Theorem tsrdir 18512

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 18495	. . . 4 ⊢ (𝐴 ∈ TosetRel → 𝐴 ∈ PosetRel)
2		psrel 18477	. . . 4 ⊢ (𝐴 ∈ PosetRel → Rel 𝐴)
3	1, 2	syl 17	. . 3 ⊢ (𝐴 ∈ TosetRel → Rel 𝐴)
4		psref2 18478	. . . . 5 ⊢ (𝐴 ∈ PosetRel → (𝐴 ∩ ◡𝐴) = ( I ↾ ∪ ∪ 𝐴))
5		inss1 4186	. . . . 5 ⊢ (𝐴 ∩ ◡𝐴) ⊆ 𝐴
6	4, 5	eqsstrrdi 3976	. . . 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 18479	. . . 4 ⊢ (𝐴 ∈ PosetRel → (𝐴 ∘ 𝐴) ⊆ 𝐴)
10	1, 9	syl 17	. . 3 ⊢ (𝐴 ∈ TosetRel → (𝐴 ∘ 𝐴) ⊆ 𝐴)
11		psdmrn 18481	. . . . . . 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 5651	. . . 4 ⊢ (𝐴 ∈ TosetRel → (dom 𝐴 × dom 𝐴) = (∪ ∪ 𝐴 × ∪ ∪ 𝐴))
15		eqid 2733	. . . . . . 7 ⊢ dom 𝐴 = dom 𝐴
16	15	istsr 18491	. . . . . 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 6229	. . . . . . . 8 ⊢ (Rel 𝐴 → (( I ↾ ∪ ∪ 𝐴) ∘ 𝐴) = 𝐴)
19	3, 18	syl 17	. . . . . . 7 ⊢ (𝐴 ∈ TosetRel → (( I ↾ ∪ ∪ 𝐴) ∘ 𝐴) = 𝐴)
20		cnvresid 6565	. . . . . . . . 9 ⊢ ◡( I ↾ ∪ ∪ 𝐴) = ( I ↾ ∪ ∪ 𝐴)
21		cnvss 5816	. . . . . . . . . 10 ⊢ (( I ↾ ∪ ∪ 𝐴) ⊆ 𝐴 → ◡( I ↾ ∪ ∪ 𝐴) ⊆ ◡𝐴)
22	7, 21	syl 17	. . . . . . . . 9 ⊢ (𝐴 ∈ TosetRel → ◡( I ↾ ∪ ∪ 𝐴) ⊆ ◡𝐴)
23	20, 22	eqsstrrid 3970	. . . . . . . 8 ⊢ (𝐴 ∈ TosetRel → ( I ↾ ∪ ∪ 𝐴) ⊆ ◡𝐴)
24		coss1 5799	. . . . . . . 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 3966	. . . . . 6 ⊢ (𝐴 ∈ TosetRel → 𝐴 ⊆ (◡𝐴 ∘ 𝐴))
27		relcnv 6057	. . . . . . . 8 ⊢ Rel ◡𝐴
28		relcoi1 6230	. . . . . . . 8 ⊢ (Rel ◡𝐴 → (◡𝐴 ∘ ( I ↾ ∪ ∪ ◡𝐴)) = ◡𝐴)
29	27, 28	ax-mp 5	. . . . . . 7 ⊢ (◡𝐴 ∘ ( I ↾ ∪ ∪ ◡𝐴)) = ◡𝐴
30		relcnvfld 6232	. . . . . . . . . . 11 ⊢ (Rel 𝐴 → ∪ ∪ 𝐴 = ∪ ∪ ◡𝐴)
31	3, 30	syl 17	. . . . . . . . . 10 ⊢ (𝐴 ∈ TosetRel → ∪ ∪ 𝐴 = ∪ ∪ ◡𝐴)
32	31	reseq2d 5932	. . . . . . . . 9 ⊢ (𝐴 ∈ TosetRel → ( I ↾ ∪ ∪ 𝐴) = ( I ↾ ∪ ∪ ◡𝐴))
33	32, 7	eqsstrrd 3966	. . . . . . . 8 ⊢ (𝐴 ∈ TosetRel → ( I ↾ ∪ ∪ ◡𝐴) ⊆ 𝐴)
34		coss2 5800	. . . . . . . 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 3970	. . . . . 6 ⊢ (𝐴 ∈ TosetRel → ◡𝐴 ⊆ (◡𝐴 ∘ 𝐴))
37	26, 36	unssd 4141	. . . . 5 ⊢ (𝐴 ∈ TosetRel → (𝐴 ∪ ◡𝐴) ⊆ (◡𝐴 ∘ 𝐴))
38	17, 37	sstrd 3941	. . . 4 ⊢ (𝐴 ∈ TosetRel → (dom 𝐴 × dom 𝐴) ⊆ (◡𝐴 ∘ 𝐴))
39	14, 38	eqsstrrd 3966	. . 3 ⊢ (𝐴 ∈ TosetRel → (∪ ∪ 𝐴 × ∪ ∪ 𝐴) ⊆ (◡𝐴 ∘ 𝐴))
40	10, 39	jca 511	. 2 ⊢ (𝐴 ∈ TosetRel → ((𝐴 ∘ 𝐴) ⊆ 𝐴 ∧ (∪ ∪ 𝐴 × ∪ ∪ 𝐴) ⊆ (◡𝐴 ∘ 𝐴)))
41		eqid 2733	. . 3 ⊢ ∪ ∪ 𝐴 = ∪ ∪ 𝐴
42	41	isdir 18506	. 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 1541   ∈ wcel 2113   ∪ cun 3896   ∩ cin 3897   ⊆ wss 3898  ∪ cuni 4858   I cid 5513   × cxp 5617  ^◡ccnv 5618  dom cdm 5619  ran crn 5620   ↾ cres 5621   ∘ ccom 5623  Rel wrel 5624  PosetRelcps 18472   TosetRel ctsr 18473  DirRelcdir 18502

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-12 2182  ax-ext 2705  ax-sep 5236  ax-nul 5246  ax-pr 5372

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-ral 3049  df-rex 3058  df-rab 3397  df-v 3439  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4283  df-if 4475  df-pw 4551  df-sn 4576  df-pr 4578  df-op 4582  df-uni 4859  df-br 5094  df-opab 5156  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-fun 6488  df-ps 18474  df-tsr 18475  df-dir 18504

This theorem is referenced by: (None)