Theorem lautset 40896

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 3479	. 2 ⊢ (𝐾 ∈ 𝐴 → 𝐾 ∈ V)
2		lautset.i	. . 3 ⊢ 𝐼 = (LAut‘𝐾)
3		fveq2 6888	. . . . . . . . 9 ⊢ (𝑘 = 𝐾 → (Base‘𝑘) = (Base‘𝐾))
4		lautset.b	. . . . . . . . 9 ⊢ 𝐵 = (Base‘𝐾)
5	3, 4	eqtr4di 2819	. . . . . . . 8 ⊢ (𝑘 = 𝐾 → (Base‘𝑘) = 𝐵)
6	5	f1oeq2d 6823	. . . . . . 7 ⊢ (𝑘 = 𝐾 → (𝑓:(Base‘𝑘)–1-1-onto→(Base‘𝑘) ↔ 𝑓:𝐵–1-1-onto→(Base‘𝑘)))
7		f1oeq3 6817	. . . . . . . 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 6888	. . . . . . . . . . 11 ⊢ (𝑘 = 𝐾 → (le‘𝑘) = (le‘𝐾))
11		lautset.l	. . . . . . . . . . 11 ⊢ ≤ = (le‘𝐾)
12	10, 11	eqtr4di 2819	. . . . . . . . . 10 ⊢ (𝑘 = 𝐾 → (le‘𝑘) = ≤ )
13	12	breqd 5125	. . . . . . . . 9 ⊢ (𝑘 = 𝐾 → (𝑥(le‘𝑘)𝑦 ↔ 𝑥 ≤ 𝑦))
14	12	breqd 5125	. . . . . . . . 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 3341	. . . . . . 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 3341	. . . . . 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 2832	. . . 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 40803	. . . 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 6902	. . . . . . . 8 ⊢ 𝐵 ∈ V
22	21, 21	mapval 8844	. . . . . . 7 ⊢ (𝐵 ↑m 𝐵) = {𝑓 ∣ 𝑓:𝐵⟶𝐵}
23		ovex 7456	. . . . . . 7 ⊢ (𝐵 ↑m 𝐵) ∈ V
24	22, 23	eqeltrri 2863	. . . . . 6 ⊢ {𝑓 ∣ 𝑓:𝐵⟶𝐵} ∈ V
25		f1of 6827	. . . . . . 7 ⊢ (𝑓:𝐵–1-1-onto→𝐵 → 𝑓:𝐵⟶𝐵)
26	25	ss2abi 4023	. . . . . 6 ⊢ {𝑓 ∣ 𝑓:𝐵–1-1-onto→𝐵} ⊆ {𝑓 ∣ 𝑓:𝐵⟶𝐵}
27	24, 26	ssexi 5298	. . . . 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 4023	. . . . 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 5298	. . . 4 ⊢ {𝑓 ∣ (𝑓:𝐵–1-1-onto→𝐵 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 ≤ 𝑦 ↔ (𝑓‘𝑥) ≤ (𝑓‘𝑦)))} ∈ V
31	19, 20, 30	fvmpt 6996	. . 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 2813	. 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 2146  {cab 2744  ∀wral 3082  Vcvv 3458   class class class wbr 5114  ⟶wf 6539  –1-1-onto→wf1o 6542  ‘cfv 6543  (class class class)co 7423   ↑_m cmap 8833  Basecbs 17294  lecple 17342  LAutclaut 40799

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 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2738  ax-sep 5262  ax-nul 5274  ax-pow 5341  ax-pr 5409  ax-un 7745

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 2570  df-eu 2600  df-clab 2745  df-cleq 2758  df-clel 2841  df-nfc 2915  df-ne 2962  df-ral 3083  df-rex 3093  df-rab 3420  df-v 3460  df-sbc 3748  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4290  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4878  df-br 5115  df-opab 5179  df-mpt 5198  df-id 5561  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-iota 6499  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7426  df-oprab 7427  df-mpo 7428  df-map 8835  df-laut 40803

This theorem is used by:  islaut  40897