Theorem psss 18501

Description: Any subset of a partially ordered set is partially ordered. (Contributed by FL, 24-Jan-2010.)

Assertion

Ref	Expression
psss	⊢ (𝑅 ∈ PosetRel → (𝑅 ∩ (𝐴 × 𝐴)) ∈ PosetRel)

Proof of Theorem psss

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		inss1 4187	. . 3 ⊢ (𝑅 ∩ (𝐴 × 𝐴)) ⊆ 𝑅
2		psrel 18490	. . 3 ⊢ (𝑅 ∈ PosetRel → Rel 𝑅)
3		relss 5729	. . 3 ⊢ ((𝑅 ∩ (𝐴 × 𝐴)) ⊆ 𝑅 → (Rel 𝑅 → Rel (𝑅 ∩ (𝐴 × 𝐴))))
4	1, 2, 3	mpsyl 68	. 2 ⊢ (𝑅 ∈ PosetRel → Rel (𝑅 ∩ (𝐴 × 𝐴)))
5		pstr2 18492	. . 3 ⊢ (𝑅 ∈ PosetRel → (𝑅 ∘ 𝑅) ⊆ 𝑅)
6		trinxp 6080	. . 3 ⊢ ((𝑅 ∘ 𝑅) ⊆ 𝑅 → ((𝑅 ∩ (𝐴 × 𝐴)) ∘ (𝑅 ∩ (𝐴 × 𝐴))) ⊆ (𝑅 ∩ (𝐴 × 𝐴)))
7	5, 6	syl 17	. 2 ⊢ (𝑅 ∈ PosetRel → ((𝑅 ∩ (𝐴 × 𝐴)) ∘ (𝑅 ∩ (𝐴 × 𝐴))) ⊆ (𝑅 ∩ (𝐴 × 𝐴)))
8		uniin 4885	. . . . . 6 ⊢ ∪ (𝑅 ∩ (𝐴 × 𝐴)) ⊆ (∪ 𝑅 ∩ ∪ (𝐴 × 𝐴))
9	8	unissi 4870	. . . . 5 ⊢ ∪ ∪ (𝑅 ∩ (𝐴 × 𝐴)) ⊆ ∪ (∪ 𝑅 ∩ ∪ (𝐴 × 𝐴))
10		uniin 4885	. . . . 5 ⊢ ∪ (∪ 𝑅 ∩ ∪ (𝐴 × 𝐴)) ⊆ (∪ ∪ 𝑅 ∩ ∪ ∪ (𝐴 × 𝐴))
11	9, 10	sstri 3941	. . . 4 ⊢ ∪ ∪ (𝑅 ∩ (𝐴 × 𝐴)) ⊆ (∪ ∪ 𝑅 ∩ ∪ ∪ (𝐴 × 𝐴))
12		elin 3915	. . . . . 6 ⊢ (𝑥 ∈ (∪ ∪ 𝑅 ∩ ∪ ∪ (𝐴 × 𝐴)) ↔ (𝑥 ∈ ∪ ∪ 𝑅 ∧ 𝑥 ∈ ∪ ∪ (𝐴 × 𝐴)))
13		unixpid 6240	. . . . . . . . 9 ⊢ ∪ ∪ (𝐴 × 𝐴) = 𝐴
14	13	eleq2i 2826	. . . . . . . 8 ⊢ (𝑥 ∈ ∪ ∪ (𝐴 × 𝐴) ↔ 𝑥 ∈ 𝐴)
15		simprr 772	. . . . . . . . . 10 ⊢ ((𝑅 ∈ PosetRel ∧ (𝑥 ∈ ∪ ∪ 𝑅 ∧ 𝑥 ∈ 𝐴)) → 𝑥 ∈ 𝐴)
16		psdmrn 18494	. . . . . . . . . . . . . . 15 ⊢ (𝑅 ∈ PosetRel → (dom 𝑅 = ∪ ∪ 𝑅 ∧ ran 𝑅 = ∪ ∪ 𝑅))
17	16	simpld 494	. . . . . . . . . . . . . 14 ⊢ (𝑅 ∈ PosetRel → dom 𝑅 = ∪ ∪ 𝑅)
18	17	eleq2d 2820	. . . . . . . . . . . . 13 ⊢ (𝑅 ∈ PosetRel → (𝑥 ∈ dom 𝑅 ↔ 𝑥 ∈ ∪ ∪ 𝑅))
19	18	biimpar 477	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ PosetRel ∧ 𝑥 ∈ ∪ ∪ 𝑅) → 𝑥 ∈ dom 𝑅)
20		eqid 2734	. . . . . . . . . . . . 13 ⊢ dom 𝑅 = dom 𝑅
21	20	psref 18495	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ PosetRel ∧ 𝑥 ∈ dom 𝑅) → 𝑥𝑅𝑥)
22	19, 21	syldan 591	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ PosetRel ∧ 𝑥 ∈ ∪ ∪ 𝑅) → 𝑥𝑅𝑥)
23	22	adantrr 717	. . . . . . . . . 10 ⊢ ((𝑅 ∈ PosetRel ∧ (𝑥 ∈ ∪ ∪ 𝑅 ∧ 𝑥 ∈ 𝐴)) → 𝑥𝑅𝑥)
