Theorem istos 18570

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 6877	. . . 4 ⊢ (𝑓 = 𝐾 → (Base‘𝑓) = (Base‘𝐾))
2		fveq2 6877	. . . . 5 ⊢ (𝑓 = 𝐾 → (le‘𝑓) = (le‘𝐾))
3	2	sbceq1d 3744	. . . 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 3746	. . 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 6890	. . . 4 ⊢ (Base‘𝐾) ∈ V
6		fvex 6890	. . . 4 ⊢ (le‘𝐾) ∈ V
7		istos.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝐾)
8		eqtr 2781	. . . . . . . . 9 ⊢ ((𝑏 = (Base‘𝐾) ∧ (Base‘𝐾) = 𝐵) → 𝑏 = 𝐵)
9		istos.l	. . . . . . . . . 10 ⊢ ≤ = (le‘𝐾)
10		eqtr 2781	. . . . . . . . . . . . 13 ⊢ ((𝑟 = (le‘𝐾) ∧ (le‘𝐾) = ≤ ) → 𝑟 = ≤ )
11		breq 5105	. . . . . . . . . . . . . . . . 17 ⊢ (𝑟 = ≤ → (𝑥𝑟𝑦 ↔ 𝑥 ≤ 𝑦))
12		breq 5105	. . . . . . . . . . . . . . . . 17 ⊢ (𝑟 = ≤ → (𝑦𝑟𝑥 ↔ 𝑦 ≤ 𝑥))
13	11, 12	orbi12d 932	. . . . . . . . . . . . . . . 16 ⊢ (𝑟 = ≤ → ((𝑥𝑟𝑦 ∨ 𝑦𝑟𝑥) ↔ (𝑥 ≤ 𝑦 ∨ 𝑦 ≤ 𝑥)))
14	13	2ralbidv 3227	. . . . . . . . . . . . . . 15 ⊢ (𝑟 = ≤ → (∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 (𝑥𝑟𝑦 ∨ 𝑦𝑟𝑥) ↔ ∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 (𝑥 ≤ 𝑦 ∨ 𝑦 ≤ 𝑥)))
15		raleq 3317	. . . . . . . . . . . . . . . 16 ⊢ (𝑏 = 𝐵 → (∀𝑦 ∈ 𝑏 (𝑥 ≤ 𝑦 ∨ 𝑦 ≤ 𝑥) ↔ ∀𝑦 ∈ 𝐵 (𝑥 ≤ 𝑦 ∨ 𝑦 ≤ 𝑥)))
16	15	raleqbi1dv 3330	. . . . . . . . . . . . . . 15 ⊢ (𝑏 = 𝐵 → (∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 (𝑥 ≤ 𝑦 ∨ 𝑦 ≤ 𝑥) ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 ≤ 𝑦 ∨ 𝑦 ≤ 𝑥)))
17	14, 16	sylan9bb 519	. . . . . . . . . . . . . 14 ⊢ ((𝑟 = ≤ ∧ 𝑏 = 𝐵) → (∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 (𝑥𝑟𝑦 ∨ 𝑦𝑟𝑥) ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 ≤ 𝑦 ∨ 𝑦 ≤ 𝑥)))
18	17	ex 418	. . . . . . . . . . . . 13 ⊢ (𝑟 = ≤ → (𝑏 = 𝐵 → (∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 (𝑥𝑟𝑦 ∨ 𝑦𝑟𝑥) ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 ≤ 𝑦 ∨ 𝑦 ≤ 𝑥))))
19	10, 18	syl 18	. . . . . . . . . . . 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 419	. . . . . . . . . . 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 2769	. . . . . . . . . 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 32	. . . . . . . 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 419	. . . . . . 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 2769	. . . . . 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 412	. . . 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 3814	. . 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 290	. 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 18569	. 2 ⊢ Toset = {𝑓 ∈ Poset ∣ [(Base‘𝑓) / 𝑏][(le‘𝑓) / 𝑟]∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 (𝑥𝑟𝑦 ∨ 𝑦𝑟𝑥)}
31	29, 30	elrab2 3649	1 ⊢ (𝐾 ∈ Toset ↔ (𝐾 ∈ Poset ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 ≤ 𝑦 ∨ 𝑦 ≤ 𝑥)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   = wceq 1570   ∈ wcel 2145  ∀wral 3077  [wsbc 3739   class class class wbr 5103  ‘cfv 6531  Basecbs 17367  lecple 17415  Posetcpo 18461  Tosetctos 18568

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2733  ax-nul 5260

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-ne 2957  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-sbc 3740  df-dif 3902  df-un 3904  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-iota 6487  df-fv 6539  df-toset 18569

This theorem is used by:  tosso  18571  tospos  18572  tleile  18573  resstos  18584  xrge0omnd  21731  zntoslem  21842  zsoring  28777  odutos  33511  xrstos  33553