Theorem islaut 40543

Description: The predicate "is a lattice automorphism". (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
islaut	⊢ (𝐾 ∈ 𝐴 → (𝐹 ∈ 𝐼 ↔ (𝐹:𝐵–1-1-onto→𝐵 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 ≤ 𝑦 ↔ (𝐹‘𝑥) ≤ (𝐹‘𝑦)))))

Distinct variable groups:   𝑥,𝑦,𝐵   𝑥,𝐹,𝑦   𝑥,𝐾,𝑦

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

Proof of Theorem islaut

Dummy variable 𝑓 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		lautset.b	. . . 4 ⊢ 𝐵 = (Base‘𝐾)
2		lautset.l	. . . 4 ⊢ ≤ = (le‘𝐾)
3		lautset.i	. . . 4 ⊢ 𝐼 = (LAut‘𝐾)
4	1, 2, 3	lautset 40542	. . 3 ⊢ (𝐾 ∈ 𝐴 → 𝐼 = {𝑓 ∣ (𝑓:𝐵–1-1-onto→𝐵 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 ≤ 𝑦 ↔ (𝑓‘𝑥) ≤ (𝑓‘𝑦)))})
5	4	eleq2d 2823	. 2 ⊢ (𝐾 ∈ 𝐴 → (𝐹 ∈ 𝐼 ↔ 𝐹 ∈ {𝑓 ∣ (𝑓:𝐵–1-1-onto→𝐵 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 ≤ 𝑦 ↔ (𝑓‘𝑥) ≤ (𝑓‘𝑦)))}))
6		f1of 6774	. . . . 5 ⊢ (𝐹:𝐵–1-1-onto→𝐵 → 𝐹:𝐵⟶𝐵)
7	1	fvexi 6848	. . . . 5 ⊢ 𝐵 ∈ V
8		fex 7174	. . . . 5 ⊢ ((𝐹:𝐵⟶𝐵 ∧ 𝐵 ∈ V) → 𝐹 ∈ V)
9	6, 7, 8	sylancl 587	. . . 4 ⊢ (𝐹:𝐵–1-1-onto→𝐵 → 𝐹 ∈ V)
10	9	adantr 480	. . 3 ⊢ ((𝐹:𝐵–1-1-onto→𝐵 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 ≤ 𝑦 ↔ (𝐹‘𝑥) ≤ (𝐹‘𝑦))) → 𝐹 ∈ V)
11		f1oeq1 6762	. . . 4 ⊢ (𝑓 = 𝐹 → (𝑓:𝐵–1-1-onto→𝐵 ↔ 𝐹:𝐵–1-1-onto→𝐵))
12		fveq1 6833	. . . . . . 7 ⊢ (𝑓 = 𝐹 → (𝑓‘𝑥) = (𝐹‘𝑥))
13		fveq1 6833	. . . . . . 7 ⊢ (𝑓 = 𝐹 → (𝑓‘𝑦) = (𝐹‘𝑦))
14	12, 13	breq12d 5099	. . . . . 6 ⊢ (𝑓 = 𝐹 → ((𝑓‘𝑥) ≤ (𝑓‘𝑦) ↔ (𝐹‘𝑥) ≤ (𝐹‘𝑦)))
15	14	bibi2d 342	. . . . 5 ⊢ (𝑓 = 𝐹 → ((𝑥 ≤ 𝑦 ↔ (𝑓‘𝑥) ≤ (𝑓‘𝑦)) ↔ (𝑥 ≤ 𝑦 ↔ (𝐹‘𝑥) ≤ (𝐹‘𝑦))))
16	15	2ralbidv 3202	. . . 4 ⊢ (𝑓 = 𝐹 → (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 ≤ 𝑦 ↔ (𝑓‘𝑥) ≤ (𝑓‘𝑦)) ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 ≤ 𝑦 ↔ (𝐹‘𝑥) ≤ (𝐹‘𝑦))))
17	11, 16	anbi12d 633	. . 3 ⊢ (𝑓 = 𝐹 → ((𝑓:𝐵–1-1-onto→𝐵 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 ≤ 𝑦 ↔ (𝑓‘𝑥) ≤ (𝑓‘𝑦))) ↔ (𝐹:𝐵–1-1-onto→𝐵 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 ≤ 𝑦 ↔ (𝐹‘𝑥) ≤ (𝐹‘𝑦)))))
18	10, 17	elab3 3630	. 2 ⊢ (𝐹 ∈ {𝑓 ∣ (𝑓:𝐵–1-1-onto→𝐵 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 ≤ 𝑦 ↔ (𝑓‘𝑥) ≤ (𝑓‘𝑦)))} ↔ (𝐹:𝐵–1-1-onto→𝐵 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 ≤ 𝑦 ↔ (𝐹‘𝑥) ≤ (𝐹‘𝑦))))
19	5, 18	bitrdi 287	1 ⊢ (𝐾 ∈ 𝐴 → (𝐹 ∈ 𝐼 ↔ (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 3430   class class class wbr 5086  ⟶wf 6488  –1-1-onto→wf1o 6491  ‘cfv 6492  Basecbs 17170  lecple 17218  LAutclaut 40445

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-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5302  ax-pr 5370  ax-un 7682

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-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-ov 7363  df-oprab 7364  df-mpo 7365  df-map 8768  df-laut 40449

This theorem is referenced by:  lautle  40544  laut1o  40545  lautcnv  40550  idlaut  40556  lautco  40557  cdleme50laut  41007