Theorem lautset 40963

Description: The set of lattice automorphisms. (Contributed by NM, 11-May-2012.)

Hypotheses

Ref	Expression
lautset.b	⊢ 𝐵 = (Base‘𝐾)
lautset.l	⊢ ≤ = (le‘𝐾)
lautset.i	⊢ 𝐼 = (LAut‘𝐾)

Assertion

Ref	Expression
lautset	⊢ (𝐾 ∈ 𝐴 → 𝐼 = {𝑓 ∣ (𝑓:𝐵–1-1-onto→𝐵 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 ≤ 𝑦 ↔ (𝑓‘𝑥) ≤ (𝑓‘𝑦)))})

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

Allowed substitution hints:   𝐴(𝑥, 𝑦, 𝑓)   𝐼(𝑥, 𝑦, 𝑓)   ≤ (𝑥, 𝑦)

Proof of Theorem lautset

Dummy variable 𝑘 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex 3474	. 2 ⊢ (𝐾 ∈ 𝐴 → 𝐾 ∈ V)
2		lautset.i	. . 3 ⊢ 𝐼 = (LAut‘𝐾)
3		fveq2 6882	. . . . . . . . 9 ⊢ (𝑘 = 𝐾 → (Base‘𝑘) = (Base‘𝐾))
4		lautset.b	. . . . . . . . 9 ⊢ 𝐵 = (Base‘𝐾)
5	3, 4	eqtr4di 2815	. . . . . . . 8 ⊢ (𝑘 = 𝐾 → (Base‘𝑘) = 𝐵)
6	5	f1oeq2d 6817	. . . . . . 7 ⊢ (𝑘 = 𝐾 → (𝑓:(Base‘𝑘)–1-1-onto→(Base‘𝑘) ↔ 𝑓:𝐵–1-1-onto→(Base‘𝑘)))
7		f1oeq3 6811	. . . . . . . 8 ⊢ ((Base‘𝑘) = 𝐵 → (𝑓:𝐵–1-1-onto→(Base‘𝑘) ↔ 𝑓:𝐵–1-1-onto→𝐵))
8	5, 7	syl 18	. . . . . . 7 ⊢ (𝑘 = 𝐾 → (𝑓:𝐵–1-1-onto→(Base‘𝑘) ↔ 𝑓:𝐵–1-1-onto→𝐵))
9	6, 8	bitrd 282	. . . . . 6 ⊢ (𝑘 = 𝐾 → (𝑓:(Base‘𝑘)–1-1-onto→(Base‘𝑘) ↔ 𝑓:𝐵–1-1-onto→𝐵))
10		fveq2 6882	. . . . . . . . . . 11 ⊢ (𝑘 = 𝐾 → (le‘𝑘) = (le‘𝐾))
11		lautset.l	. . . . . . . . . . 11 ⊢ ≤ = (le‘𝐾)
12	10, 11	eqtr4di 2815	. . . . . . . . . 10 ⊢ (𝑘 = 𝐾 → (le‘𝑘) = ≤ )
13	12	breqd 5118	. . . . . . . . 9 ⊢ (𝑘 = 𝐾 → (𝑥(le‘𝑘)𝑦 ↔ 𝑥 ≤ 𝑦))
14	12	breqd 5118	. . . . . . . . 9 ⊢ (𝑘 = 𝐾 → ((𝑓‘𝑥)(le‘𝑘)(𝑓‘𝑦) ↔ (𝑓‘𝑥) ≤ (𝑓‘𝑦)))
15	13, 14	bibi12d 348	. . . . . . . 8 ⊢ (𝑘 = 𝐾 → ((𝑥(le‘𝑘)𝑦 ↔ (𝑓‘𝑥)(le‘𝑘)(𝑓‘𝑦)) ↔ (𝑥 ≤ 𝑦 ↔ (𝑓‘𝑥) ≤ (𝑓‘𝑦))))
16	5, 15	raleqbidv 3336	. . . . . . 7 ⊢ (𝑘 = 𝐾 → (∀𝑦 ∈ (Base‘𝑘)(𝑥(le‘𝑘)𝑦 ↔ (𝑓‘𝑥)(le‘𝑘)(𝑓‘𝑦)) ↔ ∀𝑦 ∈ 𝐵 (𝑥 ≤ 𝑦 ↔ (𝑓‘𝑥) ≤ (𝑓‘𝑦))))
17	5, 16	raleqbidv 3336	. . . . . 6 ⊢ (𝑘 = 𝐾 → (∀𝑥 ∈ (Base‘𝑘)∀𝑦 ∈ (Base‘𝑘)(𝑥(le‘𝑘)𝑦 ↔ (𝑓‘𝑥)(le‘𝑘)(𝑓‘𝑦)) ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 ≤ 𝑦 ↔ (𝑓‘𝑥) ≤ (𝑓‘𝑦))))
18	9, 17	anbi12d 644	. . . . 5 ⊢ (𝑘 = 𝐾 → ((𝑓:(Base‘𝑘)–1-1-onto→(Base‘𝑘) ∧ ∀𝑥 ∈ (Base‘𝑘)∀𝑦 ∈ (Base‘𝑘)(𝑥(le‘𝑘)𝑦 ↔ (𝑓‘𝑥)(le‘𝑘)(𝑓‘𝑦))) ↔ (𝑓:𝐵–1-1-onto→𝐵 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 ≤ 𝑦 ↔ (𝑓‘𝑥) ≤ (𝑓‘𝑦)))))
19	18	abbidv 2828	. . . 4 ⊢ (𝑘 = 𝐾 → {𝑓 ∣ (𝑓:(Base‘𝑘)–1-1-onto→(Base‘𝑘) ∧ ∀𝑥 ∈ (Base‘𝑘)∀𝑦 ∈ (Base‘𝑘)(𝑥(le‘𝑘)𝑦 ↔ (𝑓‘𝑥)(le‘𝑘)(𝑓‘𝑦)))} = {𝑓 ∣ (𝑓:𝐵–1-1-onto→𝐵 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 ≤ 𝑦 ↔ (𝑓‘𝑥) ≤ (𝑓‘𝑦)))})