24		brinxp2 5700	. . . . . . . . . 10 ⊢ (𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑥 ↔ ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴) ∧ 𝑥𝑅𝑥))
25	15, 15, 23, 24	syl21anbrc 1345	. . . . . . . . 9 ⊢ ((𝑅 ∈ PosetRel ∧ (𝑥 ∈ ∪ ∪ 𝑅 ∧ 𝑥 ∈ 𝐴)) → 𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑥)
26	25	expr 456	. . . . . . . 8 ⊢ ((𝑅 ∈ PosetRel ∧ 𝑥 ∈ ∪ ∪ 𝑅) → (𝑥 ∈ 𝐴 → 𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑥))
27	14, 26	biimtrid 242	. . . . . . 7 ⊢ ((𝑅 ∈ PosetRel ∧ 𝑥 ∈ ∪ ∪ 𝑅) → (𝑥 ∈ ∪ ∪ (𝐴 × 𝐴) → 𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑥))
28	27	expimpd 453	. . . . . 6 ⊢ (𝑅 ∈ PosetRel → ((𝑥 ∈ ∪ ∪ 𝑅 ∧ 𝑥 ∈ ∪ ∪ (𝐴 × 𝐴)) → 𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑥))
29	12, 28	biimtrid 242	. . . . 5 ⊢ (𝑅 ∈ PosetRel → (𝑥 ∈ (∪ ∪ 𝑅 ∩ ∪ ∪ (𝐴 × 𝐴)) → 𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑥))
30	29	ralrimiv 3125	. . . 4 ⊢ (𝑅 ∈ PosetRel → ∀𝑥 ∈ (∪ ∪ 𝑅 ∩ ∪ ∪ (𝐴 × 𝐴))𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑥)
31		ssralv 4000	. . . 4 ⊢ (∪ ∪ (𝑅 ∩ (𝐴 × 𝐴)) ⊆ (∪ ∪ 𝑅 ∩ ∪ ∪ (𝐴 × 𝐴)) → (∀𝑥 ∈ (∪ ∪ 𝑅 ∩ ∪ ∪ (𝐴 × 𝐴))𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑥 → ∀𝑥 ∈ ∪ ∪ (𝑅 ∩ (𝐴 × 𝐴))𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑥))
32	11, 30, 31	mpsyl 68	. . 3 ⊢ (𝑅 ∈ PosetRel → ∀𝑥 ∈ ∪ ∪ (𝑅 ∩ (𝐴 × 𝐴))𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑥)
33	1	ssbri 5141	. . . . 5 ⊢ (𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑦 → 𝑥𝑅𝑦)
34	1	ssbri 5141	. . . . 5 ⊢ (𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥 → 𝑦𝑅𝑥)
35		psasym 18497	. . . . . 6 ⊢ ((𝑅 ∈ PosetRel ∧ 𝑥𝑅𝑦 ∧ 𝑦𝑅𝑥) → 𝑥 = 𝑦)
36	35	3expib 1122	. . . . 5 ⊢ (𝑅 ∈ PosetRel → ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑥) → 𝑥 = 𝑦))
37	33, 34, 36	syl2ani 607	. . . 4 ⊢ (𝑅 ∈ PosetRel → ((𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑦 ∧ 𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥) → 𝑥 = 𝑦))
