Theorem asymref 4633

Description: Two ways of saying a relation is antisymmetric and reflexive. ∪ ∪ 𝑅 is the field of a relation by relfld 4769. (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 3735	. . . . . . . . . . 11 ⊢ (x𝑅y ↔ ⟨x, y⟩ ∈ 𝑅)
2		vex 2534	. . . . . . . . . . . 12 ⊢ x ∈ V
3		vex 2534	. . . . . . . . . . . 12 ⊢ y ∈ V
4	2, 3	opeluu 4128	. . . . . . . . . . 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 3099	. . . . . . . 8 ⊢ (⟨x, y⟩ ∈ (𝑅 ∩ ◡𝑅) ↔ (⟨x, y⟩ ∈ 𝑅 ∧ ⟨x, y⟩ ∈ ◡𝑅))
11	2, 3	brcnv 4441	. . . . . . . . . 10 ⊢ (x◡𝑅y ↔ y𝑅x)
12		df-br 3735	. . . . . . . . . 10 ⊢ (x◡𝑅y ↔ ⟨x, y⟩ ∈ ◡𝑅)
13	11, 12	bitr3i 175	. . . . . . . . 9 ⊢ (y𝑅x ↔ ⟨x, y⟩ ∈ ◡𝑅)
14	1, 13	anbi12i 436	. . . . . . . 8 ⊢ ((x𝑅y ∧ y𝑅x) ↔ (⟨x, y⟩ ∈ 𝑅 ∧ ⟨x, y⟩ ∈ ◡𝑅))
15	10, 14	bitr4i 176	. . . . . . 7 ⊢ (⟨x, y⟩ ∈ (𝑅 ∩ ◡𝑅) ↔ (x𝑅y ∧ y𝑅x))
16	3	opelres 4540	. . . . . . . 8 ⊢ (⟨x, y⟩ ∈ ( I ↾ ∪ ∪ 𝑅) ↔ (⟨x, y⟩ ∈ I ∧ x ∈ ∪ ∪ 𝑅))
17		df-br 3735	. . . . . . . . . 10 ⊢ (x I y ↔ ⟨x, y⟩ ∈ I )
18	3	ideq 4411	. . . . . . . . . 10 ⊢ (x I y ↔ x = y)
19	17, 18	bitr3i 175	. . . . . . . . 9 ⊢ (⟨x, y⟩ ∈ I ↔ x = y)
20	19	anbi2ci 435	. . . . . . . 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		pm5.32 429	. . . . . 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 1335	. . . 4 ⊢ (∀y(⟨x, y⟩ ∈ (𝑅 ∩ ◡𝑅) ↔ ⟨x, y⟩ ∈ ( I ↾ ∪ ∪ 𝑅)) ↔ ∀y(x ∈ ∪ ∪ 𝑅 → ((x𝑅y ∧ y𝑅x) ↔ x = y)))
26		19.21v 1731	. . . 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 1335	. 2 ⊢ (∀x∀y(⟨x, y⟩ ∈ (𝑅 ∩ ◡𝑅) ↔ ⟨x, y⟩ ∈ ( I ↾ ∪ ∪ 𝑅)) ↔ ∀x(x ∈ ∪ ∪ 𝑅 → ∀y((x𝑅y ∧ y𝑅x) ↔ x = y)))
29		relcnv 4626	. . . 4 ⊢ Rel ◡𝑅
30		relin2 4379	. . . 4 ⊢ (Rel ◡𝑅 → Rel (𝑅 ∩ ◡𝑅))
31	29, 30	ax-mp 7	. . 3 ⊢ Rel (𝑅 ∩ ◡𝑅)
32		relres 4562	. . 3 ⊢ Rel ( I ↾ ∪ ∪ 𝑅)
33		eqrel 4352	. . 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 404	. 2 ⊢ ((𝑅 ∩ ◡𝑅) = ( I ↾ ∪ ∪ 𝑅) ↔ ∀x∀y(⟨x, y⟩ ∈ (𝑅 ∩ ◡𝑅) ↔ ⟨x, y⟩ ∈ ( I ↾ ∪ ∪ 𝑅)))
35		df-ral 2285	. 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 1224   = wceq 1226   ∈ wcel 1370  ∀wral 2280   ∩ cin 2889  ⟨cop 3349  ∪ cuni 3550   class class class wbr 3734   I cid 3995  ^◡ccnv 4267   ↾ cres 4270  Rel wrel 4273

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 617  ax-5 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-10 1373  ax-11 1374  ax-i12 1375  ax-bnd 1376  ax-4 1377  ax-14 1382  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-i5r 1406  ax-ext 2000  ax-sep 3845  ax-pow 3897  ax-pr 3914

This theorem depends on definitions:  df-bi 110  df-3an 873  df-tru 1229  df-nf 1326  df-sb 1624  df-eu 1881  df-mo 1882  df-clab 2005  df-cleq 2011  df-clel 2014  df-nfc 2145  df-ral 2285  df-rex 2286  df-v 2533  df-un 2895  df-in 2897  df-ss 2904  df-pw 3332  df-sn 3352  df-pr 3353  df-op 3355  df-uni 3551  df-br 3735  df-opab 3789  df-id 4000  df-xp 4274  df-rel 4275  df-cnv 4276  df-res 4280

This theorem is referenced by: (None)