Theorem istos 18371

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 6892	. . . 4 ⊢ (𝑓 = 𝐾 → (Base‘𝑓) = (Base‘𝐾))
2		fveq2 6892	. . . . 5 ⊢ (𝑓 = 𝐾 → (le‘𝑓) = (le‘𝐾))
3	2	sbceq1d 3783	. . . 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 3785	. . 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 6905	. . . 4 ⊢ (Base‘𝐾) ∈ V
6		fvex 6905	. . . 4 ⊢ (le‘𝐾) ∈ V
7		istos.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝐾)
8		eqtr 2756	. . . . . . . . 9 ⊢ ((𝑏 = (Base‘𝐾) ∧ (Base‘𝐾) = 𝐵) → 𝑏 = 𝐵)
9		istos.l	. . . . . . . . . 10 ⊢ ≤ = (le‘𝐾)
10		eqtr 2756	. . . . . . . . . . . . 13 ⊢ ((𝑟 = (le‘𝐾) ∧ (le‘𝐾) = ≤ ) → 𝑟 = ≤ )
11		breq 5151	. . . . . . . . . . . . . . . . 17 ⊢ (𝑟 = ≤ → (𝑥𝑟𝑦 ↔ 𝑥 ≤ 𝑦))
12		breq 5151	. . . . . . . . . . . . . . . . 17 ⊢ (𝑟 = ≤ → (𝑦𝑟𝑥 ↔ 𝑦 ≤ 𝑥))
13	11, 12	orbi12d 918	. . . . . . . . . . . . . . . 16 ⊢ (𝑟 = ≤ → ((𝑥𝑟𝑦 ∨ 𝑦𝑟𝑥) ↔ (𝑥 ≤ 𝑦 ∨ 𝑦 ≤ 𝑥)))
14	13	2ralbidv 3219	. . . . . . . . . . . . . . 15 ⊢ (𝑟 = ≤ → (∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 (𝑥𝑟𝑦 ∨ 𝑦𝑟𝑥) ↔ ∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 (𝑥 ≤ 𝑦 ∨ 𝑦 ≤ 𝑥)))
15		raleq 3323	. . . . . . . . . . . . . . . 16 ⊢ (𝑏 = 𝐵 → (∀𝑦 ∈ 𝑏 (𝑥 ≤ 𝑦 ∨ 𝑦 ≤ 𝑥) ↔ ∀𝑦 ∈ 𝐵 (𝑥 ≤ 𝑦 ∨ 𝑦 ≤ 𝑥)))
16	15	raleqbi1dv 3334	. . . . . . . . . . . . . . 15 ⊢ (𝑏 = 𝐵 → (∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 (𝑥 ≤ 𝑦 ∨ 𝑦 ≤ 𝑥) ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 ≤ 𝑦 ∨ 𝑦 ≤ 𝑥)))
17	14, 16	sylan9bb 511	. . . . . . . . . . . . . 14 ⊢ ((𝑟 = ≤ ∧ 𝑏 = 𝐵) → (∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 (𝑥𝑟𝑦 ∨ 𝑦𝑟𝑥) ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 ≤ 𝑦 ∨ 𝑦 ≤ 𝑥)))
18	17	ex 414	. . . . . . . . . . . . 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 415	. . . . . . . . . . 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 2741	. . . . . . . . . 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 415	. . . . . . 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 2741	. . . . . 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 408	. . . 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 3861	. . 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 18370	. 2 ⊢ Toset = {𝑓 ∈ Poset ∣ [(Base‘𝑓) / 𝑏][(le‘𝑓) / 𝑟]∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 (𝑥𝑟𝑦 ∨ 𝑦𝑟𝑥)}
31	29, 30	elrab2 3687	1 ⊢ (𝐾 ∈ Toset ↔ (𝐾 ∈ Poset ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 ≤ 𝑦 ∨ 𝑦 ≤ 𝑥)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 397   ∨ wo 846   = wceq 1542   ∈ wcel 2107  ∀wral 3062  [wsbc 3778   class class class wbr 5149  ‘cfv 6544  Basecbs 17144  lecple 17204  Posetcpo 18260  Tosetctos 18369

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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-nul 5307

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-sbc 3779  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-iota 6496  df-fv 6552  df-toset 18370

This theorem is referenced by:  tosso  18372  tospos  18373  tleile  18374  zntoslem  21112  resstos  32137  odutos  32138  xrstos  32180  xrge0omnd  32229