Theorem asymref 6116

Description: Two ways of saying a relation is antisymmetric and reflexive. ∪ ∪ 𝑅 is the field of a relation by relfld 6273. (Contributed by NM, 6-May-2008.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)

Assertion

Ref	Expression
asymref	⊢ ((𝑅 ∩ ◡𝑅) = ( I ↾ ∪ ∪ 𝑅) ↔ ∀𝑥 ∈ ∪ ∪ 𝑅∀𝑦((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑥) ↔ 𝑥 = 𝑦))

Distinct variable group:   𝑥,𝑦,𝑅

Proof of Theorem asymref

Step	Hyp	Ref	Expression
1		df-br 5143	. . . . . . . . . . 11 ⊢ (𝑥𝑅𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑅)
2		vex 3473	. . . . . . . . . . . 12 ⊢ 𝑥 ∈ V
3		vex 3473	. . . . . . . . . . . 12 ⊢ 𝑦 ∈ V
4	2, 3	opeluu 5466	. . . . . . . . . . 11 ⊢ (⟨𝑥, 𝑦⟩ ∈ 𝑅 → (𝑥 ∈ ∪ ∪ 𝑅 ∧ 𝑦 ∈ ∪ ∪ 𝑅))
5	1, 4	sylbi 216	. . . . . . . . . 10 ⊢ (𝑥𝑅𝑦 → (𝑥 ∈ ∪ ∪ 𝑅 ∧ 𝑦 ∈ ∪ ∪ 𝑅))
6	5	simpld 494	. . . . . . . . 9 ⊢ (𝑥𝑅𝑦 → 𝑥 ∈ ∪ ∪ 𝑅)
7	6	adantr 480	. . . . . . . 8 ⊢ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑥) → 𝑥 ∈ ∪ ∪ 𝑅)
8	7	pm4.71ri 560	. . . . . . 7 ⊢ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑥) ↔ (𝑥 ∈ ∪ ∪ 𝑅 ∧ (𝑥𝑅𝑦 ∧ 𝑦𝑅𝑥)))
9	8	bibi1i 338	. . . . . 6 ⊢ (((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑥) ↔ (𝑥 ∈ ∪ ∪ 𝑅 ∧ 𝑥 = 𝑦)) ↔ ((𝑥 ∈ ∪ ∪ 𝑅 ∧ (𝑥𝑅𝑦 ∧ 𝑦𝑅𝑥)) ↔ (𝑥 ∈ ∪ ∪ 𝑅 ∧ 𝑥 = 𝑦)))
10		elin 3960	. . . . . . . 8 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝑅 ∩ ◡𝑅) ↔ (⟨𝑥, 𝑦⟩ ∈ 𝑅 ∧ ⟨𝑥, 𝑦⟩ ∈ ◡𝑅))
11	2, 3	brcnv 5879	. . . . . . . . . 10 ⊢ (𝑥◡𝑅𝑦 ↔ 𝑦𝑅𝑥)
12		df-br 5143	. . . . . . . . . 10 ⊢ (𝑥◡𝑅𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ ◡𝑅)
13	11, 12	bitr3i 277	. . . . . . . . 9 ⊢ (𝑦𝑅𝑥 ↔ ⟨𝑥, 𝑦⟩ ∈ ◡𝑅)
14	1, 13	anbi12i 626	. . . . . . . 8 ⊢ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑥) ↔ (⟨𝑥, 𝑦⟩ ∈ 𝑅 ∧ ⟨𝑥, 𝑦⟩ ∈ ◡𝑅))
15	10, 14	bitr4i 278	. . . . . . 7 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝑅 ∩ ◡𝑅) ↔ (𝑥𝑅𝑦 ∧ 𝑦𝑅𝑥))
16	3	opelresi 5987	. . . . . . . 8 ⊢ (⟨𝑥, 𝑦⟩ ∈ ( I ↾ ∪ ∪ 𝑅) ↔ (𝑥 ∈ ∪ ∪ 𝑅 ∧ ⟨𝑥, 𝑦⟩ ∈ I ))
17		df-br 5143	. . . . . . . . . 10 ⊢ (𝑥 I 𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ I )
18	3	ideq 5849	. . . . . . . . . 10 ⊢ (𝑥 I 𝑦 ↔ 𝑥 = 𝑦)
19	17, 18	bitr3i 277	. . . . . . . . 9 ⊢ (⟨𝑥, 𝑦⟩ ∈ I ↔ 𝑥 = 𝑦)
20	19	anbi2i 622	. . . . . . . 8 ⊢ ((𝑥 ∈ ∪ ∪ 𝑅 ∧ ⟨𝑥, 𝑦⟩ ∈ I ) ↔ (𝑥 ∈ ∪ ∪ 𝑅 ∧ 𝑥 = 𝑦))
21	16, 20	bitri 275	. . . . . . 7 ⊢ (⟨𝑥, 𝑦⟩ ∈ ( I ↾ ∪ ∪ 𝑅) ↔ (𝑥 ∈ ∪ ∪ 𝑅 ∧ 𝑥 = 𝑦))
22	15, 21	bibi12i 339	. . . . . 6 ⊢ ((⟨𝑥, 𝑦⟩ ∈ (𝑅 ∩ ◡𝑅) ↔ ⟨𝑥, 𝑦⟩ ∈ ( I ↾ ∪ ∪ 𝑅)) ↔ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑥) ↔ (𝑥 ∈ ∪ ∪ 𝑅 ∧ 𝑥 = 𝑦)))
