Theorem tsrdir 18554

Description: A totally ordered set is a directed set. (Contributed by Jeff Hankins, 25-Nov-2009.) (Revised by Mario Carneiro, 22-Nov-2013.)

Assertion

Ref	Expression
tsrdir	⊢ (𝐴 ∈ TosetRel → 𝐴 ∈ DirRel)

Proof of Theorem tsrdir

Step	Hyp	Ref	Expression
1		tsrps 18537	. . . 4 ⊢ (𝐴 ∈ TosetRel → 𝐴 ∈ PosetRel)
2		psrel 18519	. . . 4 ⊢ (𝐴 ∈ PosetRel → Rel 𝐴)
3	1, 2	syl 17	. . 3 ⊢ (𝐴 ∈ TosetRel → Rel 𝐴)
4		psref2 18520	. . . . 5 ⊢ (𝐴 ∈ PosetRel → (𝐴 ∩ ◡𝐴) = ( I ↾ ∪ ∪ 𝐴))
5		inss1 4228	. . . . 5 ⊢ (𝐴 ∩ ◡𝐴) ⊆ 𝐴
6	4, 5	eqsstrrdi 4037	. . . 4 ⊢ (𝐴 ∈ PosetRel → ( I ↾ ∪ ∪ 𝐴) ⊆ 𝐴)
7	1, 6	syl 17	. . 3 ⊢ (𝐴 ∈ TosetRel → ( I ↾ ∪ ∪ 𝐴) ⊆ 𝐴)
8	3, 7	jca 513	. 2 ⊢ (𝐴 ∈ TosetRel → (Rel 𝐴 ∧ ( I ↾ ∪ ∪ 𝐴) ⊆ 𝐴))
9		pstr2 18521	. . . 4 ⊢ (𝐴 ∈ PosetRel → (𝐴 ∘ 𝐴) ⊆ 𝐴)
10	1, 9	syl 17	. . 3 ⊢ (𝐴 ∈ TosetRel → (𝐴 ∘ 𝐴) ⊆ 𝐴)
11		psdmrn 18523	. . . . . . 7 ⊢ (𝐴 ∈ PosetRel → (dom 𝐴 = ∪ ∪ 𝐴 ∧ ran 𝐴 = ∪ ∪ 𝐴))
12	1, 11	syl 17	. . . . . 6 ⊢ (𝐴 ∈ TosetRel → (dom 𝐴 = ∪ ∪ 𝐴 ∧ ran 𝐴 = ∪ ∪ 𝐴))
13	12	simpld 496	. . . . 5 ⊢ (𝐴 ∈ TosetRel → dom 𝐴 = ∪ ∪ 𝐴)
14	13	sqxpeqd 5708	. . . 4 ⊢ (𝐴 ∈ TosetRel → (dom 𝐴 × dom 𝐴) = (∪ ∪ 𝐴 × ∪ ∪ 𝐴))
15		eqid 2733	. . . . . . 7 ⊢ dom 𝐴 = dom 𝐴
16	15	istsr 18533	. . . . . 6 ⊢ (𝐴 ∈ TosetRel ↔ (𝐴 ∈ PosetRel ∧ (dom 𝐴 × dom 𝐴) ⊆ (𝐴 ∪ ◡𝐴)))
17	16	simprbi 498	. . . . 5 ⊢ (𝐴 ∈ TosetRel → (dom 𝐴 × dom 𝐴) ⊆ (𝐴 ∪ ◡𝐴))
18		relcoi2 6274	. . . . . . . 8 ⊢ (Rel 𝐴 → (( I ↾ ∪ ∪ 𝐴) ∘ 𝐴) = 𝐴)
19	3, 18	syl 17	. . . . . . 7 ⊢ (𝐴 ∈ TosetRel → (( I ↾ ∪ ∪ 𝐴) ∘ 𝐴) = 𝐴)
20		cnvresid 6625	. . . . . . . . 9 ⊢ ◡( I ↾ ∪ ∪ 𝐴) = ( I ↾ ∪ ∪ 𝐴)
21		cnvss 5871	. . . . . . . . . 10 ⊢ (( I ↾ ∪ ∪ 𝐴) ⊆ 𝐴 → ◡( I ↾ ∪ ∪ 𝐴) ⊆ ◡𝐴)
22	7, 21	syl 17	. . . . . . . . 9 ⊢ (𝐴 ∈ TosetRel → ◡( I ↾ ∪ ∪ 𝐴) ⊆ ◡𝐴)
23	20, 22	eqsstrrid 4031	. . . . . . . 8 ⊢ (𝐴 ∈ TosetRel → ( I ↾ ∪ ∪ 𝐴) ⊆ ◡𝐴)
24		coss1 5854	. . . . . . . 8 ⊢ (( I ↾ ∪ ∪ 𝐴) ⊆ ◡𝐴 → (( I ↾ ∪ ∪ 𝐴) ∘ 𝐴) ⊆ (◡𝐴 ∘ 𝐴))
25	23, 24	syl 17	. . . . . . 7 ⊢ (𝐴 ∈ TosetRel → (( I ↾ ∪ ∪ 𝐴) ∘ 𝐴) ⊆ (◡𝐴 ∘ 𝐴))
