Theorem tsrdir 18570

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 18553	. . . 4 ⊢ (𝐴 ∈ TosetRel → 𝐴 ∈ PosetRel)
2		psrel 18535	. . . 4 ⊢ (𝐴 ∈ PosetRel → Rel 𝐴)
3	1, 2	syl 17	. . 3 ⊢ (𝐴 ∈ TosetRel → Rel 𝐴)
4		psref2 18536	. . . . 5 ⊢ (𝐴 ∈ PosetRel → (𝐴 ∩ ◡𝐴) = ( I ↾ ∪ ∪ 𝐴))
5		inss1 4203	. . . . 5 ⊢ (𝐴 ∩ ◡𝐴) ⊆ 𝐴
6	4, 5	eqsstrrdi 3995	. . . 4 ⊢ (𝐴 ∈ PosetRel → ( I ↾ ∪ ∪ 𝐴) ⊆ 𝐴)
7	1, 6	syl 17	. . 3 ⊢ (𝐴 ∈ TosetRel → ( I ↾ ∪ ∪ 𝐴) ⊆ 𝐴)
8	3, 7	jca 511	. 2 ⊢ (𝐴 ∈ TosetRel → (Rel 𝐴 ∧ ( I ↾ ∪ ∪ 𝐴) ⊆ 𝐴))
9		pstr2 18537	. . . 4 ⊢ (𝐴 ∈ PosetRel → (𝐴 ∘ 𝐴) ⊆ 𝐴)
10	1, 9	syl 17	. . 3 ⊢ (𝐴 ∈ TosetRel → (𝐴 ∘ 𝐴) ⊆ 𝐴)
11		psdmrn 18539	. . . . . . 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 494	. . . . 5 ⊢ (𝐴 ∈ TosetRel → dom 𝐴 = ∪ ∪ 𝐴)
14	13	sqxpeqd 5673	. . . 4 ⊢ (𝐴 ∈ TosetRel → (dom 𝐴 × dom 𝐴) = (∪ ∪ 𝐴 × ∪ ∪ 𝐴))
15		eqid 2730	. . . . . . 7 ⊢ dom 𝐴 = dom 𝐴
16	15	istsr 18549	. . . . . 6 ⊢ (𝐴 ∈ TosetRel ↔ (𝐴 ∈ PosetRel ∧ (dom 𝐴 × dom 𝐴) ⊆ (𝐴 ∪ ◡𝐴)))
17	16	simprbi 496	. . . . 5 ⊢ (𝐴 ∈ TosetRel → (dom 𝐴 × dom 𝐴) ⊆ (𝐴 ∪ ◡𝐴))
18		relcoi2 6253	. . . . . . . 8 ⊢ (Rel 𝐴 → (( I ↾ ∪ ∪ 𝐴) ∘ 𝐴) = 𝐴)
19	3, 18	syl 17	. . . . . . 7 ⊢ (𝐴 ∈ TosetRel → (( I ↾ ∪ ∪ 𝐴) ∘ 𝐴) = 𝐴)
20		cnvresid 6598	. . . . . . . . 9 ⊢ ◡( I ↾ ∪ ∪ 𝐴) = ( I ↾ ∪ ∪ 𝐴)
21		cnvss 5839	. . . . . . . . . 10 ⊢ (( I ↾ ∪ ∪ 𝐴) ⊆ 𝐴 → ◡( I ↾ ∪ ∪ 𝐴) ⊆ ◡𝐴)
22	7, 21	syl 17	. . . . . . . . 9 ⊢ (𝐴 ∈ TosetRel → ◡( I ↾ ∪ ∪ 𝐴) ⊆ ◡𝐴)
23	20, 22	eqsstrrid 3989	. . . . . . . 8 ⊢ (𝐴 ∈ TosetRel → ( I ↾ ∪ ∪ 𝐴) ⊆ ◡𝐴)
24		coss1 5822	. . . . . . . 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 3985	. . . . . 6 ⊢ (𝐴 ∈ TosetRel → 𝐴 ⊆ (◡𝐴 ∘ 𝐴))
27		relcnv 6078	. . . . . . . 8 ⊢ Rel ◡𝐴
28		relcoi1 6254	. . . . . . . 8 ⊢ (Rel ◡𝐴 → (◡𝐴 ∘ ( I ↾ ∪ ∪ ◡𝐴)) = ◡𝐴)