38	37	alrimivv 1929	. . 3 ⊢ (𝑅 ∈ PosetRel → ∀𝑥∀𝑦((𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑦 ∧ 𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥) → 𝑥 = 𝑦))
39		asymref2 6072	. . 3 ⊢ (((𝑅 ∩ (𝐴 × 𝐴)) ∩ ◡(𝑅 ∩ (𝐴 × 𝐴))) = ( I ↾ ∪ ∪ (𝑅 ∩ (𝐴 × 𝐴))) ↔ (∀𝑥 ∈ ∪ ∪ (𝑅 ∩ (𝐴 × 𝐴))𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑥 ∧ ∀𝑥∀𝑦((𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑦 ∧ 𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥) → 𝑥 = 𝑦)))
40	32, 38, 39	sylanbrc 583	. 2 ⊢ (𝑅 ∈ PosetRel → ((𝑅 ∩ (𝐴 × 𝐴)) ∩ ◡(𝑅 ∩ (𝐴 × 𝐴))) = ( I ↾ ∪ ∪ (𝑅 ∩ (𝐴 × 𝐴))))
41		inex1g 5262	. . 3 ⊢ (𝑅 ∈ PosetRel → (𝑅 ∩ (𝐴 × 𝐴)) ∈ V)
42		isps 18489	. . 3 ⊢ ((𝑅 ∩ (𝐴 × 𝐴)) ∈ V → ((𝑅 ∩ (𝐴 × 𝐴)) ∈ PosetRel ↔ (Rel (𝑅 ∩ (𝐴 × 𝐴)) ∧ ((𝑅 ∩ (𝐴 × 𝐴)) ∘ (𝑅 ∩ (𝐴 × 𝐴))) ⊆ (𝑅 ∩ (𝐴 × 𝐴)) ∧ ((𝑅 ∩ (𝐴 × 𝐴)) ∩ ◡(𝑅 ∩ (𝐴 × 𝐴))) = ( I ↾ ∪ ∪ (𝑅 ∩ (𝐴 × 𝐴))))))
43	41, 42	syl 17	. 2 ⊢ (𝑅 ∈ PosetRel → ((𝑅 ∩ (𝐴 × 𝐴)) ∈ PosetRel ↔ (Rel (𝑅 ∩ (𝐴 × 𝐴)) ∧ ((𝑅 ∩ (𝐴 × 𝐴)) ∘ (𝑅 ∩ (𝐴 × 𝐴))) ⊆ (𝑅 ∩ (𝐴 × 𝐴)) ∧ ((𝑅 ∩ (𝐴 × 𝐴)) ∩ ◡(𝑅 ∩ (𝐴 × 𝐴))) = ( I ↾ ∪ ∪ (𝑅 ∩ (𝐴 × 𝐴))))))
44	4, 7, 40, 43	mpbir3and 1343	1 ⊢ (𝑅 ∈ PosetRel → (𝑅 ∩ (𝐴 × 𝐴)) ∈ PosetRel)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086  ∀wal 1539   = wceq 1541   ∈ wcel 2113  ∀wral 3049  Vcvv 3438   ∩ cin 3898   ⊆ wss 3899  ∪ cuni 4861   class class class wbr 5096   I cid 5516   × cxp 5620  ^◡ccnv 5621  dom cdm 5622  ran crn 5623   ↾ cres 5624   ∘ ccom 5626  Rel wrel 5627  PosetRelcps 18485

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 2115  ax-9 2123  ax-11 2162  ax-ext 2706  ax-sep 5239  ax-nul 5249  ax-pr 5375

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 2068  df-clab 2713  df-cleq 2726  df-clel 2809  df-ne 2931  df-ral 3050  df-rex 3059  df-rab 3398  df-v 3440  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4862  df-br 5097  df-opab 5159  df-id 5517  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-ps 18487

This theorem is referenced by:  tsrss  18510  ordtrest2  23146