Theorem asymref2 6061

Description: Two ways of saying a relation is antisymmetric and reflexive. (Contributed by NM, 6-May-2008.) (Proof shortened by Mario Carneiro, 4-Dec-2016.)

Assertion

Ref	Expression
asymref2	⊢ ((𝑅 ∩ ◡𝑅) = ( I ↾ ∪ ∪ 𝑅) ↔ (∀𝑥 ∈ ∪ ∪ 𝑅𝑥𝑅𝑥 ∧ ∀𝑥∀𝑦((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑥) → 𝑥 = 𝑦)))

Distinct variable group:   𝑥,𝑦,𝑅

Proof of Theorem asymref2

Step	Hyp	Ref	Expression
1		asymref 6060	. 2 ⊢ ((𝑅 ∩ ◡𝑅) = ( I ↾ ∪ ∪ 𝑅) ↔ ∀𝑥 ∈ ∪ ∪ 𝑅∀𝑦((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑥) ↔ 𝑥 = 𝑦))
2		albiim 1890	. . 3 ⊢ (∀𝑦((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑥) ↔ 𝑥 = 𝑦) ↔ (∀𝑦((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑥) → 𝑥 = 𝑦) ∧ ∀𝑦(𝑥 = 𝑦 → (𝑥𝑅𝑦 ∧ 𝑦𝑅𝑥))))
3	2	ralbii 3076	. 2 ⊢ (∀𝑥 ∈ ∪ ∪ 𝑅∀𝑦((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑥) ↔ 𝑥 = 𝑦) ↔ ∀𝑥 ∈ ∪ ∪ 𝑅(∀𝑦((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑥) → 𝑥 = 𝑦) ∧ ∀𝑦(𝑥 = 𝑦 → (𝑥𝑅𝑦 ∧ 𝑦𝑅𝑥))))
4		r19.26 3090	. . 3 ⊢ (∀𝑥 ∈ ∪ ∪ 𝑅(∀𝑦((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑥) → 𝑥 = 𝑦) ∧ ∀𝑦(𝑥 = 𝑦 → (𝑥𝑅𝑦 ∧ 𝑦𝑅𝑥))) ↔ (∀𝑥 ∈ ∪ ∪ 𝑅∀𝑦((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑥) → 𝑥 = 𝑦) ∧ ∀𝑥 ∈ ∪ ∪ 𝑅∀𝑦(𝑥 = 𝑦 → (𝑥𝑅𝑦 ∧ 𝑦𝑅𝑥))))
5		ancom 460	. . 3 ⊢ ((∀𝑥 ∈ ∪ ∪ 𝑅∀𝑦((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑥) → 𝑥 = 𝑦) ∧ ∀𝑥 ∈ ∪ ∪ 𝑅∀𝑦(𝑥 = 𝑦 → (𝑥𝑅𝑦 ∧ 𝑦𝑅𝑥))) ↔ (∀𝑥 ∈ ∪ ∪ 𝑅∀𝑦(𝑥 = 𝑦 → (𝑥𝑅𝑦 ∧ 𝑦𝑅𝑥)) ∧ ∀𝑥 ∈ ∪ ∪ 𝑅∀𝑦((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑥) → 𝑥 = 𝑦)))
6		equcom 2019	. . . . . . . 8 ⊢ (𝑥 = 𝑦 ↔ 𝑦 = 𝑥)
7	6	imbi1i 349	. . . . . . 7 ⊢ ((𝑥 = 𝑦 → (𝑥𝑅𝑦 ∧ 𝑦𝑅𝑥)) ↔ (𝑦 = 𝑥 → (𝑥𝑅𝑦 ∧ 𝑦𝑅𝑥)))
8	7	albii 1820	. . . . . 6 ⊢ (∀𝑦(𝑥 = 𝑦 → (𝑥𝑅𝑦 ∧ 𝑦𝑅𝑥)) ↔ ∀𝑦(𝑦 = 𝑥 → (𝑥𝑅𝑦 ∧ 𝑦𝑅𝑥)))
9		breq2 5093	. . . . . . . . 9 ⊢ (𝑦 = 𝑥 → (𝑥𝑅𝑦 ↔ 𝑥𝑅𝑥))
10		breq1 5092	. . . . . . . . 9 ⊢ (𝑦 = 𝑥 → (𝑦𝑅𝑥 ↔ 𝑥𝑅𝑥))
11	9, 10	anbi12d 632	. . . . . . . 8 ⊢ (𝑦 = 𝑥 → ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑥) ↔ (𝑥𝑅𝑥 ∧ 𝑥𝑅𝑥)))
12		anidm 564	. . . . . . . 8 ⊢ ((𝑥𝑅𝑥 ∧ 𝑥𝑅𝑥) ↔ 𝑥𝑅𝑥)
