Theorem isdrs 17536

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 6645	. . . . . 6 ⊢ (𝑓 = 𝐾 → (Base‘𝑓) = (Base‘𝐾))
2		isdrs.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝐾)
3	1, 2	eqtr4di 2851	. . . . 5 ⊢ (𝑓 = 𝐾 → (Base‘𝑓) = 𝐵)
4		fveq2 6645	. . . . . . 7 ⊢ (𝑓 = 𝐾 → (le‘𝑓) = (le‘𝐾))
5		isdrs.l	. . . . . . 7 ⊢ ≤ = (le‘𝐾)
6	4, 5	eqtr4di 2851	. . . . . 6 ⊢ (𝑓 = 𝐾 → (le‘𝑓) = ≤ )
7	6	sbceq1d 3725	. . . . 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 3727	. . . 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 6659	. . . . 5 ⊢ 𝐵 ∈ V
10	5	fvexi 6659	. . . . 5 ⊢ ≤ ∈ V
11		neeq1 3049	. . . . . . 7 ⊢ (𝑏 = 𝐵 → (𝑏 ≠ ∅ ↔ 𝐵 ≠ ∅))
12	11	adantr 484	. . . . . 6 ⊢ ((𝑏 = 𝐵 ∧ 𝑟 = ≤ ) → (𝑏 ≠ ∅ ↔ 𝐵 ≠ ∅))
13		rexeq 3359	. . . . . . . . 9 ⊢ (𝑏 = 𝐵 → (∃𝑧 ∈ 𝑏 (𝑥𝑟𝑧 ∧ 𝑦𝑟𝑧) ↔ ∃𝑧 ∈ 𝐵 (𝑥𝑟𝑧 ∧ 𝑦𝑟𝑧)))
14	13	raleqbi1dv 3356	. . . . . . . 8 ⊢ (𝑏 = 𝐵 → (∀𝑦 ∈ 𝑏 ∃𝑧 ∈ 𝑏 (𝑥𝑟𝑧 ∧ 𝑦𝑟𝑧) ↔ ∀𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐵 (𝑥𝑟𝑧 ∧ 𝑦𝑟𝑧)))
15	14	raleqbi1dv 3356	. . . . . . 7 ⊢ (𝑏 = 𝐵 → (∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 ∃𝑧 ∈ 𝑏 (𝑥𝑟𝑧 ∧ 𝑦𝑟𝑧) ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐵 (𝑥𝑟𝑧 ∧ 𝑦𝑟𝑧)))
16		breq 5032	. . . . . . . . . 10 ⊢ (𝑟 = ≤ → (𝑥𝑟𝑧 ↔ 𝑥 ≤ 𝑧))
17		breq 5032	. . . . . . . . . 10 ⊢ (𝑟 = ≤ → (𝑦𝑟𝑧 ↔ 𝑦 ≤ 𝑧))
18	16, 17	anbi12d 633	. . . . . . . . 9 ⊢ (𝑟 = ≤ → ((𝑥𝑟𝑧 ∧ 𝑦𝑟𝑧) ↔ (𝑥 ≤ 𝑧 ∧ 𝑦 ≤ 𝑧)))
19	18	rexbidv 3256	. . . . . . . 8 ⊢ (𝑟 = ≤ → (∃𝑧 ∈ 𝐵 (𝑥𝑟𝑧 ∧ 𝑦𝑟𝑧) ↔ ∃𝑧 ∈ 𝐵 (𝑥 ≤ 𝑧 ∧ 𝑦 ≤ 𝑧)))
20	19	2ralbidv 3164	. . . . . . 7 ⊢ (𝑟 = ≤ → (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐵 (𝑥𝑟𝑧 ∧ 𝑦𝑟𝑧) ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐵 (𝑥 ≤ 𝑧 ∧ 𝑦 ≤ 𝑧)))
21	15, 20	sylan9bb 513	. . . . . 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 633	. . . . 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 3796	. . . 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	syl6bb 290	. . 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 17531	. . 3 ⊢ Dirset = {𝑓 ∈ Proset ∣ [(Base‘𝑓) / 𝑏][(le‘𝑓) / 𝑟](𝑏 ≠ ∅ ∧ ∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 ∃𝑧 ∈ 𝑏 (𝑥𝑟𝑧 ∧ 𝑦𝑟𝑧))}
26	24, 25	elrab2 3631	. 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 281	1 ⊢ (𝐾 ∈ Dirset ↔ (𝐾 ∈ Proset ∧ 𝐵 ≠ ∅ ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐵 (𝑥 ≤ 𝑧 ∧ 𝑦 ≤ 𝑧)))

Syntax hints:   ↔ wb 209   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111   ≠ wne 2987  ∀wral 3106  ∃wrex 3107  [wsbc 3720  ∅c0 4243   class class class wbr 5030  ‘cfv 6324  Basecbs 16475  lecple 16564   Proset cproset 17528  Dirsetcdrs 17529

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-nul 5174

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-iota 6283  df-fv 6332  df-drs 17531

This theorem is referenced by:  drsdir  17537  drsprs  17538  drsbn0  17539  isdrs2  17541  isipodrs  17763