Theorem asymref 5073

Description: Two ways of saying a relation is antisymmetric and reflexive. ∪ ∪ 𝑅 is the field of a relation by relfld 5216. (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 4048	. . . . . . . . . . 11 ⊢ (𝑥𝑅𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑅)
2		vex 2776	. . . . . . . . . . . 12 ⊢ 𝑥 ∈ V
3		vex 2776	. . . . . . . . . . . 12 ⊢ 𝑦 ∈ V
4	2, 3	opeluu 4501	. . . . . . . . . . 11 ⊢ (⟨𝑥, 𝑦⟩ ∈ 𝑅 → (𝑥 ∈ ∪ ∪ 𝑅 ∧ 𝑦 ∈ ∪ ∪ 𝑅))
5	1, 4	sylbi 121	. . . . . . . . . 10 ⊢ (𝑥𝑅𝑦 → (𝑥 ∈ ∪ ∪ 𝑅 ∧ 𝑦 ∈ ∪ ∪ 𝑅))
6	5	simpld 112	. . . . . . . . 9 ⊢ (𝑥𝑅𝑦 → 𝑥 ∈ ∪ ∪ 𝑅)
7	6	adantr 276	. . . . . . . 8 ⊢ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑥) → 𝑥 ∈ ∪ ∪ 𝑅)
8	7	pm4.71ri 392	. . . . . . 7 ⊢ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑥) ↔ (𝑥 ∈ ∪ ∪ 𝑅 ∧ (𝑥𝑅𝑦 ∧ 𝑦𝑅𝑥)))
9	8	bibi1i 228	. . . . . 6 ⊢ (((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑥) ↔ (𝑥 ∈ ∪ ∪ 𝑅 ∧ 𝑥 = 𝑦)) ↔ ((𝑥 ∈ ∪ ∪ 𝑅 ∧ (𝑥𝑅𝑦 ∧ 𝑦𝑅𝑥)) ↔ (𝑥 ∈ ∪ ∪ 𝑅 ∧ 𝑥 = 𝑦)))
10		elin 3357	. . . . . . . 8 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝑅 ∩ ◡𝑅) ↔ (⟨𝑥, 𝑦⟩ ∈ 𝑅 ∧ ⟨𝑥, 𝑦⟩ ∈ ◡𝑅))
11	2, 3	brcnv 4865	. . . . . . . . . 10 ⊢ (𝑥◡𝑅𝑦 ↔ 𝑦𝑅𝑥)
12		df-br 4048	. . . . . . . . . 10 ⊢ (𝑥◡𝑅𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ ◡𝑅)
13	11, 12	bitr3i 186	. . . . . . . . 9 ⊢ (𝑦𝑅𝑥 ↔ ⟨𝑥, 𝑦⟩ ∈ ◡𝑅)
14	1, 13	anbi12i 460	. . . . . . . 8 ⊢ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑥) ↔ (⟨𝑥, 𝑦⟩ ∈ 𝑅 ∧ ⟨𝑥, 𝑦⟩ ∈ ◡𝑅))
15	10, 14	bitr4i 187	. . . . . . 7 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝑅 ∩ ◡𝑅) ↔ (𝑥𝑅𝑦 ∧ 𝑦𝑅𝑥))
16	3	opelres 4969	. . . . . . . 8 ⊢ (⟨𝑥, 𝑦⟩ ∈ ( I ↾ ∪ ∪ 𝑅) ↔ (⟨𝑥, 𝑦⟩ ∈ I ∧ 𝑥 ∈ ∪ ∪ 𝑅))
17		df-br 4048	. . . . . . . . . 10 ⊢ (𝑥 I 𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ I )
18	3	ideq 4834	. . . . . . . . . 10 ⊢ (𝑥 I 𝑦 ↔ 𝑥 = 𝑦)
19	17, 18	bitr3i 186	. . . . . . . . 9 ⊢ (⟨𝑥, 𝑦⟩ ∈ I ↔ 𝑥 = 𝑦)
20	19	anbi2ci 459	. . . . . . . 8 ⊢ ((⟨𝑥, 𝑦⟩ ∈ I ∧ 𝑥 ∈ ∪ ∪ 𝑅) ↔ (𝑥 ∈ ∪ ∪ 𝑅 ∧ 𝑥 = 𝑦))
21	16, 20	bitri 184	. . . . . . 7 ⊢ (⟨𝑥, 𝑦⟩ ∈ ( I ↾ ∪ ∪ 𝑅) ↔ (𝑥 ∈ ∪ ∪ 𝑅 ∧ 𝑥 = 𝑦))
22	15, 21	bibi12i 229	. . . . . 6 ⊢ ((⟨𝑥, 𝑦⟩ ∈ (𝑅 ∩ ◡𝑅) ↔ ⟨𝑥, 𝑦⟩ ∈ ( I ↾ ∪ ∪ 𝑅)) ↔ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑥) ↔ (𝑥 ∈ ∪ ∪ 𝑅 ∧ 𝑥 = 𝑦)))