13	11, 12	bitrdi 287	. . . . . . 7 ⊢ (𝑦 = 𝑥 → ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑥) ↔ 𝑥𝑅𝑥))
14	13	equsalvw 2005	. . . . . 6 ⊢ (∀𝑦(𝑦 = 𝑥 → (𝑥𝑅𝑦 ∧ 𝑦𝑅𝑥)) ↔ 𝑥𝑅𝑥)
15	8, 14	bitri 275	. . . . 5 ⊢ (∀𝑦(𝑥 = 𝑦 → (𝑥𝑅𝑦 ∧ 𝑦𝑅𝑥)) ↔ 𝑥𝑅𝑥)
16	15	ralbii 3076	. . . 4 ⊢ (∀𝑥 ∈ ∪ ∪ 𝑅∀𝑦(𝑥 = 𝑦 → (𝑥𝑅𝑦 ∧ 𝑦𝑅𝑥)) ↔ ∀𝑥 ∈ ∪ ∪ 𝑅𝑥𝑅𝑥)
17		df-ral 3046	. . . . 5 ⊢ (∀𝑥 ∈ ∪ ∪ 𝑅∀𝑦((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑥) → 𝑥 = 𝑦) ↔ ∀𝑥(𝑥 ∈ ∪ ∪ 𝑅 → ∀𝑦((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑥) → 𝑥 = 𝑦)))
18		df-br 5090	. . . . . . . . . . . . 13 ⊢ (𝑥𝑅𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑅)
19		vex 3438	. . . . . . . . . . . . . . 15 ⊢ 𝑥 ∈ V
20		vex 3438	. . . . . . . . . . . . . . 15 ⊢ 𝑦 ∈ V
21	19, 20	opeluu 5408	. . . . . . . . . . . . . 14 ⊢ (⟨𝑥, 𝑦⟩ ∈ 𝑅 → (𝑥 ∈ ∪ ∪ 𝑅 ∧ 𝑦 ∈ ∪ ∪ 𝑅))
22	21	simpld 494	. . . . . . . . . . . . 13 ⊢ (⟨𝑥, 𝑦⟩ ∈ 𝑅 → 𝑥 ∈ ∪ ∪ 𝑅)
23	18, 22	sylbi 217	. . . . . . . . . . . 12 ⊢ (𝑥𝑅𝑦 → 𝑥 ∈ ∪ ∪ 𝑅)
24	23	adantr 480	. . . . . . . . . . 11 ⊢ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑥) → 𝑥 ∈ ∪ ∪ 𝑅)
25	24	pm2.24d 151	. . . . . . . . . 10 ⊢ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑥) → (¬ 𝑥 ∈ ∪ ∪ 𝑅 → 𝑥 = 𝑦))
26	25	com12 32	. . . . . . . . 9 ⊢ (¬ 𝑥 ∈ ∪ ∪ 𝑅 → ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑥) → 𝑥 = 𝑦))
27	26	alrimiv 1928	. . . . . . . 8 ⊢ (¬ 𝑥 ∈ ∪ ∪ 𝑅 → ∀𝑦((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑥) → 𝑥 = 𝑦))
28		id 22	. . . . . . . 8 ⊢ (∀𝑦((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑥) → 𝑥 = 𝑦) → ∀𝑦((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑥) → 𝑥 = 𝑦))
29	27, 28	ja 186	. . . . . . 7 ⊢ ((𝑥 ∈ ∪ ∪ 𝑅 → ∀𝑦((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑥) → 𝑥 = 𝑦)) → ∀𝑦((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑥) → 𝑥 = 𝑦))
30		ax-1 6	. . . . . . 7 ⊢ (∀𝑦((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑥) → 𝑥 = 𝑦) → (𝑥 ∈ ∪ ∪ 𝑅 → ∀𝑦((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑥) → 𝑥 = 𝑦)))
