Theorem istos 18136

Description: The predicate "is a toset". (Contributed by FL, 17-Nov-2014.)

Hypotheses

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

Assertion

Ref	Expression
istos	⊢ (𝐾 ∈ Toset ↔ (𝐾 ∈ Poset ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 ≤ 𝑦 ∨ 𝑦 ≤ 𝑥)))

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

Allowed substitution hints:   𝐾(𝑥,𝑦)

Proof of Theorem istos

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

Step	Hyp	Ref	Expression
1		fveq2 6774	. . . 4 ⊢ (𝑓 = 𝐾 → (Base‘𝑓) = (Base‘𝐾))
2		fveq2 6774	. . . . 5 ⊢ (𝑓 = 𝐾 → (le‘𝑓) = (le‘𝐾))
3	2	sbceq1d 3721	. . . 4 ⊢ (𝑓 = 𝐾 → ([(le‘𝑓) / 𝑟]∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 (𝑥𝑟𝑦 ∨ 𝑦𝑟𝑥) ↔ [(le‘𝐾) / 𝑟]∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 (𝑥𝑟𝑦 ∨ 𝑦𝑟𝑥)))
4	1, 3	sbceqbid 3723	. . 3 ⊢ (𝑓 = 𝐾 → ([(Base‘𝑓) / 𝑏][(le‘𝑓) / 𝑟]∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 (𝑥𝑟𝑦 ∨ 𝑦𝑟𝑥) ↔ [(Base‘𝐾) / 𝑏][(le‘𝐾) / 𝑟]∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 (𝑥𝑟𝑦 ∨ 𝑦𝑟𝑥)))
5		fvex 6787	. . . 4 ⊢ (Base‘𝐾) ∈ V
6		fvex 6787	. . . 4 ⊢ (le‘𝐾) ∈ V
7		istos.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝐾)
8		eqtr 2761	. . . . . . . . 9 ⊢ ((𝑏 = (Base‘𝐾) ∧ (Base‘𝐾) = 𝐵) → 𝑏 = 𝐵)
9		istos.l	. . . . . . . . . 10 ⊢ ≤ = (le‘𝐾)
10		eqtr 2761	. . . . . . . . . . . . 13 ⊢ ((𝑟 = (le‘𝐾) ∧ (le‘𝐾) = ≤ ) → 𝑟 = ≤ )
11		breq 5076	. . . . . . . . . . . . . . . . 17 ⊢ (𝑟 = ≤ → (𝑥𝑟𝑦 ↔ 𝑥 ≤ 𝑦))
12		breq 5076	. . . . . . . . . . . . . . . . 17 ⊢ (𝑟 = ≤ → (𝑦𝑟𝑥 ↔ 𝑦 ≤ 𝑥))
13	11, 12	orbi12d 916	. . . . . . . . . . . . . . . 16 ⊢ (𝑟 = ≤ → ((𝑥𝑟𝑦 ∨ 𝑦𝑟𝑥) ↔ (𝑥 ≤ 𝑦 ∨ 𝑦 ≤ 𝑥)))
14	13	2ralbidv 3129	. . . . . . . . . . . . . . 15 ⊢ (𝑟 = ≤ → (∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 (𝑥𝑟𝑦 ∨ 𝑦𝑟𝑥) ↔ ∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 (𝑥 ≤ 𝑦 ∨ 𝑦 ≤ 𝑥)))
15		raleq 3342	. . . . . . . . . . . . . . . 16 ⊢ (𝑏 = 𝐵 → (∀𝑦 ∈ 𝑏 (𝑥 ≤ 𝑦 ∨ 𝑦 ≤ 𝑥) ↔ ∀𝑦 ∈ 𝐵 (𝑥 ≤ 𝑦 ∨ 𝑦 ≤ 𝑥)))
16	15	raleqbi1dv 3340	. . . . . . . . . . . . . . 15 ⊢ (𝑏 = 𝐵 → (∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 (𝑥 ≤ 𝑦 ∨ 𝑦 ≤ 𝑥) ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 ≤ 𝑦 ∨ 𝑦 ≤ 𝑥)))
17	14, 16	sylan9bb 510	. . . . . . . . . . . . . 14 ⊢ ((𝑟 = ≤ ∧ 𝑏 = 𝐵) → (∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 (𝑥𝑟𝑦 ∨ 𝑦𝑟𝑥) ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 ≤ 𝑦 ∨ 𝑦 ≤ 𝑥)))
18	17	ex 413	. . . . . . . . . . . . 13 ⊢ (𝑟 = ≤ → (𝑏 = 𝐵 → (∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 (𝑥𝑟𝑦 ∨ 𝑦𝑟𝑥) ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 ≤ 𝑦 ∨ 𝑦 ≤ 𝑥))))
19	10, 18	syl 17	. . . . . . . . . . . 12 ⊢ ((𝑟 = (le‘𝐾) ∧ (le‘𝐾) = ≤ ) → (𝑏 = 𝐵 → (∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 (𝑥𝑟𝑦 ∨ 𝑦𝑟𝑥) ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 ≤ 𝑦 ∨ 𝑦 ≤ 𝑥))))
