Theorem psss 18298

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 4162	. . 3 ⊢ (𝑅 ∩ (𝐴 × 𝐴)) ⊆ 𝑅
2		psrel 18287	. . 3 ⊢ (𝑅 ∈ PosetRel → Rel 𝑅)
3		relss 5692	. . 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 18289	. . 3 ⊢ (𝑅 ∈ PosetRel → (𝑅 ∘ 𝑅) ⊆ 𝑅)
6		trinxp 6030	. . 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 4865	. . . . . 6 ⊢ ∪ (𝑅 ∩ (𝐴 × 𝐴)) ⊆ (∪ 𝑅 ∩ ∪ (𝐴 × 𝐴))
9	8	unissi 4848	. . . . 5 ⊢ ∪ ∪ (𝑅 ∩ (𝐴 × 𝐴)) ⊆ ∪ (∪ 𝑅 ∩ ∪ (𝐴 × 𝐴))
10		uniin 4865	. . . . 5 ⊢ ∪ (∪ 𝑅 ∩ ∪ (𝐴 × 𝐴)) ⊆ (∪ ∪ 𝑅 ∩ ∪ ∪ (𝐴 × 𝐴))
11	9, 10	sstri 3930	. . . 4 ⊢ ∪ ∪ (𝑅 ∩ (𝐴 × 𝐴)) ⊆ (∪ ∪ 𝑅 ∩ ∪ ∪ (𝐴 × 𝐴))
12		elin 3903	. . . . . 6 ⊢ (𝑥 ∈ (∪ ∪ 𝑅 ∩ ∪ ∪ (𝐴 × 𝐴)) ↔ (𝑥 ∈ ∪ ∪ 𝑅 ∧ 𝑥 ∈ ∪ ∪ (𝐴 × 𝐴)))
13		unixpid 6187	. . . . . . . . 9 ⊢ ∪ ∪ (𝐴 × 𝐴) = 𝐴
14	13	eleq2i 2830	. . . . . . . 8 ⊢ (𝑥 ∈ ∪ ∪ (𝐴 × 𝐴) ↔ 𝑥 ∈ 𝐴)
15		simprr 770	. . . . . . . . . 10 ⊢ ((𝑅 ∈ PosetRel ∧ (𝑥 ∈ ∪ ∪ 𝑅 ∧ 𝑥 ∈ 𝐴)) → 𝑥 ∈ 𝐴)
16		psdmrn 18291	. . . . . . . . . . . . . . 15 ⊢ (𝑅 ∈ PosetRel → (dom 𝑅 = ∪ ∪ 𝑅 ∧ ran 𝑅 = ∪ ∪ 𝑅))
17	16	simpld 495	. . . . . . . . . . . . . 14 ⊢ (𝑅 ∈ PosetRel → dom 𝑅 = ∪ ∪ 𝑅)
18	17	eleq2d 2824	. . . . . . . . . . . . 13 ⊢ (𝑅 ∈ PosetRel → (𝑥 ∈ dom 𝑅 ↔ 𝑥 ∈ ∪ ∪ 𝑅))
19	18	biimpar 478	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ PosetRel ∧ 𝑥 ∈ ∪ ∪ 𝑅) → 𝑥 ∈ dom 𝑅)
20		eqid 2738	. . . . . . . . . . . . 13 ⊢ dom 𝑅 = dom 𝑅
21	20	psref 18292	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ PosetRel ∧ 𝑥 ∈ dom 𝑅) → 𝑥𝑅𝑥)
22	19, 21	syldan 591	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ PosetRel ∧ 𝑥 ∈ ∪ ∪ 𝑅) → 𝑥𝑅𝑥)
23	22	adantrr 714	. . . . . . . . . 10 ⊢ ((𝑅 ∈ PosetRel ∧ (𝑥 ∈ ∪ ∪ 𝑅 ∧ 𝑥 ∈ 𝐴)) → 𝑥𝑅𝑥)
24		brinxp2 5664	. . . . . . . . . 10 ⊢ (𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑥 ↔ ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴) ∧ 𝑥𝑅𝑥))
25	15, 15, 23, 24	syl21anbrc 1343	. . . . . . . . 9 ⊢ ((𝑅 ∈ PosetRel ∧ (𝑥 ∈ ∪ ∪ 𝑅 ∧ 𝑥 ∈ 𝐴)) → 𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑥)
26	25	expr 457	. . . . . . . 8 ⊢ ((𝑅 ∈ PosetRel ∧ 𝑥 ∈ ∪ ∪ 𝑅) → (𝑥 ∈ 𝐴 → 𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑥))
27	14, 26	syl5bi 241	. . . . . . 7 ⊢ ((𝑅 ∈ PosetRel ∧ 𝑥 ∈ ∪ ∪ 𝑅) → (𝑥 ∈ ∪ ∪ (𝐴 × 𝐴) → 𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑥))
28	27	expimpd 454	. . . . . 6 ⊢ (𝑅 ∈ PosetRel → ((𝑥 ∈ ∪ ∪ 𝑅 ∧ 𝑥 ∈ ∪ ∪ (𝐴 × 𝐴)) → 𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑥))
29	12, 28	syl5bi 241	. . . . 5 ⊢ (𝑅 ∈ PosetRel → (𝑥 ∈ (∪ ∪ 𝑅 ∩ ∪ ∪ (𝐴 × 𝐴)) → 𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑥))
30	29	ralrimiv 3102	. . . 4 ⊢ (𝑅 ∈ PosetRel → ∀𝑥 ∈ (∪ ∪ 𝑅 ∩ ∪ ∪ (𝐴 × 𝐴))𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑥)
31		ssralv 3987	. . . 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 5119	. . . . 5 ⊢ (𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑦 → 𝑥𝑅𝑦)
34	1	ssbri 5119	. . . . 5 ⊢ (𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥 → 𝑦𝑅𝑥)
35		psasym 18294	. . . . . 6 ⊢ ((𝑅 ∈ PosetRel ∧ 𝑥𝑅𝑦 ∧ 𝑦𝑅𝑥) → 𝑥 = 𝑦)
36	35	3expib 1121	. . . . 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 1931	. . 3 ⊢ (𝑅 ∈ PosetRel → ∀𝑥∀𝑦((𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑦 ∧ 𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥) → 𝑥 = 𝑦))
39		asymref2 6022	. . 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 5243	. . 3 ⊢ (𝑅 ∈ PosetRel → (𝑅 ∩ (𝐴 × 𝐴)) ∈ V)
42		isps 18286	. . 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 1341	1 ⊢ (𝑅 ∈ PosetRel → (𝑅 ∩ (𝐴 × 𝐴)) ∈ PosetRel)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1086  ∀wal 1537   = wceq 1539   ∈ wcel 2106  ∀wral 3064  Vcvv 3432   ∩ cin 3886   ⊆ wss 3887  ∪ cuni 4839   class class class wbr 5074   I cid 5488   × cxp 5587  ^◡ccnv 5588  dom cdm 5589  ran crn 5590   ↾ cres 5591   ∘ ccom 5593  Rel wrel 5594  PosetRelcps 18282

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ps 18284

This theorem is referenced by:  tsrss  18307  ordtrest2  22355