Theorem isdrs 18258

Description: Property of being a directed set. (Contributed by Stefan O'Rear, 1-Feb-2015.)

Hypotheses

Ref	Expression
isdrs.b	⊢ 𝐵 = (Base‘𝐾)
isdrs.l	⊢ ≤ = (le‘𝐾)

Assertion

Ref	Expression
isdrs	⊢ (𝐾 ∈ Dirset ↔ (𝐾 ∈ Proset ∧ 𝐵 ≠ ∅ ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐵 (𝑥 ≤ 𝑧 ∧ 𝑦 ≤ 𝑧)))

Distinct variable groups:   𝑥,𝐾,𝑦,𝑧   𝑥,𝐵,𝑦,𝑧   𝑥, ≤ ,𝑦,𝑧

Proof of Theorem isdrs

Dummy variables 𝑓 𝑏 𝑟 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 6882	. . . . . 6 ⊢ (𝑓 = 𝐾 → (Base‘𝑓) = (Base‘𝐾))
2		isdrs.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝐾)
3	1, 2	eqtr4di 2782	. . . . 5 ⊢ (𝑓 = 𝐾 → (Base‘𝑓) = 𝐵)
4		fveq2 6882	. . . . . . 7 ⊢ (𝑓 = 𝐾 → (le‘𝑓) = (le‘𝐾))
5		isdrs.l	. . . . . . 7 ⊢ ≤ = (le‘𝐾)
6	4, 5	eqtr4di 2782	. . . . . 6 ⊢ (𝑓 = 𝐾 → (le‘𝑓) = ≤ )
7	6	sbceq1d 3775	. . . . 5 ⊢ (𝑓 = 𝐾 → ([(le‘𝑓) / 𝑟](𝑏 ≠ ∅ ∧ ∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 ∃𝑧 ∈ 𝑏 (𝑥𝑟𝑧 ∧ 𝑦𝑟𝑧)) ↔ [ ≤ / 𝑟](𝑏 ≠ ∅ ∧ ∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 ∃𝑧 ∈ 𝑏 (𝑥𝑟𝑧 ∧ 𝑦𝑟𝑧))))
8	3, 7	sbceqbid 3777	. . . 4 ⊢ (𝑓 = 𝐾 → ([(Base‘𝑓) / 𝑏][(le‘𝑓) / 𝑟](𝑏 ≠ ∅ ∧ ∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 ∃𝑧 ∈ 𝑏 (𝑥𝑟𝑧 ∧ 𝑦𝑟𝑧)) ↔ [𝐵 / 𝑏][ ≤ / 𝑟](𝑏 ≠ ∅ ∧ ∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 ∃𝑧 ∈ 𝑏 (𝑥𝑟𝑧 ∧ 𝑦𝑟𝑧))))
9	2	fvexi 6896	. . . . 5 ⊢ 𝐵 ∈ V
10	5	fvexi 6896	. . . . 5 ⊢ ≤ ∈ V
11		neeq1 2995	. . . . . . 7 ⊢ (𝑏 = 𝐵 → (𝑏 ≠ ∅ ↔ 𝐵 ≠ ∅))
12	11	adantr 480	. . . . . 6 ⊢ ((𝑏 = 𝐵 ∧ 𝑟 = ≤ ) → (𝑏 ≠ ∅ ↔ 𝐵 ≠ ∅))
13		rexeq 3313	. . . . . . . . 9 ⊢ (𝑏 = 𝐵 → (∃𝑧 ∈ 𝑏 (𝑥𝑟𝑧 ∧ 𝑦𝑟𝑧) ↔ ∃𝑧 ∈ 𝐵 (𝑥𝑟𝑧 ∧ 𝑦𝑟𝑧)))
14	13	raleqbi1dv 3325	. . . . . . . 8 ⊢ (𝑏 = 𝐵 → (∀𝑦 ∈ 𝑏 ∃𝑧 ∈ 𝑏 (𝑥𝑟𝑧 ∧ 𝑦𝑟𝑧) ↔ ∀𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐵 (𝑥𝑟𝑧 ∧ 𝑦𝑟𝑧)))
15	14	raleqbi1dv 3325	. . . . . . 7 ⊢ (𝑏 = 𝐵 → (∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 ∃𝑧 ∈ 𝑏 (𝑥𝑟𝑧 ∧ 𝑦𝑟𝑧) ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐵 (𝑥𝑟𝑧 ∧ 𝑦𝑟𝑧)))
16		breq 5141	. . . . . . . . . 10 ⊢ (𝑟 = ≤ → (𝑥𝑟𝑧 ↔ 𝑥 ≤ 𝑧))
17		breq 5141	. . . . . . . . . 10 ⊢ (𝑟 = ≤ → (𝑦𝑟𝑧 ↔ 𝑦 ≤ 𝑧))
18	16, 17	anbi12d 630	. . . . . . . . 9 ⊢ (𝑟 = ≤ → ((𝑥𝑟𝑧 ∧ 𝑦𝑟𝑧) ↔ (𝑥 ≤ 𝑧 ∧ 𝑦 ≤ 𝑧)))
19	18	rexbidv 3170	. . . . . . . 8 ⊢ (𝑟 = ≤ → (∃𝑧 ∈ 𝐵 (𝑥𝑟𝑧 ∧ 𝑦𝑟𝑧) ↔ ∃𝑧 ∈ 𝐵 (𝑥 ≤ 𝑧 ∧ 𝑦 ≤ 𝑧)))