31	29, 30	impbii 209	. . . . . 6 ⊢ ((𝑥 ∈ ∪ ∪ 𝑅 → ∀𝑦((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑥) → 𝑥 = 𝑦)) ↔ ∀𝑦((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑥) → 𝑥 = 𝑦))
32	31	albii 1820	. . . . 5 ⊢ (∀𝑥(𝑥 ∈ ∪ ∪ 𝑅 → ∀𝑦((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑥) → 𝑥 = 𝑦)) ↔ ∀𝑥∀𝑦((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑥) → 𝑥 = 𝑦))
33	17, 32	bitri 275	. . . 4 ⊢ (∀𝑥 ∈ ∪ ∪ 𝑅∀𝑦((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑥) → 𝑥 = 𝑦) ↔ ∀𝑥∀𝑦((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑥) → 𝑥 = 𝑦))
34	16, 33	anbi12i 628	. . 3 ⊢ ((∀𝑥 ∈ ∪ ∪ 𝑅∀𝑦(𝑥 = 𝑦 → (𝑥𝑅𝑦 ∧ 𝑦𝑅𝑥)) ∧ ∀𝑥 ∈ ∪ ∪ 𝑅∀𝑦((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑥) → 𝑥 = 𝑦)) ↔ (∀𝑥 ∈ ∪ ∪ 𝑅𝑥𝑅𝑥 ∧ ∀𝑥∀𝑦((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑥) → 𝑥 = 𝑦)))
35	4, 5, 34	3bitri 297	. 2 ⊢ (∀𝑥 ∈ ∪ ∪ 𝑅(∀𝑦((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑥) → 𝑥 = 𝑦) ∧ ∀𝑦(𝑥 = 𝑦 → (𝑥𝑅𝑦 ∧ 𝑦𝑅𝑥))) ↔ (∀𝑥 ∈ ∪ ∪ 𝑅𝑥𝑅𝑥 ∧ ∀𝑥∀𝑦((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑥) → 𝑥 = 𝑦)))
36	1, 3, 35	3bitri 297	1 ⊢ ((𝑅 ∩ ◡𝑅) = ( I ↾ ∪ ∪ 𝑅) ↔ (∀𝑥 ∈ ∪ ∪ 𝑅𝑥𝑅𝑥 ∧ ∀𝑥∀𝑦((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑥) → 𝑥 = 𝑦)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395  ∀wal 1539   = wceq 1541   ∈ wcel 2110  ∀wral 3045   ∩ cin 3899  ⟨cop 4580  ∪ cuni 4857   class class class wbr 5089   I cid 5508  ^◡ccnv 5613   ↾ cres 5616

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2112  ax-9 2120  ax-ext 2702  ax-sep 5232  ax-nul 5242  ax-pr 5368

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2067  df-clab 2709  df-cleq 2722  df-clel 2804  df-ral 3046  df-rex 3055  df-rab 3394  df-v 3436  df-dif 3903  df-un 3905  df-in 3907  df-ss 3917  df-nul 4282  df-if 4474  df-sn 4575  df-pr 4577  df-op 4581  df-uni 4858  df-br 5090  df-opab 5152  df-id 5509  df-xp 5620  df-rel 5621  df-cnv 5622  df-res 5626

This theorem is referenced by:  pslem  18470  psss  18478