Theorem lautset 40452

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 3463	. 2 ⊢ (𝐾 ∈ 𝐴 → 𝐾 ∈ V)
2		lautset.i	. . 3 ⊢ 𝐼 = (LAut‘𝐾)
3		fveq2 6842	. . . . . . . . 9 ⊢ (𝑘 = 𝐾 → (Base‘𝑘) = (Base‘𝐾))
4		lautset.b	. . . . . . . . 9 ⊢ 𝐵 = (Base‘𝐾)
5	3, 4	eqtr4di 2790	. . . . . . . 8 ⊢ (𝑘 = 𝐾 → (Base‘𝑘) = 𝐵)
6	5	f1oeq2d 6778	. . . . . . 7 ⊢ (𝑘 = 𝐾 → (𝑓:(Base‘𝑘)–1-1-onto→(Base‘𝑘) ↔ 𝑓:𝐵–1-1-onto→(Base‘𝑘)))
7		f1oeq3 6772	. . . . . . . 8 ⊢ ((Base‘𝑘) = 𝐵 → (𝑓:𝐵–1-1-onto→(Base‘𝑘) ↔ 𝑓:𝐵–1-1-onto→𝐵))
8	5, 7	syl 17	. . . . . . 7 ⊢ (𝑘 = 𝐾 → (𝑓:𝐵–1-1-onto→(Base‘𝑘) ↔ 𝑓:𝐵–1-1-onto→𝐵))
9	6, 8	bitrd 279	. . . . . 6 ⊢ (𝑘 = 𝐾 → (𝑓:(Base‘𝑘)–1-1-onto→(Base‘𝑘) ↔ 𝑓:𝐵–1-1-onto→𝐵))
10		fveq2 6842	. . . . . . . . . . 11 ⊢ (𝑘 = 𝐾 → (le‘𝑘) = (le‘𝐾))
11		lautset.l	. . . . . . . . . . 11 ⊢ ≤ = (le‘𝐾)
12	10, 11	eqtr4di 2790	. . . . . . . . . 10 ⊢ (𝑘 = 𝐾 → (le‘𝑘) = ≤ )
13	12	breqd 5111	. . . . . . . . 9 ⊢ (𝑘 = 𝐾 → (𝑥(le‘𝑘)𝑦 ↔ 𝑥 ≤ 𝑦))
14	12	breqd 5111	. . . . . . . . 9 ⊢ (𝑘 = 𝐾 → ((𝑓‘𝑥)(le‘𝑘)(𝑓‘𝑦) ↔ (𝑓‘𝑥) ≤ (𝑓‘𝑦)))
15	13, 14	bibi12d 345	. . . . . . . 8 ⊢ (𝑘 = 𝐾 → ((𝑥(le‘𝑘)𝑦 ↔ (𝑓‘𝑥)(le‘𝑘)(𝑓‘𝑦)) ↔ (𝑥 ≤ 𝑦 ↔ (𝑓‘𝑥) ≤ (𝑓‘𝑦))))
16	5, 15	raleqbidv 3318	. . . . . . 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 3318	. . . . . 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 633	. . . . 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 2803	. . . 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 40359	. . . 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 6856	. . . . . . . 8 ⊢ 𝐵 ∈ V
22	21, 21	mapval 8787	. . . . . . 7 ⊢ (𝐵 ↑m 𝐵) = {𝑓 ∣ 𝑓:𝐵⟶𝐵}
23		ovex 7401	. . . . . . 7 ⊢ (𝐵 ↑m 𝐵) ∈ V
24	22, 23	eqeltrri 2834	. . . . . 6 ⊢ {𝑓 ∣ 𝑓:𝐵⟶𝐵} ∈ V
25		f1of 6782	. . . . . . 7 ⊢ (𝑓:𝐵–1-1-onto→𝐵 → 𝑓:𝐵⟶𝐵)
26	25	ss2abi 4020	. . . . . 6 ⊢ {𝑓 ∣ 𝑓:𝐵–1-1-onto→𝐵} ⊆ {𝑓 ∣ 𝑓:𝐵⟶𝐵}
27	24, 26	ssexi 5269	. . . . 5 ⊢ {𝑓 ∣ 𝑓:𝐵–1-1-onto→𝐵} ∈ V
28		simpl 482	. . . . . 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 4020	. . . . 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 5269	. . . 4 ⊢ {𝑓 ∣ (𝑓:𝐵–1-1-onto→𝐵 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 ≤ 𝑦 ↔ (𝑓‘𝑥) ≤ (𝑓‘𝑦)))} ∈ V
31	19, 20, 30	fvmpt 6949	. . 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 2784	. 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 17	1 ⊢ (𝐾 ∈ 𝐴 → 𝐼 = {𝑓 ∣ (𝑓:𝐵–1-1-onto→𝐵 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 ≤ 𝑦 ↔ (𝑓‘𝑥) ≤ (𝑓‘𝑦)))})

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114  {cab 2715  ∀wral 3052  Vcvv 3442   class class class wbr 5100  ⟶wf 6496  –1-1-onto→wf1o 6499  ‘cfv 6500  (class class class)co 7368   ↑_m cmap 8775  Basecbs 17148  lecple 17196  LAutclaut 40355

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-sbc 3743  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5527  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-ov 7371  df-oprab 7372  df-mpo 7373  df-map 8777  df-laut 40359

This theorem is referenced by:  islaut  40453