20	19	2ralbidv 3210	. . . . . . 7 ⊢ (𝑟 = ≤ → (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐵 (𝑥𝑟𝑧 ∧ 𝑦𝑟𝑧) ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐵 (𝑥 ≤ 𝑧 ∧ 𝑦 ≤ 𝑧)))
21	15, 20	sylan9bb 509	. . . . . 6 ⊢ ((𝑏 = 𝐵 ∧ 𝑟 = ≤ ) → (∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 ∃𝑧 ∈ 𝑏 (𝑥𝑟𝑧 ∧ 𝑦𝑟𝑧) ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐵 (𝑥 ≤ 𝑧 ∧ 𝑦 ≤ 𝑧)))
22	12, 21	anbi12d 630	. . . . 5 ⊢ ((𝑏 = 𝐵 ∧ 𝑟 = ≤ ) → ((𝑏 ≠ ∅ ∧ ∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 ∃𝑧 ∈ 𝑏 (𝑥𝑟𝑧 ∧ 𝑦𝑟𝑧)) ↔ (𝐵 ≠ ∅ ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐵 (𝑥 ≤ 𝑧 ∧ 𝑦 ≤ 𝑧))))
23	9, 10, 22	sbc2ie 3853	. . . 4 ⊢ ([𝐵 / 𝑏][ ≤ / 𝑟](𝑏 ≠ ∅ ∧ ∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 ∃𝑧 ∈ 𝑏 (𝑥𝑟𝑧 ∧ 𝑦𝑟𝑧)) ↔ (𝐵 ≠ ∅ ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐵 (𝑥 ≤ 𝑧 ∧ 𝑦 ≤ 𝑧)))
24	8, 23	bitrdi 287	. . 3 ⊢ (𝑓 = 𝐾 → ([(Base‘𝑓) / 𝑏][(le‘𝑓) / 𝑟](𝑏 ≠ ∅ ∧ ∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 ∃𝑧 ∈ 𝑏 (𝑥𝑟𝑧 ∧ 𝑦𝑟𝑧)) ↔ (𝐵 ≠ ∅ ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐵 (𝑥 ≤ 𝑧 ∧ 𝑦 ≤ 𝑧))))
25		df-drs 18253	. . 3 ⊢ Dirset = {𝑓 ∈ Proset ∣ [(Base‘𝑓) / 𝑏][(le‘𝑓) / 𝑟](𝑏 ≠ ∅ ∧ ∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 ∃𝑧 ∈ 𝑏 (𝑥𝑟𝑧 ∧ 𝑦𝑟𝑧))}
26	24, 25	elrab2 3679	. 2 ⊢ (𝐾 ∈ Dirset ↔ (𝐾 ∈ Proset ∧ (𝐵 ≠ ∅ ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐵 (𝑥 ≤ 𝑧 ∧ 𝑦 ≤ 𝑧))))
27		3anass 1092	. 2 ⊢ ((𝐾 ∈ Proset ∧ 𝐵 ≠ ∅ ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐵 (𝑥 ≤ 𝑧 ∧ 𝑦 ≤ 𝑧)) ↔ (𝐾 ∈ Proset ∧ (𝐵 ≠ ∅ ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐵 (𝑥 ≤ 𝑧 ∧ 𝑦 ≤ 𝑧))))
28	26, 27	bitr4i 278	1 ⊢ (𝐾 ∈ Dirset ↔ (𝐾 ∈ Proset ∧ 𝐵 ≠ ∅ ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐵 (𝑥 ≤ 𝑧 ∧ 𝑦 ≤ 𝑧)))

Syntax hints:   ↔ wb 205   ∧ wa 395   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098   ≠ wne 2932  ∀wral 3053  ∃wrex 3062  [wsbc 3770  ∅c0 4315   class class class wbr 5139  ‘cfv 6534  Basecbs 17145  lecple 17205   Proset cproset 18250  Dirsetcdrs 18251

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2695  ax-nul 5297

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-ne 2933  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-sbc 3771  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-br 5140  df-iota 6486  df-fv 6542  df-drs 18253

This theorem is referenced by:  drsdir  18259  drsprs  18260  drsbn0  18261  isdrs2  18263  isipodrs  18494