23		pm5.32 573	. . . . . 6 ⊢ ((𝑥 ∈ ∪ ∪ 𝑅 → ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑥) ↔ 𝑥 = 𝑦)) ↔ ((𝑥 ∈ ∪ ∪ 𝑅 ∧ (𝑥𝑅𝑦 ∧ 𝑦𝑅𝑥)) ↔ (𝑥 ∈ ∪ ∪ 𝑅 ∧ 𝑥 = 𝑦)))
24	9, 22, 23	3bitr4i 303	. . . . 5 ⊢ ((⟨𝑥, 𝑦⟩ ∈ (𝑅 ∩ ◡𝑅) ↔ ⟨𝑥, 𝑦⟩ ∈ ( I ↾ ∪ ∪ 𝑅)) ↔ (𝑥 ∈ ∪ ∪ 𝑅 → ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑥) ↔ 𝑥 = 𝑦)))
25	24	albii 1814	. . . 4 ⊢ (∀𝑦(⟨𝑥, 𝑦⟩ ∈ (𝑅 ∩ ◡𝑅) ↔ ⟨𝑥, 𝑦⟩ ∈ ( I ↾ ∪ ∪ 𝑅)) ↔ ∀𝑦(𝑥 ∈ ∪ ∪ 𝑅 → ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑥) ↔ 𝑥 = 𝑦)))
26		19.21v 1935	. . . 4 ⊢ (∀𝑦(𝑥 ∈ ∪ ∪ 𝑅 → ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑥) ↔ 𝑥 = 𝑦)) ↔ (𝑥 ∈ ∪ ∪ 𝑅 → ∀𝑦((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑥) ↔ 𝑥 = 𝑦)))
27	25, 26	bitri 275	. . 3 ⊢ (∀𝑦(⟨𝑥, 𝑦⟩ ∈ (𝑅 ∩ ◡𝑅) ↔ ⟨𝑥, 𝑦⟩ ∈ ( I ↾ ∪ ∪ 𝑅)) ↔ (𝑥 ∈ ∪ ∪ 𝑅 → ∀𝑦((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑥) ↔ 𝑥 = 𝑦)))
28	27	albii 1814	. 2 ⊢ (∀𝑥∀𝑦(⟨𝑥, 𝑦⟩ ∈ (𝑅 ∩ ◡𝑅) ↔ ⟨𝑥, 𝑦⟩ ∈ ( I ↾ ∪ ∪ 𝑅)) ↔ ∀𝑥(𝑥 ∈ ∪ ∪ 𝑅 → ∀𝑦((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑥) ↔ 𝑥 = 𝑦)))
29		relcnv 6102	. . . 4 ⊢ Rel ◡𝑅
30		relin2 5809	. . . 4 ⊢ (Rel ◡𝑅 → Rel (𝑅 ∩ ◡𝑅))
31	29, 30	ax-mp 5	. . 3 ⊢ Rel (𝑅 ∩ ◡𝑅)
32		relres 6008	. . 3 ⊢ Rel ( I ↾ ∪ ∪ 𝑅)
33		eqrel 5780	. . 3 ⊢ ((Rel (𝑅 ∩ ◡𝑅) ∧ Rel ( I ↾ ∪ ∪ 𝑅)) → ((𝑅 ∩ ◡𝑅) = ( I ↾ ∪ ∪ 𝑅) ↔ ∀𝑥∀𝑦(⟨𝑥, 𝑦⟩ ∈ (𝑅 ∩ ◡𝑅) ↔ ⟨𝑥, 𝑦⟩ ∈ ( I ↾ ∪ ∪ 𝑅))))
34	31, 32, 33	mp2an 691	. 2 ⊢ ((𝑅 ∩ ◡𝑅) = ( I ↾ ∪ ∪ 𝑅) ↔ ∀𝑥∀𝑦(⟨𝑥, 𝑦⟩ ∈ (𝑅 ∩ ◡𝑅) ↔ ⟨𝑥, 𝑦⟩ ∈ ( I ↾ ∪ ∪ 𝑅)))
35		df-ral 3057	. 2 ⊢ (∀𝑥 ∈ ∪ ∪ 𝑅∀𝑦((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑥) ↔ 𝑥 = 𝑦) ↔ ∀𝑥(𝑥 ∈ ∪ ∪ 𝑅 → ∀𝑦((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑥) ↔ 𝑥 = 𝑦)))
36	28, 34, 35	3bitr4i 303	1 ⊢ ((𝑅 ∩ ◡𝑅) = ( I ↾ ∪ ∪ 𝑅) ↔ ∀𝑥 ∈ ∪ ∪ 𝑅∀𝑦((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑥) ↔ 𝑥 = 𝑦))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395  ∀wal 1532   = wceq 1534   ∈ wcel 2099  ∀wral 3056   ∩ cin 3943  ⟨cop 4630  ∪ cuni 4903   class class class wbr 5142   I cid 5569  ^◡ccnv 5671   ↾ cres 5674  Rel wrel 5677

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2698  ax-sep 5293  ax-nul 5300  ax-pr 5423

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2705  df-cleq 2719  df-clel 2805  df-ral 3057  df-rex 3066  df-rab 3428  df-v 3471  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5143  df-opab 5205  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-res 5684

This theorem is referenced by:  asymref2  6117  letsr  18576