Theorem tsrdir 17850

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 17833	. . . 4 ⊢ (𝐴 ∈ TosetRel → 𝐴 ∈ PosetRel)
2		psrel 17815	. . . 4 ⊢ (𝐴 ∈ PosetRel → Rel 𝐴)
3	1, 2	syl 17	. . 3 ⊢ (𝐴 ∈ TosetRel → Rel 𝐴)
4		psref2 17816	. . . . 5 ⊢ (𝐴 ∈ PosetRel → (𝐴 ∩ ◡𝐴) = ( I ↾ ∪ ∪ 𝐴))
5		inss1 4207	. . . . 5 ⊢ (𝐴 ∩ ◡𝐴) ⊆ 𝐴
6	4, 5	eqsstrrdi 4024	. . . 4 ⊢ (𝐴 ∈ PosetRel → ( I ↾ ∪ ∪ 𝐴) ⊆ 𝐴)
7	1, 6	syl 17	. . 3 ⊢ (𝐴 ∈ TosetRel → ( I ↾ ∪ ∪ 𝐴) ⊆ 𝐴)
8	3, 7	jca 514	. 2 ⊢ (𝐴 ∈ TosetRel → (Rel 𝐴 ∧ ( I ↾ ∪ ∪ 𝐴) ⊆ 𝐴))
9		pstr2 17817	. . . 4 ⊢ (𝐴 ∈ PosetRel → (𝐴 ∘ 𝐴) ⊆ 𝐴)
10	1, 9	syl 17	. . 3 ⊢ (𝐴 ∈ TosetRel → (𝐴 ∘ 𝐴) ⊆ 𝐴)
11		psdmrn 17819	. . . . . . 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 497	. . . . 5 ⊢ (𝐴 ∈ TosetRel → dom 𝐴 = ∪ ∪ 𝐴)
14	13	sqxpeqd 5589	. . . 4 ⊢ (𝐴 ∈ TosetRel → (dom 𝐴 × dom 𝐴) = (∪ ∪ 𝐴 × ∪ ∪ 𝐴))
15		eqid 2823	. . . . . . 7 ⊢ dom 𝐴 = dom 𝐴
16	15	istsr 17829	. . . . . 6 ⊢ (𝐴 ∈ TosetRel ↔ (𝐴 ∈ PosetRel ∧ (dom 𝐴 × dom 𝐴) ⊆ (𝐴 ∪ ◡𝐴)))
17	16	simprbi 499	. . . . 5 ⊢ (𝐴 ∈ TosetRel → (dom 𝐴 × dom 𝐴) ⊆ (𝐴 ∪ ◡𝐴))
18		relcoi2 6130	. . . . . . . 8 ⊢ (Rel 𝐴 → (( I ↾ ∪ ∪ 𝐴) ∘ 𝐴) = 𝐴)
19	3, 18	syl 17	. . . . . . 7 ⊢ (𝐴 ∈ TosetRel → (( I ↾ ∪ ∪ 𝐴) ∘ 𝐴) = 𝐴)
20		cnvresid 6435	. . . . . . . . 9 ⊢ ◡( I ↾ ∪ ∪ 𝐴) = ( I ↾ ∪ ∪ 𝐴)
21		cnvss 5745	. . . . . . . . . 10 ⊢ (( I ↾ ∪ ∪ 𝐴) ⊆ 𝐴 → ◡( I ↾ ∪ ∪ 𝐴) ⊆ ◡𝐴)
22	7, 21	syl 17	. . . . . . . . 9 ⊢ (𝐴 ∈ TosetRel → ◡( I ↾ ∪ ∪ 𝐴) ⊆ ◡𝐴)
23	20, 22	eqsstrrid 4018	. . . . . . . 8 ⊢ (𝐴 ∈ TosetRel → ( I ↾ ∪ ∪ 𝐴) ⊆ ◡𝐴)
24		coss1 5728	. . . . . . . 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 4008	. . . . . 6 ⊢ (𝐴 ∈ TosetRel → 𝐴 ⊆ (◡𝐴 ∘ 𝐴))
27		relcnv 5969	. . . . . . . 8 ⊢ Rel ◡𝐴
28		relcoi1 6131	. . . . . . . 8 ⊢ (Rel ◡𝐴 → (◡𝐴 ∘ ( I ↾ ∪ ∪ ◡𝐴)) = ◡𝐴)
29	27, 28	ax-mp 5	. . . . . . 7 ⊢ (◡𝐴 ∘ ( I ↾ ∪ ∪ ◡𝐴)) = ◡𝐴