23		pm5.32 453	. . . . . 6 ⊢ ((𝑥 ∈ ∪ ∪ 𝑅 → ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑥) ↔ 𝑥 = 𝑦)) ↔ ((𝑥 ∈ ∪ ∪ 𝑅 ∧ (𝑥𝑅𝑦 ∧ 𝑦𝑅𝑥)) ↔ (𝑥 ∈ ∪ ∪ 𝑅 ∧ 𝑥 = 𝑦)))
24	9, 22, 23	3bitr4i 212	. . . . 5 ⊢ ((⟨𝑥, 𝑦⟩ ∈ (𝑅 ∩ ◡𝑅) ↔ ⟨𝑥, 𝑦⟩ ∈ ( I ↾ ∪ ∪ 𝑅)) ↔ (𝑥 ∈ ∪ ∪ 𝑅 → ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑥) ↔ 𝑥 = 𝑦)))
25	24	albii 1494	. . . 4 ⊢ (∀𝑦(⟨𝑥, 𝑦⟩ ∈ (𝑅 ∩ ◡𝑅) ↔ ⟨𝑥, 𝑦⟩ ∈ ( I ↾ ∪ ∪ 𝑅)) ↔ ∀𝑦(𝑥 ∈ ∪ ∪ 𝑅 → ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑥) ↔ 𝑥 = 𝑦)))
26		19.21v 1897	. . . 4 ⊢ (∀𝑦(𝑥 ∈ ∪ ∪ 𝑅 → ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑥) ↔ 𝑥 = 𝑦)) ↔ (𝑥 ∈ ∪ ∪ 𝑅 → ∀𝑦((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑥) ↔ 𝑥 = 𝑦)))
27	25, 26	bitri 184	. . 3 ⊢ (∀𝑦(⟨𝑥, 𝑦⟩ ∈ (𝑅 ∩ ◡𝑅) ↔ ⟨𝑥, 𝑦⟩ ∈ ( I ↾ ∪ ∪ 𝑅)) ↔ (𝑥 ∈ ∪ ∪ 𝑅 → ∀𝑦((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑥) ↔ 𝑥 = 𝑦)))
28	27	albii 1494	. 2 ⊢ (∀𝑥∀𝑦(⟨𝑥, 𝑦⟩ ∈ (𝑅 ∩ ◡𝑅) ↔ ⟨𝑥, 𝑦⟩ ∈ ( I ↾ ∪ ∪ 𝑅)) ↔ ∀𝑥(𝑥 ∈ ∪ ∪ 𝑅 → ∀𝑦((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑥) ↔ 𝑥 = 𝑦)))
29		relcnv 5065	. . . 4 ⊢ Rel ◡𝑅
30		relin2 4798	. . . 4 ⊢ (Rel ◡𝑅 → Rel (𝑅 ∩ ◡𝑅))
31	29, 30	ax-mp 5	. . 3 ⊢ Rel (𝑅 ∩ ◡𝑅)
32		relres 4992	. . 3 ⊢ Rel ( I ↾ ∪ ∪ 𝑅)
33		eqrel 4768	. . 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 426	. 2 ⊢ ((𝑅 ∩ ◡𝑅) = ( I ↾ ∪ ∪ 𝑅) ↔ ∀𝑥∀𝑦(⟨𝑥, 𝑦⟩ ∈ (𝑅 ∩ ◡𝑅) ↔ ⟨𝑥, 𝑦⟩ ∈ ( I ↾ ∪ ∪ 𝑅)))
35		df-ral 2490	. 2 ⊢ (∀𝑥 ∈ ∪ ∪ 𝑅∀𝑦((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑥) ↔ 𝑥 = 𝑦) ↔ ∀𝑥(𝑥 ∈ ∪ ∪ 𝑅 → ∀𝑦((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑥) ↔ 𝑥 = 𝑦)))
36	28, 34, 35	3bitr4i 212	1 ⊢ ((𝑅 ∩ ◡𝑅) = ( I ↾ ∪ ∪ 𝑅) ↔ ∀𝑥 ∈ ∪ ∪ 𝑅∀𝑦((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑥) ↔ 𝑥 = 𝑦))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105  ∀wal 1371   = wceq 1373   ∈ wcel 2177  ∀wral 2485   ∩ cin 3166  ⟨cop 3637  ∪ cuni 3852   class class class wbr 4047   I cid 4339  ^◡ccnv 4678   ↾ cres 4681  Rel wrel 4684

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-14 2180  ax-ext 2188  ax-sep 4166  ax-pow 4222  ax-pr 4257

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ral 2490  df-rex 2491  df-v 2775  df-un 3171  df-in 3173  df-ss 3180  df-pw 3619  df-sn 3640  df-pr 3641  df-op 3643  df-uni 3853  df-br 4048  df-opab 4110  df-id 4344  df-xp 4685  df-rel 4686  df-cnv 4687  df-res 4691

This theorem is referenced by: (None)