20		df-laut 40870	. . . 4 ⊢ LAut = (𝑘 ∈ V ↦ {𝑓 ∣ (𝑓:(Base‘𝑘)–1-1-onto→(Base‘𝑘) ∧ ∀𝑥 ∈ (Base‘𝑘)∀𝑦 ∈ (Base‘𝑘)(𝑥(le‘𝑘)𝑦 ↔ (𝑓‘𝑥)(le‘𝑘)(𝑓‘𝑦)))})
21	4	fvexi 6896	. . . . . . . 8 ⊢ 𝐵 ∈ V
22	21, 21	mapval 8841	. . . . . . 7 ⊢ (𝐵 ↑m 𝐵) = {𝑓 ∣ 𝑓:𝐵⟶𝐵}
23		ovex 7450	. . . . . . 7 ⊢ (𝐵 ↑m 𝐵) ∈ V
24	22, 23	eqeltrri 2859	. . . . . 6 ⊢ {𝑓 ∣ 𝑓:𝐵⟶𝐵} ∈ V
25		f1of 6821	. . . . . . 7 ⊢ (𝑓:𝐵–1-1-onto→𝐵 → 𝑓:𝐵⟶𝐵)
26	25	ss2abi 4017	. . . . . 6 ⊢ {𝑓 ∣ 𝑓:𝐵–1-1-onto→𝐵} ⊆ {𝑓 ∣ 𝑓:𝐵⟶𝐵}
27	24, 26	ssexi 5291	. . . . 5 ⊢ {𝑓 ∣ 𝑓:𝐵–1-1-onto→𝐵} ∈ V
28		simpl 488	. . . . . 6 ⊢ ((𝑓:𝐵–1-1-onto→𝐵 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 ≤ 𝑦 ↔ (𝑓‘𝑥) ≤ (𝑓‘𝑦))) → 𝑓:𝐵–1-1-onto→𝐵)
29	28	ss2abi 4017	. . . . 5 ⊢ {𝑓 ∣ (𝑓:𝐵–1-1-onto→𝐵 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 ≤ 𝑦 ↔ (𝑓‘𝑥) ≤ (𝑓‘𝑦)))} ⊆ {𝑓 ∣ 𝑓:𝐵–1-1-onto→𝐵}
30	27, 29	ssexi 5291	. . . 4 ⊢ {𝑓 ∣ (𝑓:𝐵–1-1-onto→𝐵 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 ≤ 𝑦 ↔ (𝑓‘𝑥) ≤ (𝑓‘𝑦)))} ∈ V
31	19, 20, 30	fvmpt 6990	. . 3 ⊢ (𝐾 ∈ V → (LAut‘𝐾) = {𝑓 ∣ (𝑓:𝐵–1-1-onto→𝐵 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 ≤ 𝑦 ↔ (𝑓‘𝑥) ≤ (𝑓‘𝑦)))})
32	2, 31	eqtrid 2809	. 2 ⊢ (𝐾 ∈ V → 𝐼 = {𝑓 ∣ (𝑓:𝐵–1-1-onto→𝐵 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 ≤ 𝑦 ↔ (𝑓‘𝑥) ≤ (𝑓‘𝑦)))})
33	1, 32	syl 18	1 ⊢ (𝐾 ∈ 𝐴 → 𝐼 = {𝑓 ∣ (𝑓:𝐵–1-1-onto→𝐵 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 ≤ 𝑦 ↔ (𝑓‘𝑥) ≤ (𝑓‘𝑦)))})

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145  {cab 2740  ∀wral 3078  Vcvv 3453   class class class wbr 5107  ⟶wf 6533  –1-1-onto→wf1o 6536  ‘cfv 6537  (class class class)co 7417   ↑_m cmap 8830  Basecbs 17307  lecple 17355  LAutclaut 40866

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-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-sep 5255  ax-nul 5267  ax-pow 5334  ax-pr 5402  ax-un 7740

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-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-rab 3415  df-v 3455  df-sbc 3743  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-br 5108  df-opab 5172  df-mpt 5191  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7420  df-oprab 7421  df-mpo 7422  df-map 8832  df-laut 40870

This theorem is used by:  islaut  40964