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Theorem restidsingOLD 5617
Description: Obsolete proof of restidsing 5616 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 5584 . 2 Rel ( I ↾ {𝐴})
2 relxp 5283 . 2 Rel ({𝐴} × {𝐴})
3 df-br 4805 . . . . . 6 (𝑥 I 𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ I )
43bicomi 214 . . . . 5 (⟨𝑥, 𝑦⟩ ∈ I ↔ 𝑥 I 𝑦)
54anbi1i 733 . . . 4 ((⟨𝑥, 𝑦⟩ ∈ I ∧ 𝑥 ∈ {𝐴}) ↔ (𝑥 I 𝑦𝑥 ∈ {𝐴}))
6 simpr 479 . . . . . 6 ((𝑥 I 𝑦𝑥 ∈ {𝐴}) → 𝑥 ∈ {𝐴})
7 velsn 4337 . . . . . . . 8 (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴)
8 vex 3343 . . . . . . . . . 10 𝑦 ∈ V
9 ideqg 5429 . . . . . . . . . . 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 2780 . . . . . . . . . . 11 ((𝑥 = 𝑦𝑥 = 𝐴) → 𝑦 = 𝐴)
1312ex 449 . . . . . . . . . 10 (𝑥 = 𝑦 → (𝑥 = 𝐴𝑦 = 𝐴))
14 velsn 4337 . . . . . . . . . 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 444 . . . . . 6 ((𝑥 I 𝑦𝑥 ∈ {𝐴}) → 𝑦 ∈ {𝐴})
196, 18jca 555 . . . . 5 ((𝑥 I 𝑦𝑥 ∈ {𝐴}) → (𝑥 ∈ {𝐴} ∧ 𝑦 ∈ {𝐴}))
20 eqtr3 2781 . . . . . . . . . . . 12 ((𝑦 = 𝐴𝑥 = 𝐴) → 𝑦 = 𝑥)
218ideq 5430 . . . . . . . . . . . . 13 (𝑥 I 𝑦𝑥 = 𝑦)
22 equcom 2100 . . . . . . . . . . . . 13 (𝑥 = 𝑦𝑦 = 𝑥)
2321, 22bitri 264 . . . . . . . . . . . 12 (𝑥 I 𝑦𝑦 = 𝑥)
2420, 23sylibr 224 . . . . . . . . . . 11 ((𝑦 = 𝐴𝑥 = 𝐴) → 𝑥 I 𝑦)
2524ex 449 . . . . . . . . . 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 444 . . . . . 6 ((𝑥 ∈ {𝐴} ∧ 𝑦 ∈ {𝐴}) → 𝑥 I 𝑦)
30 simpl 474 . . . . . 6 ((𝑥 ∈ {𝐴} ∧ 𝑦 ∈ {𝐴}) → 𝑥 ∈ {𝐴})
3129, 30jca 555 . . . . 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 5559 . . 3 (⟨𝑥, 𝑦⟩ ∈ ( I ↾ {𝐴}) ↔ (⟨𝑥, 𝑦⟩ ∈ I ∧ 𝑥 ∈ {𝐴}))
35 opelxp 5303 . . 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 5371 1 ( I ↾ {𝐴}) = ({𝐴} × {𝐴})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1632  wcel 2139  Vcvv 3340  {csn 4321  cop 4327   class class class wbr 4804   I cid 5173   × cxp 5264  cres 5268
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-br 4805  df-opab 4865  df-id 5174  df-xp 5272  df-rel 5273  df-res 5278
This theorem is referenced by: (None)
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