Theorem psss 18213

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 4159	. . 3 ⊢ (𝑅 ∩ (𝐴 × 𝐴)) ⊆ 𝑅
2		psrel 18202	. . 3 ⊢ (𝑅 ∈ PosetRel → Rel 𝑅)
3		relss 5682	. . 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 18204	. . 3 ⊢ (𝑅 ∈ PosetRel → (𝑅 ∘ 𝑅) ⊆ 𝑅)
6		trinxp 6019	. . 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 4862	. . . . . 6 ⊢ ∪ (𝑅 ∩ (𝐴 × 𝐴)) ⊆ (∪ 𝑅 ∩ ∪ (𝐴 × 𝐴))
9	8	unissi 4845	. . . . 5 ⊢ ∪ ∪ (𝑅 ∩ (𝐴 × 𝐴)) ⊆ ∪ (∪ 𝑅 ∩ ∪ (𝐴 × 𝐴))
10		uniin 4862	. . . . 5 ⊢ ∪ (∪ 𝑅 ∩ ∪ (𝐴 × 𝐴)) ⊆ (∪ ∪ 𝑅 ∩ ∪ ∪ (𝐴 × 𝐴))
11	9, 10	sstri 3926	. . . 4 ⊢ ∪ ∪ (𝑅 ∩ (𝐴 × 𝐴)) ⊆ (∪ ∪ 𝑅 ∩ ∪ ∪ (𝐴 × 𝐴))
12		elin 3899	. . . . . 6 ⊢ (𝑥 ∈ (∪ ∪ 𝑅 ∩ ∪ ∪ (𝐴 × 𝐴)) ↔ (𝑥 ∈ ∪ ∪ 𝑅 ∧ 𝑥 ∈ ∪ ∪ (𝐴 × 𝐴)))
13		unixpid 6176	. . . . . . . . 9 ⊢ ∪ ∪ (𝐴 × 𝐴) = 𝐴
14	13	eleq2i 2830	. . . . . . . 8 ⊢ (𝑥 ∈ ∪ ∪ (𝐴 × 𝐴) ↔ 𝑥 ∈ 𝐴)
15		simprr 769	. . . . . . . . . 10 ⊢ ((𝑅 ∈ PosetRel ∧ (𝑥 ∈ ∪ ∪ 𝑅 ∧ 𝑥 ∈ 𝐴)) → 𝑥 ∈ 𝐴)
16		psdmrn 18206	. . . . . . . . . . . . . . 15 ⊢ (𝑅 ∈ PosetRel → (dom 𝑅 = ∪ ∪ 𝑅 ∧ ran 𝑅 = ∪ ∪ 𝑅))
17	16	simpld 494	. . . . . . . . . . . . . 14 ⊢ (𝑅 ∈ PosetRel → dom 𝑅 = ∪ ∪ 𝑅)
18	17	eleq2d 2824	. . . . . . . . . . . . 13 ⊢ (𝑅 ∈ PosetRel → (𝑥 ∈ dom 𝑅 ↔ 𝑥 ∈ ∪ ∪ 𝑅))
19	18	biimpar 477	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ PosetRel ∧ 𝑥 ∈ ∪ ∪ 𝑅) → 𝑥 ∈ dom 𝑅)
20		eqid 2738	. . . . . . . . . . . . 13 ⊢ dom 𝑅 = dom 𝑅
21	20	psref 18207	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ PosetRel ∧ 𝑥 ∈ dom 𝑅) → 𝑥𝑅𝑥)
22	19, 21	syldan 590	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ PosetRel ∧ 𝑥 ∈ ∪ ∪ 𝑅) → 𝑥𝑅𝑥)
23	22	adantrr 713	. . . . . . . . . 10 ⊢ ((𝑅 ∈ PosetRel ∧ (𝑥 ∈ ∪ ∪ 𝑅 ∧ 𝑥 ∈ 𝐴)) → 𝑥𝑅𝑥)
24		brinxp2 5655	. . . . . . . . . 10 ⊢ (𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑥 ↔ ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴) ∧ 𝑥𝑅𝑥))
25	15, 15, 23, 24	syl21anbrc 1342	. . . . . . . . 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	syl5bi 241	. . . . . . 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	syl5bi 241	. . . . 5 ⊢ (𝑅 ∈ PosetRel → (𝑥 ∈ (∪ ∪ 𝑅 ∩ ∪ ∪ (𝐴 × 𝐴)) → 𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑥))
30	29	ralrimiv 3106	. . . 4 ⊢ (𝑅 ∈ PosetRel → ∀𝑥 ∈ (∪ ∪ 𝑅 ∩ ∪ ∪ (𝐴 × 𝐴))𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑥)
31		ssralv 3983	. . . 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 5115	. . . . 5 ⊢ (𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑦 → 𝑥𝑅𝑦)
34	1	ssbri 5115	. . . . 5 ⊢ (𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥 → 𝑦𝑅𝑥)
35		psasym 18209	. . . . . 6 ⊢ ((𝑅 ∈ PosetRel ∧ 𝑥𝑅𝑦 ∧ 𝑦𝑅𝑥) → 𝑥 = 𝑦)
36	35	3expib 1120	. . . . 5 ⊢ (𝑅 ∈ PosetRel → ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑥) → 𝑥 = 𝑦))
37	33, 34, 36	syl2ani 606	. . . 4 ⊢ (𝑅 ∈ PosetRel → ((𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑦 ∧ 𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥) → 𝑥 = 𝑦))
38	37	alrimivv 1932	. . 3 ⊢ (𝑅 ∈ PosetRel → ∀𝑥∀𝑦((𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑦 ∧ 𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥) → 𝑥 = 𝑦))
39		asymref2 6011	. . 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 582	. 2 ⊢ (𝑅 ∈ PosetRel → ((𝑅 ∩ (𝐴 × 𝐴)) ∩ ◡(𝑅 ∩ (𝐴 × 𝐴))) = ( I ↾ ∪ ∪ (𝑅 ∩ (𝐴 × 𝐴))))
41		inex1g 5238	. . 3 ⊢ (𝑅 ∈ PosetRel → (𝑅 ∩ (𝐴 × 𝐴)) ∈ V)
42		isps 18201	. . 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 1340	1 ⊢ (𝑅 ∈ PosetRel → (𝑅 ∩ (𝐴 × 𝐴)) ∈ PosetRel)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1085  ∀wal 1537   = wceq 1539   ∈ wcel 2108  ∀wral 3063  Vcvv 3422   ∩ cin 3882   ⊆ wss 3883  ∪ cuni 4836   class class class wbr 5070   I cid 5479   × cxp 5578  ^◡ccnv 5579  dom cdm 5580  ran crn 5581   ↾ cres 5582   ∘ ccom 5584  Rel wrel 5585  PosetRelcps 18197

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-opab 5133  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ps 18199

This theorem is referenced by:  tsrss  18222  ordtrest2  22263