Theorem psss 18545

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 4202	. . 3 ⊢ (𝑅 ∩ (𝐴 × 𝐴)) ⊆ 𝑅
2		psrel 18534	. . 3 ⊢ (𝑅 ∈ PosetRel → Rel 𝑅)
3		relss 5746	. . 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 18536	. . 3 ⊢ (𝑅 ∈ PosetRel → (𝑅 ∘ 𝑅) ⊆ 𝑅)
6		trinxp 6100	. . 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 4897	. . . . . 6 ⊢ ∪ (𝑅 ∩ (𝐴 × 𝐴)) ⊆ (∪ 𝑅 ∩ ∪ (𝐴 × 𝐴))
9	8	unissi 4882	. . . . 5 ⊢ ∪ ∪ (𝑅 ∩ (𝐴 × 𝐴)) ⊆ ∪ (∪ 𝑅 ∩ ∪ (𝐴 × 𝐴))
10		uniin 4897	. . . . 5 ⊢ ∪ (∪ 𝑅 ∩ ∪ (𝐴 × 𝐴)) ⊆ (∪ ∪ 𝑅 ∩ ∪ ∪ (𝐴 × 𝐴))
11	9, 10	sstri 3958	. . . 4 ⊢ ∪ ∪ (𝑅 ∩ (𝐴 × 𝐴)) ⊆ (∪ ∪ 𝑅 ∩ ∪ ∪ (𝐴 × 𝐴))
12		elin 3932	. . . . . 6 ⊢ (𝑥 ∈ (∪ ∪ 𝑅 ∩ ∪ ∪ (𝐴 × 𝐴)) ↔ (𝑥 ∈ ∪ ∪ 𝑅 ∧ 𝑥 ∈ ∪ ∪ (𝐴 × 𝐴)))
13		unixpid 6259	. . . . . . . . 9 ⊢ ∪ ∪ (𝐴 × 𝐴) = 𝐴
14	13	eleq2i 2821	. . . . . . . 8 ⊢ (𝑥 ∈ ∪ ∪ (𝐴 × 𝐴) ↔ 𝑥 ∈ 𝐴)
15		simprr 772	. . . . . . . . . 10 ⊢ ((𝑅 ∈ PosetRel ∧ (𝑥 ∈ ∪ ∪ 𝑅 ∧ 𝑥 ∈ 𝐴)) → 𝑥 ∈ 𝐴)
16		psdmrn 18538	. . . . . . . . . . . . . . 15 ⊢ (𝑅 ∈ PosetRel → (dom 𝑅 = ∪ ∪ 𝑅 ∧ ran 𝑅 = ∪ ∪ 𝑅))
17	16	simpld 494	. . . . . . . . . . . . . 14 ⊢ (𝑅 ∈ PosetRel → dom 𝑅 = ∪ ∪ 𝑅)
18	17	eleq2d 2815	. . . . . . . . . . . . 13 ⊢ (𝑅 ∈ PosetRel → (𝑥 ∈ dom 𝑅 ↔ 𝑥 ∈ ∪ ∪ 𝑅))
19	18	biimpar 477	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ PosetRel ∧ 𝑥 ∈ ∪ ∪ 𝑅) → 𝑥 ∈ dom 𝑅)
20		eqid 2730	. . . . . . . . . . . . 13 ⊢ dom 𝑅 = dom 𝑅
21	20	psref 18539	. . . . . . . . . . . 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 5718	. . . . . . . . . 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 4017	. . . 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 5154	. . . . 5 ⊢ (𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑦 → 𝑥𝑅𝑦)
34	1	ssbri 5154	. . . . 5 ⊢ (𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥 → 𝑦𝑅𝑥)
35		psasym 18541	. . . . . 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 1928	. . 3 ⊢ (𝑅 ∈ PosetRel → ∀𝑥∀𝑦((𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑦 ∧ 𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥) → 𝑥 = 𝑦))
39		asymref2 6092	. . 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 5276	. . 3 ⊢ (𝑅 ∈ PosetRel → (𝑅 ∩ (𝐴 × 𝐴)) ∈ V)
42		isps 18533	. . 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 1538   = wceq 1540   ∈ wcel 2109  ∀wral 3045  Vcvv 3450   ∩ cin 3915   ⊆ wss 3916  ∪ cuni 4873   class class class wbr 5109   I cid 5534   × cxp 5638  ^◡ccnv 5639  dom cdm 5640  ran crn 5641   ↾ cres 5642   ∘ ccom 5644  Rel wrel 5645  PosetRelcps 18529

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-11 2158  ax-12 2178  ax-ext 2702  ax-sep 5253  ax-nul 5263  ax-pr 5389

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-ne 2927  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-dif 3919  df-un 3921  df-in 3923  df-ss 3933  df-nul 4299  df-if 4491  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-br 5110  df-opab 5172  df-id 5535  df-xp 5646  df-rel 5647  df-cnv 5648  df-co 5649  df-dm 5650  df-rn 5651  df-res 5652  df-ps 18531

This theorem is referenced by:  tsrss  18554  ordtrest2  23097