Theorem lautset 37378

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 3459	. 2 ⊢ (𝐾 ∈ 𝐴 → 𝐾 ∈ V)
2		lautset.i	. . 3 ⊢ 𝐼 = (LAut‘𝐾)
3		fveq2 6645	. . . . . . . . 9 ⊢ (𝑘 = 𝐾 → (Base‘𝑘) = (Base‘𝐾))
4		lautset.b	. . . . . . . . 9 ⊢ 𝐵 = (Base‘𝐾)
5	3, 4	eqtr4di 2851	. . . . . . . 8 ⊢ (𝑘 = 𝐾 → (Base‘𝑘) = 𝐵)
6	5	f1oeq2d 6586	. . . . . . 7 ⊢ (𝑘 = 𝐾 → (𝑓:(Base‘𝑘)–1-1-onto→(Base‘𝑘) ↔ 𝑓:𝐵–1-1-onto→(Base‘𝑘)))
7		f1oeq3 6581	. . . . . . . 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 282	. . . . . 6 ⊢ (𝑘 = 𝐾 → (𝑓:(Base‘𝑘)–1-1-onto→(Base‘𝑘) ↔ 𝑓:𝐵–1-1-onto→𝐵))
10		fveq2 6645	. . . . . . . . . . 11 ⊢ (𝑘 = 𝐾 → (le‘𝑘) = (le‘𝐾))
11		lautset.l	. . . . . . . . . . 11 ⊢ ≤ = (le‘𝐾)
12	10, 11	eqtr4di 2851	. . . . . . . . . 10 ⊢ (𝑘 = 𝐾 → (le‘𝑘) = ≤ )
13	12	breqd 5041	. . . . . . . . 9 ⊢ (𝑘 = 𝐾 → (𝑥(le‘𝑘)𝑦 ↔ 𝑥 ≤ 𝑦))
14	12	breqd 5041	. . . . . . . . 9 ⊢ (𝑘 = 𝐾 → ((𝑓‘𝑥)(le‘𝑘)(𝑓‘𝑦) ↔ (𝑓‘𝑥) ≤ (𝑓‘𝑦)))
15	13, 14	bibi12d 349	. . . . . . . 8 ⊢ (𝑘 = 𝐾 → ((𝑥(le‘𝑘)𝑦 ↔ (𝑓‘𝑥)(le‘𝑘)(𝑓‘𝑦)) ↔ (𝑥 ≤ 𝑦 ↔ (𝑓‘𝑥) ≤ (𝑓‘𝑦))))
16	5, 15	raleqbidv 3354	. . . . . . 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 3354	. . . . . 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 2862	. . . 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 37285	. . . 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 6659	. . . . . . . 8 ⊢ 𝐵 ∈ V
22	21, 21	mapval 8401	. . . . . . 7 ⊢ (𝐵 ↑m 𝐵) = {𝑓 ∣ 𝑓:𝐵⟶𝐵}
23		ovex 7168	. . . . . . 7 ⊢ (𝐵 ↑m 𝐵) ∈ V
24	22, 23	eqeltrri 2887	. . . . . 6 ⊢ {𝑓 ∣ 𝑓:𝐵⟶𝐵} ∈ V
25		f1of 6590	. . . . . . 7 ⊢ (𝑓:𝐵–1-1-onto→𝐵 → 𝑓:𝐵⟶𝐵)
26	25	ss2abi 3994	. . . . . 6 ⊢ {𝑓 ∣ 𝑓:𝐵–1-1-onto→𝐵} ⊆ {𝑓 ∣ 𝑓:𝐵⟶𝐵}
27	24, 26	ssexi 5190	. . . . 5 ⊢ {𝑓 ∣ 𝑓:𝐵–1-1-onto→𝐵} ∈ V
28		simpl 486	. . . . . 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 3994	. . . . 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 5190	. . . 4 ⊢ {𝑓 ∣ (𝑓:𝐵–1-1-onto→𝐵 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 ≤ 𝑦 ↔ (𝑓‘𝑥) ≤ (𝑓‘𝑦)))} ∈ V
31	19, 20, 30	fvmpt 6745	. . 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	syl5eq 2845	. 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 209   ∧ wa 399   = wceq 1538   ∈ wcel 2111  {cab 2776  ∀wral 3106  Vcvv 3441   class class class wbr 5030  ⟶wf 6320  –1-1-onto→wf1o 6323  ‘cfv 6324  (class class class)co 7135   ↑_m cmap 8389  Basecbs 16475  lecple 16564  LAutclaut 37281

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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-ov 7138  df-oprab 7139  df-mpo 7140  df-map 8391  df-laut 37285

This theorem is referenced by:  islaut  37379