Theorem asymref 4653

Description: Two ways of saying a relation is antisymmetric and reflexive. ∪ ∪ 𝑅 is the field of a relation by relfld 4789. (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 3756	. . . . . . . . . . 11 ⊢ (x𝑅y ↔ ⟨x, y⟩ ∈ 𝑅)
2		vex 2554	. . . . . . . . . . . 12 ⊢ x ∈ V
3		vex 2554	. . . . . . . . . . . 12 ⊢ y ∈ V
4	2, 3	opeluu 4148	. . . . . . . . . . 11 ⊢ (⟨x, y⟩ ∈ 𝑅 → (x ∈ ∪ ∪ 𝑅 ∧ y ∈ ∪ ∪ 𝑅))
5	1, 4	sylbi 114	. . . . . . . . . 10 ⊢ (x𝑅y → (x ∈ ∪ ∪ 𝑅 ∧ y ∈ ∪ ∪ 𝑅))
6	5	simpld 105	. . . . . . . . 9 ⊢ (x𝑅y → x ∈ ∪ ∪ 𝑅)
7	6	adantr 261	. . . . . . . 8 ⊢ ((x𝑅y ∧ y𝑅x) → x ∈ ∪ ∪ 𝑅)
8	7	pm4.71ri 372	. . . . . . 7 ⊢ ((x𝑅y ∧ y𝑅x) ↔ (x ∈ ∪ ∪ 𝑅 ∧ (x𝑅y ∧ y𝑅x)))
9	8	bibi1i 217	. . . . . 6 ⊢ (((x𝑅y ∧ y𝑅x) ↔ (x ∈ ∪ ∪ 𝑅 ∧ x = y)) ↔ ((x ∈ ∪ ∪ 𝑅 ∧ (x𝑅y ∧ y𝑅x)) ↔ (x ∈ ∪ ∪ 𝑅 ∧ x = y)))
10		elin 3120	. . . . . . . 8 ⊢ (⟨x, y⟩ ∈ (𝑅 ∩ ◡𝑅) ↔ (⟨x, y⟩ ∈ 𝑅 ∧ ⟨x, y⟩ ∈ ◡𝑅))
11	2, 3	brcnv 4461	. . . . . . . . . 10 ⊢ (x◡𝑅y ↔ y𝑅x)
12		df-br 3756	. . . . . . . . . 10 ⊢ (x◡𝑅y ↔ ⟨x, y⟩ ∈ ◡𝑅)
13	11, 12	bitr3i 175	. . . . . . . . 9 ⊢ (y𝑅x ↔ ⟨x, y⟩ ∈ ◡𝑅)
14	1, 13	anbi12i 433	. . . . . . . 8 ⊢ ((x𝑅y ∧ y𝑅x) ↔ (⟨x, y⟩ ∈ 𝑅 ∧ ⟨x, y⟩ ∈ ◡𝑅))
15	10, 14	bitr4i 176	. . . . . . 7 ⊢ (⟨x, y⟩ ∈ (𝑅 ∩ ◡𝑅) ↔ (x𝑅y ∧ y𝑅x))
16	3	opelres 4560	. . . . . . . 8 ⊢ (⟨x, y⟩ ∈ ( I ↾ ∪ ∪ 𝑅) ↔ (⟨x, y⟩ ∈ I ∧ x ∈ ∪ ∪ 𝑅))
17		df-br 3756	. . . . . . . . . 10 ⊢ (x I y ↔ ⟨x, y⟩ ∈ I )
18	3	ideq 4431	. . . . . . . . . 10 ⊢ (x I y ↔ x = y)
19	17, 18	bitr3i 175	. . . . . . . . 9 ⊢ (⟨x, y⟩ ∈ I ↔ x = y)
20	19	anbi2ci 432	. . . . . . . 8 ⊢ ((⟨x, y⟩ ∈ I ∧ x ∈ ∪ ∪ 𝑅) ↔ (x ∈ ∪ ∪ 𝑅 ∧ x = y))
21	16, 20	bitri 173	. . . . . . 7 ⊢ (⟨x, y⟩ ∈ ( I ↾ ∪ ∪ 𝑅) ↔ (x ∈ ∪ ∪ 𝑅 ∧ x = y))
22	15, 21	bibi12i 218	. . . . . 6 ⊢ ((⟨x, y⟩ ∈ (𝑅 ∩ ◡𝑅) ↔ ⟨x, y⟩ ∈ ( I ↾ ∪ ∪ 𝑅)) ↔ ((x𝑅y ∧ y𝑅x) ↔ (x ∈ ∪ ∪ 𝑅 ∧ x = y)))
