Theorem asymref 6093

Description: Two ways of saying a relation is antisymmetric and reflexive. ∪ ∪ 𝑅 is the field of a relation by relfld 6251. (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 5095	. . . . . . . . . . 11 ⊢ (𝑥𝑅𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑅)
2		vex 3452	. . . . . . . . . . . 12 ⊢ 𝑥 ∈ V
3		vex 3452	. . . . . . . . . . . 12 ⊢ 𝑦 ∈ V
4	2, 3	opeluu 5432	. . . . . . . . . . 11 ⊢ (⟨𝑥, 𝑦⟩ ∈ 𝑅 → (𝑥 ∈ ∪ ∪ 𝑅 ∧ 𝑦 ∈ ∪ ∪ 𝑅))
5	1, 4	sylbi 219	. . . . . . . . . 10 ⊢ (𝑥𝑅𝑦 → (𝑥 ∈ ∪ ∪ 𝑅 ∧ 𝑦 ∈ ∪ ∪ 𝑅))
6	5	simpld 497	. . . . . . . . 9 ⊢ (𝑥𝑅𝑦 → 𝑥 ∈ ∪ ∪ 𝑅)
7	6	adantr 483	. . . . . . . 8 ⊢ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑥) → 𝑥 ∈ ∪ ∪ 𝑅)
8	7	pm4.71ri 567	. . . . . . 7 ⊢ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑥) ↔ (𝑥 ∈ ∪ ∪ 𝑅 ∧ (𝑥𝑅𝑦 ∧ 𝑦𝑅𝑥)))
9	8	bibi1i 340	. . . . . 6 ⊢ (((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑥) ↔ (𝑥 ∈ ∪ ∪ 𝑅 ∧ 𝑥 = 𝑦)) ↔ ((𝑥 ∈ ∪ ∪ 𝑅 ∧ (𝑥𝑅𝑦 ∧ 𝑦𝑅𝑥)) ↔ (𝑥 ∈ ∪ ∪ 𝑅 ∧ 𝑥 = 𝑦)))
10		elin 3915	. . . . . . . 8 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝑅 ∩ ◡𝑅) ↔ (⟨𝑥, 𝑦⟩ ∈ 𝑅 ∧ ⟨𝑥, 𝑦⟩ ∈ ◡𝑅))
11	2, 3	brcnv 5847	. . . . . . . . . 10 ⊢ (𝑥◡𝑅𝑦 ↔ 𝑦𝑅𝑥)
12		df-br 5095	. . . . . . . . . 10 ⊢ (𝑥◡𝑅𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ ◡𝑅)
13	11, 12	bitr3i 279	. . . . . . . . 9 ⊢ (𝑦𝑅𝑥 ↔ ⟨𝑥, 𝑦⟩ ∈ ◡𝑅)
14	1, 13	anbi12i 636	. . . . . . . 8 ⊢ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑥) ↔ (⟨𝑥, 𝑦⟩ ∈ 𝑅 ∧ ⟨𝑥, 𝑦⟩ ∈ ◡𝑅))
15	10, 14	bitr4i 280	. . . . . . 7 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝑅 ∩ ◡𝑅) ↔ (𝑥𝑅𝑦 ∧ 𝑦𝑅𝑥))
16	3	opelresi 5966	. . . . . . . 8 ⊢ (⟨𝑥, 𝑦⟩ ∈ ( I ↾ ∪ ∪ 𝑅) ↔ (𝑥 ∈ ∪ ∪ 𝑅 ∧ ⟨𝑥, 𝑦⟩ ∈ I ))
17		df-br 5095	. . . . . . . . . 10 ⊢ (𝑥 I 𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ I )
18	3	ideq 5817	. . . . . . . . . 10 ⊢ (𝑥 I 𝑦 ↔ 𝑥 = 𝑦)
19	17, 18	bitr3i 279	. . . . . . . . 9 ⊢ (⟨𝑥, 𝑦⟩ ∈ I ↔ 𝑥 = 𝑦)
20	19	anbi2i 631	. . . . . . . 8 ⊢ ((𝑥 ∈ ∪ ∪ 𝑅 ∧ ⟨𝑥, 𝑦⟩ ∈ I ) ↔ (𝑥 ∈ ∪ ∪ 𝑅 ∧ 𝑥 = 𝑦))
21	16, 20	bitri 277	. . . . . . 7 ⊢ (⟨𝑥, 𝑦⟩ ∈ ( I ↾ ∪ ∪ 𝑅) ↔ (𝑥 ∈ ∪ ∪ 𝑅 ∧ 𝑥 = 𝑦))
22	15, 21	bibi12i 341	. . . . . 6 ⊢ ((⟨𝑥, 𝑦⟩ ∈ (𝑅 ∩ ◡𝑅) ↔ ⟨𝑥, 𝑦⟩ ∈ ( I ↾ ∪ ∪ 𝑅)) ↔ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑥) ↔ (𝑥 ∈ ∪ ∪ 𝑅 ∧ 𝑥 = 𝑦)))
23		pm5.32 580	. . . . . 6 ⊢ ((𝑥 ∈ ∪ ∪ 𝑅 → ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑥) ↔ 𝑥 = 𝑦)) ↔ ((𝑥 ∈ ∪ ∪ 𝑅 ∧ (𝑥𝑅𝑦 ∧ 𝑦𝑅𝑥)) ↔ (𝑥 ∈ ∪ ∪ 𝑅 ∧ 𝑥 = 𝑦)))
