Theorem restidsingOLD 5459

Description: Obsolete proof of restidsing 5458 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.

Step	Hyp	Ref	Expression
1		relres 5426	. 2 ⊢ Rel ( I ↾ {𝐴})
2		relxp 5227	. 2 ⊢ Rel ({𝐴} × {𝐴})
3		df-br 4654	. . . . . 6 ⊢ (𝑥 I 𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ I )
4	3	bicomi 214	. . . . 5 ⊢ (⟨𝑥, 𝑦⟩ ∈ I ↔ 𝑥 I 𝑦)
5	4	anbi1i 731	. . . 4 ⊢ ((⟨𝑥, 𝑦⟩ ∈ I ∧ 𝑥 ∈ {𝐴}) ↔ (𝑥 I 𝑦 ∧ 𝑥 ∈ {𝐴}))
6		simpr 477	. . . . . 6 ⊢ ((𝑥 I 𝑦 ∧ 𝑥 ∈ {𝐴}) → 𝑥 ∈ {𝐴})
7		velsn 4193	. . . . . . . 8 ⊢ (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴)
8		vex 3203	. . . . . . . . . 10 ⊢ 𝑦 ∈ V
9		ideqg 5273	. . . . . . . . . . 11 ⊢ (𝑦 ∈ V → (𝑥 I 𝑦 ↔ 𝑥 = 𝑦))
10	9	biimpd 219	. . . . . . . . . 10 ⊢ (𝑦 ∈ V → (𝑥 I 𝑦 → 𝑥 = 𝑦))
11	8, 10	ax-mp 5	. . . . . . . . 9 ⊢ (𝑥 I 𝑦 → 𝑥 = 𝑦)
12		eqtr2 2642	. . . . . . . . . . 11 ⊢ ((𝑥 = 𝑦 ∧ 𝑥 = 𝐴) → 𝑦 = 𝐴)
13	12	ex 450	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → (𝑥 = 𝐴 → 𝑦 = 𝐴))
14		velsn 4193	. . . . . . . . . 10 ⊢ (𝑦 ∈ {𝐴} ↔ 𝑦 = 𝐴)
15	13, 14	syl6ibr 242	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (𝑥 = 𝐴 → 𝑦 ∈ {𝐴}))
16	11, 15	syl 17	. . . . . . . 8 ⊢ (𝑥 I 𝑦 → (𝑥 = 𝐴 → 𝑦 ∈ {𝐴}))
17	7, 16	syl5bi 232	. . . . . . 7 ⊢ (𝑥 I 𝑦 → (𝑥 ∈ {𝐴} → 𝑦 ∈ {𝐴}))
18	17	imp 445	. . . . . 6 ⊢ ((𝑥 I 𝑦 ∧ 𝑥 ∈ {𝐴}) → 𝑦 ∈ {𝐴})
19	6, 18	jca 554	. . . . 5 ⊢ ((𝑥 I 𝑦 ∧ 𝑥 ∈ {𝐴}) → (𝑥 ∈ {𝐴} ∧ 𝑦 ∈ {𝐴}))
20		eqtr3 2643	. . . . . . . . . . . 12 ⊢ ((𝑦 = 𝐴 ∧ 𝑥 = 𝐴) → 𝑦 = 𝑥)
21	8	ideq 5274	. . . . . . . . . . . . 13 ⊢ (𝑥 I 𝑦 ↔ 𝑥 = 𝑦)
22		equcom 1945	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑦 ↔ 𝑦 = 𝑥)
23	21, 22	bitri 264	. . . . . . . . . . . 12 ⊢ (𝑥 I 𝑦 ↔ 𝑦 = 𝑥)
24	20, 23	sylibr 224	. . . . . . . . . . 11 ⊢ ((𝑦 = 𝐴 ∧ 𝑥 = 𝐴) → 𝑥 I 𝑦)
25	24	ex 450	. . . . . . . . . 10 ⊢ (𝑦 = 𝐴 → (𝑥 = 𝐴 → 𝑥 I 𝑦))
26	14, 25	sylbi 207	. . . . . . . . 9 ⊢ (𝑦 ∈ {𝐴} → (𝑥 = 𝐴 → 𝑥 I 𝑦))
27	26	com12 32	. . . . . . . 8 ⊢ (𝑥 = 𝐴 → (𝑦 ∈ {𝐴} → 𝑥 I 𝑦))
28	7, 27	sylbi 207	. . . . . . 7 ⊢ (𝑥 ∈ {𝐴} → (𝑦 ∈ {𝐴} → 𝑥 I 𝑦))
29	28	imp 445	. . . . . 6 ⊢ ((𝑥 ∈ {𝐴} ∧ 𝑦 ∈ {𝐴}) → 𝑥 I 𝑦)
30		simpl 473	. . . . . 6 ⊢ ((𝑥 ∈ {𝐴} ∧ 𝑦 ∈ {𝐴}) → 𝑥 ∈ {𝐴})
31	29, 30	jca 554	. . . . 5 ⊢ ((𝑥 ∈ {𝐴} ∧ 𝑦 ∈ {𝐴}) → (𝑥 I 𝑦 ∧ 𝑥 ∈ {𝐴}))
32	19, 31	impbii 199	. . . 4 ⊢ ((𝑥 I 𝑦 ∧ 𝑥 ∈ {𝐴}) ↔ (𝑥 ∈ {𝐴} ∧ 𝑦 ∈ {𝐴}))
33	5, 32	bitri 264	. . 3 ⊢ ((⟨𝑥, 𝑦⟩ ∈ I ∧ 𝑥 ∈ {𝐴}) ↔ (𝑥 ∈ {𝐴} ∧ 𝑦 ∈ {𝐴}))
34	8	opelres 5401	. . 3 ⊢ (⟨𝑥, 𝑦⟩ ∈ ( I ↾ {𝐴}) ↔ (⟨𝑥, 𝑦⟩ ∈ I ∧ 𝑥 ∈ {𝐴}))
35		opelxp 5146	. . 3 ⊢ (⟨𝑥, 𝑦⟩ ∈ ({𝐴} × {𝐴}) ↔ (𝑥 ∈ {𝐴} ∧ 𝑦 ∈ {𝐴}))
36	33, 34, 35	3bitr4i 292	. 2 ⊢ (⟨𝑥, 𝑦⟩ ∈ ( I ↾ {𝐴}) ↔ ⟨𝑥, 𝑦⟩ ∈ ({𝐴} × {𝐴}))
37	1, 2, 36	eqrelriiv 5214	1 ⊢ ( I ↾ {𝐴}) = ({𝐴} × {𝐴})

Syntax hints:   → wi 4   ∧ wa 384   = wceq 1483   ∈ wcel 1990  Vcvv 3200  {csn 4177  ⟨cop 4183   class class class wbr 4653   I cid 5023   × cxp 5112   ↾ cres 5116

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-res 5126

This theorem is referenced by: (None)