Theorem islaut 35369

Description: The predictate "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 35368	. . 3 ⊢ (𝐾 ∈ 𝐴 → 𝐼 = {𝑓 ∣ (𝑓:𝐵–1-1-onto→𝐵 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 ≤ 𝑦 ↔ (𝑓‘𝑥) ≤ (𝑓‘𝑦)))})
5	4	eleq2d 2687	. 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 6137	. . . . 5 ⊢ (𝐹:𝐵–1-1-onto→𝐵 → 𝐹:𝐵⟶𝐵)
7		fvex 6201	. . . . . 6 ⊢ (Base‘𝐾) ∈ V
8	1, 7	eqeltri 2697	. . . . 5 ⊢ 𝐵 ∈ V
9		fex 6490	. . . . 5 ⊢ ((𝐹:𝐵⟶𝐵 ∧ 𝐵 ∈ V) → 𝐹 ∈ V)
10	6, 8, 9	sylancl 694	. . . 4 ⊢ (𝐹:𝐵–1-1-onto→𝐵 → 𝐹 ∈ V)
11	10	adantr 481	. . 3 ⊢ ((𝐹:𝐵–1-1-onto→𝐵 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 ≤ 𝑦 ↔ (𝐹‘𝑥) ≤ (𝐹‘𝑦))) → 𝐹 ∈ V)
12		f1oeq1 6127	. . . 4 ⊢ (𝑓 = 𝐹 → (𝑓:