24	9, 22, 23	3bitr4i 305	. . . . 5 ⊢ ((⟨𝑥, 𝑦⟩ ∈ (𝑅 ∩ ◡𝑅) ↔ ⟨𝑥, 𝑦⟩ ∈ ( I ↾ ∪ ∪ 𝑅)) ↔ (𝑥 ∈ ∪ ∪ 𝑅 → ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑥) ↔ 𝑥 = 𝑦)))
25	24	albii 1833	. . . 4 ⊢ (∀𝑦(⟨𝑥, 𝑦⟩ ∈ (𝑅 ∩ ◡𝑅) ↔ ⟨𝑥, 𝑦⟩ ∈ ( I ↾ ∪ ∪ 𝑅)) ↔ ∀𝑦(𝑥 ∈ ∪ ∪ 𝑅 → ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑥) ↔ 𝑥 = 𝑦)))
26		19.21v 1953	. . . 4 ⊢ (∀𝑦(𝑥 ∈ ∪ ∪ 𝑅 → ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑥) ↔ 𝑥 = 𝑦)) ↔ (𝑥 ∈ ∪ ∪ 𝑅 → ∀𝑦((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑥) ↔ 𝑥 = 𝑦)))
27	25, 26	bitri 277	. . 3 ⊢ (∀𝑦(⟨𝑥, 𝑦⟩ ∈ (𝑅 ∩ ◡𝑅) ↔ ⟨𝑥, 𝑦⟩ ∈ ( I ↾ ∪ ∪ 𝑅)) ↔ (𝑥 ∈ ∪ ∪ 𝑅 → ∀𝑦((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑥) ↔ 𝑥 = 𝑦)))
28	27	albii 1833	. 2 ⊢ (∀𝑥∀𝑦(⟨𝑥, 𝑦⟩ ∈ (𝑅 ∩ ◡𝑅) ↔ ⟨𝑥, 𝑦⟩ ∈ ( I ↾ ∪ ∪ 𝑅)) ↔ ∀𝑥(𝑥 ∈ ∪ ∪ 𝑅 → ∀𝑦((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑥) ↔ 𝑥 = 𝑦)))
29		relcnv 6083	. . . 4 ⊢ Rel ◡𝑅
30		relin2 5779	. . . 4 ⊢ (Rel ◡𝑅 → Rel (𝑅 ∩ ◡𝑅))
31	29, 30	ax-mp 5	. . 3 ⊢ Rel (𝑅 ∩ ◡𝑅)
32		relres 5984	. . 3 ⊢ Rel ( I ↾ ∪ ∪ 𝑅)
33		eqrel 5749	. . 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 700	. 2 ⊢ ((𝑅 ∩ ◡𝑅) = ( I ↾ ∪ ∪ 𝑅) ↔ ∀𝑥∀𝑦(⟨𝑥, 𝑦⟩ ∈ (𝑅 ∩ ◡𝑅) ↔ ⟨𝑥, 𝑦⟩ ∈ ( I ↾ ∪ ∪ 𝑅)))
35		df-ral 3071	. 2 ⊢ (∀𝑥 ∈ ∪ ∪ 𝑅∀𝑦((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑥) ↔ 𝑥 = 𝑦) ↔ ∀𝑥(𝑥 ∈ ∪ ∪ 𝑅 → ∀𝑦((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑥) ↔ 𝑥 = 𝑦)))
36	28, 34, 35	3bitr4i 305	1 ⊢ ((𝑅 ∩ ◡𝑅) = ( I ↾ ∪ ∪ 𝑅) ↔ ∀𝑥 ∈ ∪ ∪ 𝑅∀𝑦((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑥) ↔ 𝑥 = 𝑦))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398  ∀wal 1552   = wceq 1554   ∈ wcel 2136  ∀wral 3070   ∩ cin 3898  ⟨cop 4582  ∪ cuni 4859   class class class wbr 5094   I cid 5534  ^◡ccnv 5639   ↾ cres 5642  Rel wrel 5645

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1809  ax-4 1823  ax-5 1924  ax-6 1981  ax-7 2022  ax-8 2138  ax-9 2146  ax-ext 2728  ax-sep 5240  ax-pr 5384

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 857  df-3an 1097  df-tru 1557  df-fal 1567  df-ex 1794  df-sb 2085  df-clab 2735  df-cleq 2748  df-clel 2831  df-ral 3071  df-rex 3081  df-rab 3409  df-v 3450  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4281  df-if 4475  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-br 5095  df-opab 5157  df-id 5535  df-xp 5646  df-rel 5647  df-cnv 5648  df-res 5652

This theorem is referenced by:  asymref2  6094  letsr  18601