𝐴}) ↔ (𝑥 I 𝑦𝑥 ∈ {𝐴}))
6 simpr 476 . . . . . 6 ((𝑥 I 𝑦𝑥 ∈ {𝐴}) → 𝑥 ∈ {𝐴})
7 velsn 4141 . . . . . . . 8 (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴)
8 vex 3176 . . . . . . . . . 10 𝑦 ∈ V
9 ideqg 5195 . . . . . . . . . . 11 (𝑦 ∈ V → (𝑥 I 𝑦𝑥 = 𝑦))
109biimpd 218 . . . . . . . . . 10 (𝑦 ∈ V → (𝑥 I 𝑦𝑥 = 𝑦))
118, 10ax-mp 5 . . . . . . . . 9 (𝑥 I 𝑦𝑥 = 𝑦)
12 eqtr2 2630 . . . . . . . . . . 11 ((𝑥 = 𝑦𝑥 = 𝐴) → 𝑦 = 𝐴)
1312ex 449 . . . . . . . . . 10 (𝑥 = 𝑦 → (𝑥 = 𝐴𝑦 = 𝐴))
14 velsn 4141 . . . . . . . . . 10 (𝑦 ∈ {𝐴} ↔ 𝑦 = 𝐴)
1513, 14syl6ibr 241 . . . . . . . . 9 (𝑥 = 𝑦 → (𝑥 = 𝐴𝑦 ∈ {𝐴}))
1611, 15syl 17 . . . . . . . 8 (𝑥 I 𝑦 → (𝑥 = 𝐴𝑦 ∈ {𝐴}))
177, 16syl5bi 231 . . . . . . 7 (𝑥 I 𝑦 → (𝑥 ∈ {𝐴} → 𝑦 ∈ {𝐴}))
1817imp 444 . . . . . 6 ((𝑥 I 𝑦𝑥 ∈ {𝐴}) → 𝑦 ∈ {𝐴})
196, 18jca 553 . . . . 5 ((𝑥 I 𝑦𝑥 ∈ {𝐴}) → (𝑥 ∈ {𝐴} ∧ 𝑦 ∈ {𝐴}))
20 eqtr3 2631 . . . . . . . . . . . 12 ((𝑦 = 𝐴𝑥 = 𝐴) → 𝑦 = 𝑥)
218ideq 5196 . . . . . . . . . . . . 13 (𝑥 I 𝑦𝑥 = 𝑦)
22 equcom 1932 . . . . . . . . . . . . 13 (𝑥 = 𝑦𝑦 = 𝑥)
2321, 22bitri 263 . . . . . . . . . . . 12 (𝑥 I 𝑦𝑦 = 𝑥)
2420, 23sylibr 223 . . . . . . . . . . 11 ((𝑦 = 𝐴𝑥 = 𝐴) → 𝑥 I 𝑦)
2524ex 449 . . . . . . . . . 10 (𝑦 = 𝐴 → (𝑥 = 𝐴𝑥 I 𝑦))
2614, 25sylbi 206 . . . . . . . . 9 (𝑦 ∈ {𝐴} → (𝑥 = 𝐴𝑥 I 𝑦))
2726com12 32 . . . . . . . 8 (𝑥 = 𝐴 → (𝑦 ∈ {𝐴} → 𝑥 I 𝑦))
287, 27sylbi 206 . . . . . . 7 (𝑥 ∈ {𝐴} → (𝑦 ∈ {𝐴} → 𝑥 I 𝑦))
2928imp 444 . . . . . 6 ((𝑥 ∈ {𝐴} ∧ 𝑦 ∈ {𝐴}) → 𝑥 I 𝑦)
30 simpl 472 . . . . . 6 ((𝑥 ∈ {𝐴} ∧ 𝑦 ∈ {𝐴}) → 𝑥 ∈ {𝐴})
3129, 30jca 553 . . . . 5 ((𝑥 ∈ {𝐴} ∧ 𝑦 ∈ {𝐴}) → (𝑥 I 𝑦𝑥 ∈ {𝐴}))
3219, 31impbii 198 . . . 4 ((𝑥 I 𝑦𝑥 ∈ {𝐴}) ↔ (𝑥 ∈ {𝐴} ∧ 𝑦 ∈ {𝐴}))
335, 32bitri 263 . . 3 ((⟨𝑥, 𝑦⟩ ∈ I ∧ 𝑥 ∈ {𝐴}) ↔ (𝑥 ∈ {𝐴} ∧ 𝑦 ∈ {𝐴}))
348opelres 5322 . . 3 (⟨𝑥, 𝑦⟩ ∈ ( I ↾ {𝐴}) ↔ (⟨𝑥, 𝑦⟩ ∈ I ∧ 𝑥 ∈ {𝐴}))
35 opelxp 5070 . . 3 (⟨𝑥, 𝑦⟩ ∈ ({𝐴} × {𝐴}) ↔ (𝑥 ∈ {𝐴} ∧ 𝑦 ∈ {𝐴}))
3633, 34, 353bitr4i 291 . 2 (⟨𝑥, 𝑦⟩ ∈ ( I ↾ {𝐴}) ↔ ⟨𝑥, 𝑦⟩ ∈ ({𝐴} × {𝐴}))
371, 2, 36eqrelriiv 5137 1 ( I ↾ {𝐴}) = ({𝐴} × {𝐴})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1475  wcel 1977  Vcvv 3173  {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: (None)
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