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Theorem restidsingOLD 5423
Description: Obsolete proof of restidsing 5422 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 5390 . 2 Rel ( I ↾ {𝐴})
2 relxp 5193 . 2 Rel ({𝐴} × {𝐴})
3 df-br 4619 . . . . . 6 (𝑥 I 𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ I )
43bicomi 214 . . . . 5 (⟨𝑥, 𝑦⟩ ∈ I ↔ 𝑥 I 𝑦)
54anbi1i 730 . . . 4 ((⟨𝑥, 𝑦⟩ ∈ I ∧ 𝑥 ∈ {𝐴}) ↔ (𝑥 I 𝑦𝑥 ∈ {𝐴}))
6 simpr 477 . . . . . 6 ((𝑥 I 𝑦𝑥 ∈ {𝐴}) → 𝑥 ∈ {𝐴})
7 velsn 4169 . . . . . . . 8 (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴)
8 vex 3192 . . . . . . . . . 10 𝑦 ∈ V
9 ideqg 5238 . . . . . . . . . . 11 (𝑦 ∈ V → (𝑥 I 𝑦𝑥 = 𝑦))
109biimpd 219 . . . . . . . . . 10 (𝑦 ∈ V → (𝑥 I 𝑦𝑥 = 𝑦))
118, 10ax-mp 5 . . . . . . . . 9 (𝑥 I 𝑦𝑥 = 𝑦)
12 eqtr2 2641 . . . . . . . . . . 11 ((𝑥 = 𝑦𝑥 = 𝐴) → 𝑦 = 𝐴)
1312ex 450 . . . . . . . . . 10 (𝑥 = 𝑦 → (𝑥 = 𝐴𝑦 = 𝐴))
14 velsn 4169 . . . . . . . . . 10 (𝑦 ∈ {𝐴} ↔ 𝑦 = 𝐴)
1513, 14syl6ibr 242 . . . . . . . . 9 (𝑥 = 𝑦 → (𝑥 = 𝐴𝑦 ∈ {𝐴}))
1611, 15syl 17 . . . . . . . 8 (𝑥 I 𝑦 → (𝑥 = 𝐴𝑦 ∈ {𝐴}))
177, 16syl5bi 232 . . . . . . 7 (𝑥 I 𝑦 → (𝑥 ∈ {𝐴} → 𝑦 ∈ {𝐴}))
1817imp 445 . . . . . 6 ((𝑥 I 𝑦𝑥 ∈ {𝐴}) → 𝑦 ∈ {𝐴})
196, 18jca 554 . . . . 5 ((𝑥 I 𝑦𝑥 ∈ {𝐴}) → (𝑥 ∈ {𝐴} ∧ 𝑦 ∈ {𝐴}))
20 eqtr3 2642 . . . . . . . . . . . 12 ((𝑦 = 𝐴𝑥 = 𝐴) → 𝑦 = 𝑥)
218ideq 5239 . . . . . . . . . . . . 13 (𝑥 I 𝑦𝑥 = 𝑦)
22 equcom 1942 . . . . . . . . . . . . 13 (𝑥 = 𝑦𝑦 = 𝑥)
2321, 22bitri 264 . . . . . . . . . . . 12 (𝑥 I 𝑦𝑦 = 𝑥)
2420, 23sylibr 224 . . . . . . . . . . 11 ((𝑦 = 𝐴𝑥 = 𝐴) → 𝑥 I 𝑦)
2524ex 450 . . . . . . . . . 10 (𝑦 = 𝐴 → (𝑥 = 𝐴𝑥 I 𝑦))
2614, 25sylbi 207 . . . . . . . . 9 (𝑦 ∈ {𝐴} → (𝑥 = 𝐴𝑥 I 𝑦))
2726com12 32 . . . . . . . 8 (𝑥 = 𝐴 → (𝑦 ∈ {𝐴} → 𝑥 I 𝑦))
287, 27sylbi 207 . . . . . . 7 (𝑥 ∈ {𝐴} → (𝑦 ∈ {𝐴} → 𝑥 I 𝑦))
2928imp 445 . . . . . 6 ((𝑥 ∈ {𝐴} ∧ 𝑦 ∈ {𝐴}) → 𝑥 I 𝑦)
30 simpl 473 . . . . . 6 ((𝑥 ∈ {𝐴} ∧ 𝑦 ∈ {𝐴}) → 𝑥 ∈ {𝐴})
3129, 30jca 554 . . . . 5 ((𝑥 ∈ {𝐴} ∧ 𝑦 ∈ {𝐴}) → (𝑥 I 𝑦𝑥 ∈ {𝐴}))
3219, 31impbii 199 . . . 4 ((𝑥 I 𝑦𝑥 ∈ {𝐴}) ↔ (𝑥 ∈ {𝐴} ∧ 𝑦 ∈ {𝐴}))
335, 32bitri 264 . . 3 ((⟨𝑥, 𝑦⟩ ∈ I ∧ 𝑥 ∈ {𝐴}) ↔ (𝑥 ∈ {𝐴} ∧ 𝑦 ∈ {𝐴}))
348opelres 5366 . . 3 (⟨𝑥, 𝑦⟩ ∈ ( I ↾ {𝐴}) ↔ (⟨𝑥, 𝑦⟩ ∈ I ∧ 𝑥 ∈ {𝐴}))
35 opelxp 5111 . . 3 (⟨𝑥, 𝑦⟩ ∈ ({𝐴} × {𝐴}) ↔ (𝑥 ∈ {𝐴} ∧ 𝑦 ∈ {𝐴}))
3633, 34, 353bitr4i 292 . 2 (⟨𝑥, 𝑦⟩ ∈ ( I ↾ {𝐴}) ↔ ⟨𝑥, 𝑦⟩ ∈ ({𝐴} × {𝐴}))
371, 2, 36eqrelriiv 5180 1 ( I ↾ {𝐴}) = ({𝐴} × {𝐴})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wcel 1987  Vcvv 3189  {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: (None)
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