23		pm5.32 426	. . . . . 6 ⊢ ((x ∈ ∪ ∪ 𝑅 → ((x𝑅y ∧ y𝑅x) ↔ x = y)) ↔ ((x ∈ ∪ ∪ 𝑅 ∧ (x𝑅y ∧ y𝑅x)) ↔ (x ∈ ∪ ∪ 𝑅 ∧ x = y)))
24	9, 22, 23	3bitr4i 201	. . . . 5 ⊢ ((⟨x, y⟩ ∈ (𝑅 ∩ ◡𝑅) ↔ ⟨x, y⟩ ∈ ( I ↾ ∪ ∪ 𝑅)) ↔ (x ∈ ∪ ∪ 𝑅 → ((x𝑅y ∧ y𝑅x) ↔ x = y)))
25	24	albii 1356	. . . 4 ⊢ (∀y(⟨x, y⟩ ∈ (𝑅 ∩ ◡𝑅) ↔ ⟨x, y⟩ ∈ ( I ↾ ∪ ∪ 𝑅)) ↔ ∀y(x ∈ ∪ ∪ 𝑅 → ((x𝑅y ∧ y𝑅x) ↔ x = y)))
26		19.21v 1750	. . . 4 ⊢ (∀y(x ∈ ∪ ∪ 𝑅 → ((x𝑅y ∧ y𝑅x) ↔ x = y)) ↔ (x ∈ ∪ ∪ 𝑅 → ∀y((x𝑅y ∧ y𝑅x) ↔ x = y)))
27	25, 26	bitri 173	. . 3 ⊢ (∀y(⟨x, y⟩ ∈ (𝑅 ∩ ◡𝑅) ↔ ⟨x, y⟩ ∈ ( I ↾ ∪ ∪ 𝑅)) ↔ (x ∈ ∪ ∪ 𝑅 → ∀y((x𝑅y ∧ y𝑅x) ↔ x = y)))
28	27	albii 1356	. 2 ⊢ (∀x∀y(⟨x, y⟩ ∈ (𝑅 ∩ ◡𝑅) ↔ ⟨x, y⟩ ∈ ( I ↾ ∪ ∪ 𝑅)) ↔ ∀x(x ∈ ∪ ∪ 𝑅 → ∀y((x𝑅y ∧ y𝑅x) ↔ x = y)))
29		relcnv 4646	. . . 4 ⊢ Rel ◡𝑅
30		relin2 4399	. . . 4 ⊢ (Rel ◡𝑅 → Rel (𝑅 ∩ ◡𝑅))
31	29, 30	ax-mp 7	. . 3 ⊢ Rel (𝑅 ∩ ◡𝑅)
32		relres 4582	. . 3 ⊢ Rel ( I ↾ ∪ ∪ 𝑅)
33		eqrel 4372	. . 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 402	. 2 ⊢ ((𝑅 ∩ ◡𝑅) = ( I ↾ ∪ ∪ 𝑅) ↔ ∀x∀y(⟨x, y⟩ ∈ (𝑅 ∩ ◡𝑅) ↔ ⟨x, y⟩ ∈ ( I ↾ ∪ ∪ 𝑅)))
35		df-ral 2305	. 2 ⊢ (∀x ∈ ∪ ∪ 𝑅∀y((x𝑅y ∧ y𝑅x) ↔ x = y) ↔ ∀x(x ∈ ∪ ∪ 𝑅 → ∀y((x𝑅y ∧ y𝑅x) ↔ x = y)))
36	28, 34, 35	3bitr4i 201	1 ⊢ ((𝑅 ∩ ◡𝑅) = ( I ↾ ∪ ∪ 𝑅) ↔ ∀x ∈ ∪ ∪ 𝑅∀y((x𝑅y ∧ y𝑅x) ↔ x = y))

Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98  ∀wal 1240   = wceq 1242   ∈ wcel 1390  ∀wral 2300   ∩ cin 2910  ⟨cop 3370  ∪ cuni 3571   class class class wbr 3755   I cid 4016  ^◡ccnv 4287   ↾ cres 4290  Rel wrel 4293

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935

This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-opab 3810  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-res 4300

This theorem is referenced by: (None)