Theorem tsrdir 17964

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 17947	. . . 4 ⊢ (𝐴 ∈ TosetRel → 𝐴 ∈ PosetRel)
2		psrel 17929	. . . 4 ⊢ (𝐴 ∈ PosetRel → Rel 𝐴)
3	1, 2	syl 17	. . 3 ⊢ (𝐴 ∈ TosetRel → Rel 𝐴)
4		psref2 17930	. . . . 5 ⊢ (𝐴 ∈ PosetRel → (𝐴 ∩ ◡𝐴) = ( I ↾ ∪ ∪ 𝐴))
5		inss1 4119	. . . . 5 ⊢ (𝐴 ∩ ◡𝐴) ⊆ 𝐴
6	4, 5	eqsstrrdi 3932	. . . 4 ⊢ (𝐴 ∈ PosetRel → ( I ↾ ∪ ∪ 𝐴) ⊆ 𝐴)
7	1, 6	syl 17	. . 3 ⊢ (𝐴 ∈ TosetRel → ( I ↾ ∪ ∪ 𝐴) ⊆ 𝐴)
8	3, 7	jca 515	. 2 ⊢ (𝐴 ∈ TosetRel → (Rel 𝐴 ∧ ( I ↾ ∪ ∪ 𝐴) ⊆ 𝐴))
9		pstr2 17931	. . . 4 ⊢ (𝐴 ∈ PosetRel → (𝐴 ∘ 𝐴) ⊆ 𝐴)
10	1, 9	syl 17	. . 3 ⊢ (𝐴 ∈ TosetRel → (𝐴 ∘ 𝐴) ⊆ 𝐴)
11		psdmrn 17933	. . . . . . 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 498	. . . . 5 ⊢ (𝐴 ∈ TosetRel → dom 𝐴 = ∪ ∪ 𝐴)
14	13	sqxpeqd 5557	. . . 4 ⊢ (𝐴 ∈ TosetRel → (dom 𝐴 × dom 𝐴) = (∪ ∪ 𝐴 × ∪ ∪ 𝐴))
15		eqid 2738	. . . . . . 7 ⊢ dom 𝐴 = dom 𝐴
16	15	istsr 17943	. . . . . 6 ⊢ (𝐴 ∈ TosetRel ↔ (𝐴 ∈ PosetRel ∧ (dom 𝐴 × dom 𝐴) ⊆ (𝐴 ∪ ◡𝐴)))
17	16	simprbi 500	. . . . 5 ⊢ (𝐴 ∈ TosetRel → (dom 𝐴 × dom 𝐴) ⊆ (𝐴 ∪ ◡𝐴))
18		relcoi2 6109	. . . . . . . 8 ⊢ (Rel 𝐴 → (( I ↾ ∪ ∪ 𝐴) ∘ 𝐴) = 𝐴)
19	3, 18	syl 17	. . . . . . 7 ⊢ (𝐴 ∈ TosetRel → (( I ↾ ∪ ∪ 𝐴) ∘ 𝐴) = 𝐴)
20		cnvresid 6418	. . . . . . . . 9 ⊢ ◡( I ↾ ∪ ∪ 𝐴) = ( I ↾ ∪ ∪ 𝐴)
21		cnvss 5715	. . . . . . . . . 10 ⊢ (( I ↾ ∪ ∪ 𝐴) ⊆ 𝐴 → ◡( I ↾ ∪ ∪ 𝐴) ⊆ ◡𝐴)
22	7, 21	syl 17	. . . . . . . . 9 ⊢ (𝐴 ∈ TosetRel → ◡( I ↾ ∪ ∪ 𝐴) ⊆ ◡𝐴)
23	20, 22	eqsstrrid 3926	. . . . . . . 8 ⊢ (𝐴 ∈ TosetRel → ( I ↾ ∪ ∪ 𝐴) ⊆ ◡𝐴)
24		coss1 5698	. . . . . . . 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 3916	. . . . . 6 ⊢ (𝐴 ∈ TosetRel → 𝐴 ⊆ (◡𝐴 ∘ 𝐴))
27		relcnv 5941	. . . . . . . 8 ⊢ Rel ◡𝐴
28		relcoi1 6110	. . . . . . . 8 ⊢ (Rel ◡𝐴 → (◡𝐴 ∘ ( I ↾ ∪ ∪ ◡𝐴)) = ◡𝐴)
29	27, 28	ax-mp 5	. . . . . . 7 ⊢ (◡𝐴 ∘ ( I ↾ ∪ ∪ ◡𝐴)) = ◡𝐴
30		relcnvfld 6112	. . . . . . . . . . 11 ⊢ (Rel 𝐴 → ∪ ∪ 𝐴 = ∪ ∪ ◡𝐴)
31	3, 30	syl 17	. . . . . . . . . 10 ⊢ (𝐴 ∈ TosetRel → ∪ ∪ 𝐴 = ∪ ∪ ◡𝐴)
32	31	reseq2d 5825	. . . . . . . . 9 ⊢ (𝐴 ∈ TosetRel → ( I ↾ ∪ ∪ 𝐴) = ( I ↾ ∪ ∪ ◡𝐴))
33	32, 7	eqsstrrd 3916	. . . . . . . 8 ⊢ (𝐴 ∈ TosetRel → ( I ↾ ∪ ∪ ◡𝐴) ⊆ 𝐴)
34		coss2 5699	. . . . . . . 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 3926	. . . . . 6 ⊢ (𝐴 ∈ TosetRel → ◡𝐴 ⊆ (◡𝐴 ∘ 𝐴))
37	26, 36	unssd 4076	. . . . 5 ⊢ (𝐴 ∈ TosetRel → (𝐴 ∪ ◡𝐴) ⊆ (◡𝐴 ∘ 𝐴))
38	17, 37	sstrd 3887	. . . 4 ⊢ (𝐴 ∈ TosetRel → (dom 𝐴 × dom 𝐴) ⊆ (◡𝐴 ∘ 𝐴))
39	14, 38	eqsstrrd 3916	. . 3 ⊢ (𝐴 ∈ TosetRel → (∪ ∪ 𝐴 × ∪ ∪ 𝐴) ⊆ (◡𝐴 ∘ 𝐴))
40	10, 39	jca 515	. 2 ⊢ (𝐴 ∈ TosetRel → ((𝐴 ∘ 𝐴) ⊆ 𝐴 ∧ (∪ ∪ 𝐴 × ∪ ∪ 𝐴) ⊆ (◡𝐴 ∘ 𝐴)))
41		eqid 2738	. . 3 ⊢ ∪ ∪ 𝐴 = ∪ ∪ 𝐴
42	41	isdir 17958	. 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 399   = wceq 1542   ∈ wcel 2114   ∪ cun 3841   ∩ cin 3842   ⊆ wss 3843  ∪ cuni 4796   I cid 5428   × cxp 5523  ^◡ccnv 5524  dom cdm 5525  ran crn 5526   ↾ cres 5527   ∘ ccom 5529  Rel wrel 5530  PosetRelcps 17924   TosetRel ctsr 17925  DirRelcdir 17954

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2710  ax-sep 5167  ax-nul 5174  ax-pr 5296

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ral 3058  df-rex 3059  df-rab 3062  df-v 3400  df-dif 3846  df-un 3848  df-in 3850  df-ss 3860  df-nul 4212  df-if 4415  df-pw 4490  df-sn 4517  df-pr 4519  df-op 4523  df-uni 4797  df-br 5031  df-opab 5093  df-id 5429  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-res 5537  df-ima 5538  df-fun 6341  df-ps 17926  df-tsr 17927  df-dir 17956

This theorem is referenced by: (None)