30		relcnvfld 6133	. . . . . . . . . . 11 ⊢ (Rel 𝐴 → ∪ ∪ 𝐴 = ∪ ∪ ◡𝐴)
31	3, 30	syl 17	. . . . . . . . . 10 ⊢ (𝐴 ∈ TosetRel → ∪ ∪ 𝐴 = ∪ ∪ ◡𝐴)
32	31	reseq2d 5855	. . . . . . . . 9 ⊢ (𝐴 ∈ TosetRel → ( I ↾ ∪ ∪ 𝐴) = ( I ↾ ∪ ∪ ◡𝐴))
33	32, 7	eqsstrrd 4008	. . . . . . . 8 ⊢ (𝐴 ∈ TosetRel → ( I ↾ ∪ ∪ ◡𝐴) ⊆ 𝐴)
34		coss2 5729	. . . . . . . 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 4018	. . . . . 6 ⊢ (𝐴 ∈ TosetRel → ◡𝐴 ⊆ (◡𝐴 ∘ 𝐴))
37	26, 36	unssd 4164	. . . . 5 ⊢ (𝐴 ∈ TosetRel → (𝐴 ∪ ◡𝐴) ⊆ (◡𝐴 ∘ 𝐴))
38	17, 37	sstrd 3979	. . . 4 ⊢ (𝐴 ∈ TosetRel → (dom 𝐴 × dom 𝐴) ⊆ (◡𝐴 ∘ 𝐴))
39	14, 38	eqsstrrd 4008	. . 3 ⊢ (𝐴 ∈ TosetRel → (∪ ∪ 𝐴 × ∪ ∪ 𝐴) ⊆ (◡𝐴 ∘ 𝐴))
40	10, 39	jca 514	. 2 ⊢ (𝐴 ∈ TosetRel → ((𝐴 ∘ 𝐴) ⊆ 𝐴 ∧ (∪ ∪ 𝐴 × ∪ ∪ 𝐴) ⊆ (◡𝐴 ∘ 𝐴)))
41		eqid 2823	. . 3 ⊢ ∪ ∪ 𝐴 = ∪ ∪ 𝐴
42	41	isdir 17844	. 2 ⊢ (𝐴 ∈ TosetRel → (𝐴 ∈ DirRel ↔ ((Rel 𝐴 ∧ ( I ↾ ∪ ∪ 𝐴) ⊆ 𝐴) ∧ ((𝐴 ∘ 𝐴) ⊆ 𝐴 ∧ (∪ ∪ 𝐴 × ∪ ∪ 𝐴) ⊆ (◡𝐴 ∘ 𝐴)))))
43	8, 40, 42	mpbir2and 711	1 ⊢ (𝐴 ∈ TosetRel → 𝐴 ∈ DirRel)

Syntax hints:   → wi 4   ∧ wa 398   = wceq 1537   ∈ wcel 2114   ∪ cun 3936   ∩ cin 3937   ⊆ wss 3938  ∪ cuni 4840   I cid 5461   × cxp 5555  ^◡ccnv 5556  dom cdm 5557  ran crn 5558   ↾ cres 5559   ∘ ccom 5561  Rel wrel 5562  PosetRelcps 17810   TosetRel ctsr 17811  DirRelcdir 17840

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 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-sep 5205  ax-nul 5212  ax-pr 5332

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ral 3145  df-rex 3146  df-rab 3149  df-v 3498  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-pw 4543  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4841  df-br 5069  df-opab 5131  df-id 5462  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-res 5569  df-ima 5570  df-fun 6359  df-ps 17812  df-tsr 17813  df-dir 17842

This theorem is referenced by: (None)