Theorem isdrs 17135

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 ∈ Preset ∧ 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 6352	. . . . . 6 ⊢ (𝑓 = 𝐾 → (Base‘𝑓) = (Base‘𝐾))
2		isdrs.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝐾)
3	1, 2	syl6eqr 2812	. . . . 5 ⊢ (𝑓 = 𝐾 → (Base‘𝑓) = 𝐵)
4		fveq2 6352	. . . . . . 7 ⊢ (𝑓 = 𝐾 → (le‘𝑓) = (le‘𝐾))
5		isdrs.l	. . . . . . 7 ⊢ ≤ = (le‘𝐾)
6	4, 5	syl6eqr 2812	. . . . . 6 ⊢ (𝑓 = 𝐾 → (le‘𝑓) = ≤ )
7	6	sbceq1d 3581	. . . . 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 3583	. . . 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		fvex 6362	. . . . . 6 ⊢ (Base‘𝐾) ∈ V
10	2, 9	eqeltri 2835	. . . . 5 ⊢ 𝐵 ∈ V
11		fvex 6362	. . . . . 6 ⊢ (le‘𝐾) ∈ V
12	5, 11	eqeltri 2835	. . . . 5 ⊢ ≤ ∈ V
13		neeq1 2994	. . . . . . 7 ⊢ (𝑏 = 𝐵 → (𝑏 ≠ ∅ ↔ 𝐵 ≠ ∅))
14	13	adantr 472	. . . . . 6 ⊢ ((𝑏 = 𝐵 ∧ 𝑟 = ≤ ) → (𝑏 ≠ ∅ ↔ 𝐵 ≠ ∅))
15		rexeq 3278	. . . . . . . . 9 ⊢ (𝑏 = 𝐵 → (∃𝑧 ∈ 𝑏 (𝑥𝑟𝑧 ∧ 𝑦𝑟𝑧) ↔ ∃𝑧 ∈ 𝐵 (𝑥𝑟𝑧 ∧ 𝑦𝑟𝑧)))
16	15	raleqbi1dv 3285	. . . . . . . 8 ⊢ (𝑏 = 𝐵 → (∀𝑦 ∈ 𝑏 ∃𝑧 ∈ 𝑏 (𝑥𝑟𝑧 ∧ 𝑦𝑟𝑧) ↔ ∀𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐵 (𝑥𝑟𝑧 ∧ 𝑦𝑟𝑧)))
17	16	raleqbi1dv 3285	. . . . . . 7 ⊢ (𝑏 = 𝐵 → (∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 ∃𝑧 ∈ 𝑏 (𝑥𝑟𝑧 ∧ 𝑦𝑟𝑧) ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐵 (𝑥𝑟𝑧 ∧ 𝑦𝑟𝑧)))
18		breq 4806	. . . . . . . . . 10 ⊢ (𝑟 = ≤ → (𝑥𝑟𝑧 ↔ 𝑥 ≤ 𝑧))
19		breq 4806	. . . . . . . . . 10 ⊢ (𝑟 = ≤ → (𝑦𝑟𝑧 ↔ 𝑦 ≤ 𝑧))
20	18, 19	anbi12d 749	. . . . . . . . 9 ⊢ (𝑟 = ≤ → ((𝑥𝑟𝑧 ∧ 𝑦𝑟𝑧) ↔ (𝑥 ≤ 𝑧 ∧ 𝑦 ≤ 𝑧)))
21	20	rexbidv 3190	. . . . . . . 8 ⊢ (𝑟 = ≤ → (∃𝑧 ∈ 𝐵 (𝑥𝑟𝑧 ∧ 𝑦𝑟𝑧) ↔ ∃𝑧 ∈ 𝐵 (𝑥 ≤ 𝑧 ∧ 𝑦 ≤ 𝑧)))
22	21	2ralbidv 3127	. . . . . . 7 ⊢ (𝑟 = ≤ → (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐵 (𝑥𝑟𝑧 ∧ 𝑦𝑟𝑧) ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐵 (𝑥 ≤ 𝑧 ∧ 𝑦 ≤ 𝑧)))
23	17, 22	sylan9bb 738	. . . . . 6 ⊢ ((𝑏 = 𝐵 ∧ 𝑟 = ≤ ) → (∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 ∃𝑧 ∈ 𝑏 (𝑥𝑟𝑧 ∧ 𝑦𝑟𝑧) ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐵 (𝑥 ≤ 𝑧 ∧ 𝑦 ≤ 𝑧)))
24	14, 23	anbi12d 749	. . . . 5 ⊢ ((𝑏 = 𝐵 ∧ 𝑟 = ≤ ) → ((𝑏 ≠ ∅ ∧ ∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 ∃𝑧 ∈ 𝑏 (𝑥𝑟𝑧 ∧ 𝑦𝑟𝑧)) ↔ (𝐵 ≠ ∅ ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐵 (𝑥 ≤ 𝑧 ∧ 𝑦 ≤ 𝑧))))
25	10, 12, 24	sbc2ie 3646	. . . 4 ⊢ ([𝐵 / 𝑏][ ≤ / 𝑟](𝑏 ≠ ∅ ∧ ∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 ∃𝑧 ∈ 𝑏 (𝑥𝑟𝑧 ∧ 𝑦𝑟𝑧)) ↔ (𝐵 ≠ ∅ ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐵 (𝑥 ≤ 𝑧 ∧ 𝑦 ≤ 𝑧)))
26	8, 25	syl6bb 276	. . 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))))
27		df-drs 17130	. . 3 ⊢ Dirset = {𝑓 ∈ Preset ∣ [(Base‘𝑓) / 𝑏][(le‘𝑓) / 𝑟](𝑏 ≠ ∅ ∧ ∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 ∃𝑧 ∈ 𝑏 (𝑥𝑟𝑧 ∧ 𝑦𝑟𝑧))}
28	26, 27	elrab2 3507	. 2 ⊢ (𝐾 ∈ Dirset ↔ (𝐾 ∈ Preset ∧ (𝐵 ≠ ∅ ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐵 (𝑥 ≤ 𝑧 ∧ 𝑦 ≤ 𝑧))))
29		3anass 1081	. 2 ⊢ ((𝐾 ∈ Preset ∧ 𝐵 ≠ ∅ ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐵 (𝑥 ≤ 𝑧 ∧ 𝑦 ≤ 𝑧)) ↔ (𝐾 ∈ Preset ∧ (𝐵 ≠ ∅ ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐵 (𝑥 ≤ 𝑧 ∧ 𝑦 ≤ 𝑧))))
30	28, 29	bitr4i 267	1 ⊢ (𝐾 ∈ Dirset ↔ (𝐾 ∈ Preset ∧ 𝐵 ≠ ∅ ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐵 (𝑥 ≤ 𝑧 ∧ 𝑦 ≤ 𝑧)))

Syntax hints:   ↔ wb 196   ∧ wa 383   ∧ w3a 1072   = wceq 1632   ∈ wcel 2139   ≠ wne 2932  ∀wral 3050  ∃wrex 3051  Vcvv 3340  [wsbc 3576  ∅c0 4058   class class class wbr 4804  ‘cfv 6049  Basecbs 16059  lecple 16150   Preset cpreset 17127  Dirsetcdrs 17128

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-nul 4941

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-iota 6012  df-fv 6057  df-drs 17130

This theorem is referenced by:  drsdir  17136  drsprs  17137  drsbn0  17138  isdrs2  17140  isipodrs  17362