29	27, 28	ax-mp 5	. . . . . . 7 ⊢ (◡𝐴 ∘ ( I ↾ ∪ ∪ ◡𝐴)) = ◡𝐴
30		relcnvfld 6256	. . . . . . . . . . 11 ⊢ (Rel 𝐴 → ∪ ∪ 𝐴 = ∪ ∪ ◡𝐴)
31	3, 30	syl 17	. . . . . . . . . 10 ⊢ (𝐴 ∈ TosetRel → ∪ ∪ 𝐴 = ∪ ∪ ◡𝐴)
32	31	reseq2d 5953	. . . . . . . . 9 ⊢ (𝐴 ∈ TosetRel → ( I ↾ ∪ ∪ 𝐴) = ( I ↾ ∪ ∪ ◡𝐴))
33	32, 7	eqsstrrd 3985	. . . . . . . 8 ⊢ (𝐴 ∈ TosetRel → ( I ↾ ∪ ∪ ◡𝐴) ⊆ 𝐴)
34		coss2 5823	. . . . . . . 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 3989	. . . . . 6 ⊢ (𝐴 ∈ TosetRel → ◡𝐴 ⊆ (◡𝐴 ∘ 𝐴))
37	26, 36	unssd 4158	. . . . 5 ⊢ (𝐴 ∈ TosetRel → (𝐴 ∪ ◡𝐴) ⊆ (◡𝐴 ∘ 𝐴))
38	17, 37	sstrd 3960	. . . 4 ⊢ (𝐴 ∈ TosetRel → (dom 𝐴 × dom 𝐴) ⊆ (◡𝐴 ∘ 𝐴))
39	14, 38	eqsstrrd 3985	. . 3 ⊢ (𝐴 ∈ TosetRel → (∪ ∪ 𝐴 × ∪ ∪ 𝐴) ⊆ (◡𝐴 ∘ 𝐴))
40	10, 39	jca 511	. 2 ⊢ (𝐴 ∈ TosetRel → ((𝐴 ∘ 𝐴) ⊆ 𝐴 ∧ (∪ ∪ 𝐴 × ∪ ∪ 𝐴) ⊆ (◡𝐴 ∘ 𝐴)))
41		eqid 2730	. . 3 ⊢ ∪ ∪ 𝐴 = ∪ ∪ 𝐴
42	41	isdir 18564	. 2 ⊢ (𝐴 ∈ TosetRel → (𝐴 ∈ DirRel ↔ ((Rel 𝐴 ∧ ( I ↾ ∪ ∪ 𝐴) ⊆ 𝐴) ∧ ((𝐴 ∘ 𝐴) ⊆ 𝐴 ∧ (∪ ∪ 𝐴 × ∪ ∪ 𝐴) ⊆ (◡𝐴 ∘ 𝐴)))))
43	8, 40, 42	mpbir2and 713	1 ⊢ (𝐴 ∈ TosetRel → 𝐴 ∈ DirRel)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109   ∪ cun 3915   ∩ cin 3916   ⊆ wss 3917  ∪ cuni 4874   I cid 5535   × cxp 5639  ^◡ccnv 5640  dom cdm 5641  ran crn 5642   ↾ cres 5643   ∘ ccom 5645  Rel wrel 5646  PosetRelcps 18530   TosetRel ctsr 18531  DirRelcdir 18560

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-12 2178  ax-ext 2702  ax-sep 5254  ax-nul 5264  ax-pr 5390

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-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-br 5111  df-opab 5173  df-id 5536  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-fun 6516  df-ps 18532  df-tsr 18533  df-dir 18562

This theorem is referenced by: (None)