20	19	expcom 414	. . . . . . . . . . 11 ⊢ ((le‘𝐾) = ≤ → (𝑟 = (le‘𝐾) → (𝑏 = 𝐵 → (∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 (𝑥𝑟𝑦 ∨ 𝑦𝑟𝑥) ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 ≤ 𝑦 ∨ 𝑦 ≤ 𝑥)))))
21	20	eqcoms 2746	. . . . . . . . . 10 ⊢ ( ≤ = (le‘𝐾) → (𝑟 = (le‘𝐾) → (𝑏 = 𝐵 → (∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 (𝑥𝑟𝑦 ∨ 𝑦𝑟𝑥) ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 ≤ 𝑦 ∨ 𝑦 ≤ 𝑥)))))
22	9, 21	ax-mp 5	. . . . . . . . 9 ⊢ (𝑟 = (le‘𝐾) → (𝑏 = 𝐵 → (∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 (𝑥𝑟𝑦 ∨ 𝑦𝑟𝑥) ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 ≤ 𝑦 ∨ 𝑦 ≤ 𝑥))))
23	8, 22	syl5com 31	. . . . . . . 8 ⊢ ((𝑏 = (Base‘𝐾) ∧ (Base‘𝐾) = 𝐵) → (𝑟 = (le‘𝐾) → (∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 (𝑥𝑟𝑦 ∨ 𝑦𝑟𝑥) ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 ≤ 𝑦 ∨ 𝑦 ≤ 𝑥))))
24	23	expcom 414	. . . . . . 7 ⊢ ((Base‘𝐾) = 𝐵 → (𝑏 = (Base‘𝐾) → (𝑟 = (le‘𝐾) → (∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 (𝑥𝑟𝑦 ∨ 𝑦𝑟𝑥) ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 ≤ 𝑦 ∨ 𝑦 ≤ 𝑥)))))
25	24	eqcoms 2746	. . . . . 6 ⊢ (𝐵 = (Base‘𝐾) → (𝑏 = (Base‘𝐾) → (𝑟 = (le‘𝐾) → (∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 (𝑥𝑟𝑦 ∨ 𝑦𝑟𝑥) ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 ≤ 𝑦 ∨ 𝑦 ≤ 𝑥)))))
26	7, 25	ax-mp 5	. . . . 5 ⊢ (𝑏 = (Base‘𝐾) → (𝑟 = (le‘𝐾) → (∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 (𝑥𝑟𝑦 ∨ 𝑦𝑟𝑥) ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 ≤ 𝑦 ∨ 𝑦 ≤ 𝑥))))
27	26	imp 407	. . . 4 ⊢ ((𝑏 = (Base‘𝐾) ∧ 𝑟 = (le‘𝐾)) → (∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 (𝑥𝑟𝑦 ∨ 𝑦𝑟𝑥) ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 ≤ 𝑦 ∨ 𝑦 ≤ 𝑥)))
28	5, 6, 27	sbc2ie 3799	. . 3 ⊢ ([(Base‘𝐾) / 𝑏][(le‘𝐾) / 𝑟]∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 (𝑥𝑟𝑦 ∨ 𝑦𝑟𝑥) ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 ≤ 𝑦 ∨ 𝑦 ≤ 𝑥))
29	4, 28	bitrdi 287	. 2 ⊢ (𝑓 = 𝐾 → ([(Base‘𝑓) / 𝑏][(le‘𝑓) / 𝑟]∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 (𝑥𝑟𝑦 ∨ 𝑦𝑟𝑥) ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 ≤ 𝑦 ∨ 𝑦 ≤ 𝑥)))
30		df-toset 18135	. 2 ⊢ Toset = {𝑓 ∈ Poset ∣ [(Base‘𝑓) / 𝑏][(le‘𝑓) / 𝑟]∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 (𝑥𝑟𝑦 ∨ 𝑦𝑟𝑥)}
31	29, 30	elrab2 3627	1 ⊢ (𝐾 ∈ Toset ↔ (𝐾 ∈ Poset ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 ≤ 𝑦 ∨ 𝑦 ≤ 𝑥)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   ∨ wo 844   = wceq 1539   ∈ wcel 2106  ∀wral 3064  [wsbc 3716   class class class wbr 5074  ‘cfv 6433  Basecbs 16912  lecple 16969  Posetcpo 18025  Tosetctos 18134

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-nul 5230

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-sbc 3717  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-iota 6391  df-fv 6441  df-toset 18135

This theorem is referenced by:  tosso  18137  tospos  18138  tleile  18139  zntoslem  20764  resstos  31245  odutos  31246  xrstos  31288  xrge0omnd  31337