26	19, 25	eqsstrrd 4021	. . . . . 6 ⊢ (𝐴 ∈ TosetRel → 𝐴 ⊆ (◡𝐴 ∘ 𝐴))
27		relcnv 6101	. . . . . . . 8 ⊢ Rel ◡𝐴
28		relcoi1 6275	. . . . . . . 8 ⊢ (Rel ◡𝐴 → (◡𝐴 ∘ ( I ↾ ∪ ∪ ◡𝐴)) = ◡𝐴)
29	27, 28	ax-mp 5	. . . . . . 7 ⊢ (◡𝐴 ∘ ( I ↾ ∪ ∪ ◡𝐴)) = ◡𝐴
30		relcnvfld 6277	. . . . . . . . . . 11 ⊢ (Rel 𝐴 → ∪ ∪ 𝐴 = ∪ ∪ ◡𝐴)
31	3, 30	syl 17	. . . . . . . . . 10 ⊢ (𝐴 ∈ TosetRel → ∪ ∪ 𝐴 = ∪ ∪ ◡𝐴)
32	31	reseq2d 5980	. . . . . . . . 9 ⊢ (𝐴 ∈ TosetRel → ( I ↾ ∪ ∪ 𝐴) = ( I ↾ ∪ ∪ ◡𝐴))
33	32, 7	eqsstrrd 4021	. . . . . . . 8 ⊢ (𝐴 ∈ TosetRel → ( I ↾ ∪ ∪ ◡𝐴) ⊆ 𝐴)
34		coss2 5855	. . . . . . . 8 ⊢ (( I ↾ ∪ ∪ ◡𝐴) ⊆ 𝐴 → (◡𝐴 ∘ ( I ↾ ∪ ∪ ◡𝐴)) ⊆ (◡𝐴 ∘ 𝐴))
35	33, 34	syl 17	. . . . . . 7 ⊢ (𝐴 ∈ TosetRel → (◡𝐴 ∘ ( I ↾ ∪ ∪ ◡𝐴)) ⊆ (◡𝐴 ∘ 𝐴))
36	29, 35	eqsstrrid 4031	. . . . . 6 ⊢ (𝐴 ∈ TosetRel → ◡𝐴 ⊆ (◡𝐴 ∘ 𝐴))
37	26, 36	unssd 4186	. . . . 5 ⊢ (𝐴 ∈ TosetRel → (𝐴 ∪ ◡𝐴) ⊆ (◡𝐴 ∘ 𝐴))
38	17, 37	sstrd 3992	. . . 4 ⊢ (𝐴 ∈ TosetRel → (dom 𝐴 × dom 𝐴) ⊆ (◡𝐴 ∘ 𝐴))
39	14, 38	eqsstrrd 4021	. . 3 ⊢ (𝐴 ∈ TosetRel → (∪ ∪ 𝐴 × ∪ ∪ 𝐴) ⊆ (◡𝐴 ∘ 𝐴))
40	10, 39	jca 513	. 2 ⊢ (𝐴 ∈ TosetRel → ((𝐴 ∘ 𝐴) ⊆ 𝐴 ∧ (∪ ∪ 𝐴 × ∪ ∪ 𝐴) ⊆ (◡𝐴 ∘ 𝐴)))
41		eqid 2733	. . 3 ⊢ ∪ ∪ 𝐴 = ∪ ∪ 𝐴
42	41	isdir 18548	. 2 ⊢ (𝐴 ∈ TosetRel → (𝐴 ∈ DirRel ↔ ((Rel 𝐴 ∧ ( I ↾ ∪ ∪ 𝐴) ⊆ 𝐴) ∧ ((𝐴 ∘ 𝐴) ⊆ 𝐴 ∧ (∪ ∪ 𝐴 × ∪ ∪ 𝐴) ⊆ (◡𝐴 ∘ 𝐴)))))
43	8, 40, 42	mpbir2and 712	1 ⊢ (𝐴 ∈ TosetRel → 𝐴 ∈ DirRel)

Syntax hints:   → wi 4   ∧ wa 397   = wceq 1542   ∈ wcel 2107   ∪ cun 3946   ∩ cin 3947   ⊆ wss 3948  ∪ cuni 4908   I cid 5573   × cxp 5674  ^◡ccnv 5675  dom cdm 5676  ran crn 5677   ↾ cres 5678   ∘ ccom 5680  Rel wrel 5681  PosetRelcps 18514   TosetRel ctsr 18515  DirRelcdir 18544

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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-12 2172  ax-ext 2704  ax-sep 5299  ax-nul 5306  ax-pr 5427

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-fun 6543  df-ps 18516  df-tsr 18517  df-dir 18546

This theorem is referenced by: (None)