Theorem asymref 5171

Description: Two ways of saying a relation is antisymmetric and reflexive. ∪ ∪ 𝑅 is the field of a relation by relfld 5314. (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 4129	. . . . . . . . . . 11 ⊢ (𝑥𝑅𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑅)
2		vex 2824	. . . . . . . . . . . 12 ⊢ 𝑥 ∈ V
3		vex 2824	. . . . . . . . . . . 12 ⊢ 𝑦 ∈ V
4	2, 3	opeluu 4594	. . . . . . . . . . 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 396	. . . . . . 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 3412	. . . . . . . 8 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝑅 ∩ ◡𝑅) ↔ (⟨𝑥, 𝑦⟩ ∈ 𝑅 ∧ ⟨𝑥, 𝑦⟩ ∈ ◡𝑅))
11	2, 3	brcnv 4961	. . . . . . . . . 10 ⊢ (𝑥◡𝑅𝑦 ↔ 𝑦𝑅𝑥)
12		df-br 4129	. . . . . . . . . 10 ⊢ (𝑥◡𝑅𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ ◡𝑅)
13	11, 12	bitr3i 186	. . . . . . . . 9 ⊢ (𝑦𝑅𝑥 ↔ ⟨𝑥, 𝑦⟩ ∈ ◡𝑅)
14	1, 13	anbi12i 464	. . . . . . . 8 ⊢ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑥) ↔ (⟨𝑥, 𝑦⟩ ∈ 𝑅 ∧ ⟨𝑥, 𝑦⟩ ∈ ◡𝑅))
15	10, 14	bitr4i 187	. . . . . . 7 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝑅 ∩ ◡𝑅) ↔ (𝑥𝑅𝑦 ∧ 𝑦𝑅𝑥))
16	3	opelres 5066	. . . . . . . 8 ⊢ (⟨𝑥, 𝑦⟩ ∈ ( I ↾ ∪ ∪ 𝑅) ↔ (⟨𝑥, 𝑦⟩ ∈ I ∧ 𝑥 ∈ ∪ ∪ 𝑅))
17		df-br 4129	. . . . . . . . . 10 ⊢ (𝑥 I 𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ I )
18	3	ideq 4930	. . . . . . . . . 10 ⊢ (𝑥 I 𝑦 ↔ 𝑥 = 𝑦)
19	17, 18	bitr3i 186	. . . . . . . . 9 ⊢ (⟨𝑥, 𝑦⟩ ∈ I ↔ 𝑥 = 𝑦)
20	19	anbi2ci 463	. . . . . . . 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		pm5.32 457	. . . . . 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 1523	. . . 4 ⊢ (∀𝑦(⟨𝑥, 𝑦⟩ ∈ (𝑅 ∩ ◡𝑅) ↔ ⟨𝑥, 𝑦⟩ ∈ ( I ↾ ∪ ∪ 𝑅)) ↔ ∀𝑦(𝑥 ∈ ∪ ∪ 𝑅 → ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑥) ↔ 𝑥 = 𝑦)))
26		19.21v 1926	. . . 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 1523	. 2 ⊢ (∀𝑥∀𝑦(⟨𝑥, 𝑦⟩ ∈ (𝑅 ∩ ◡𝑅) ↔ ⟨𝑥, 𝑦⟩ ∈ ( I ↾ ∪ ∪ 𝑅)) ↔ ∀𝑥(𝑥 ∈ ∪ ∪ 𝑅 → ∀𝑦((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑥) ↔ 𝑥 = 𝑦)))
29		relcnv 5163	. . . 4 ⊢ Rel ◡𝑅
30		relin2 4894	. . . 4 ⊢ (Rel ◡𝑅 → Rel (𝑅 ∩ ◡𝑅))
31	29, 30	ax-mp 5	. . 3 ⊢ Rel (𝑅 ∩ ◡𝑅)
32		relres 5089	. . 3 ⊢ Rel ( I ↾ ∪ ∪ 𝑅)
33		eqrel 4862	. . 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 430	. 2 ⊢ ((𝑅 ∩ ◡𝑅) = ( I ↾ ∪ ∪ 𝑅) ↔ ∀𝑥∀𝑦(⟨𝑥, 𝑦⟩ ∈ (𝑅 ∩ ◡𝑅) ↔ ⟨𝑥, 𝑦⟩ ∈ ( I ↾ ∪ ∪ 𝑅)))
35		df-ral 2533	. 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 1400   = wceq 1402   ∈ wcel 2209  ∀wral 2528   ∩ cin 3219  ⟨cop 3711  ∪ cuni 3933   class class class wbr 4128   I cid 4431  ^◡ccnv 4771   ↾ cres 4774  Rel wrel 4777

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4247  ax-pow 4309  ax-pr 4344

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3714  df-pr 3715  df-op 3717  df-uni 3934  df-br 4129  df-opab 4191  df-id 4436  df-xp 4778  df-rel 4779  df-cnv 4780  df-res 4784

This theorem is referenced by: (None)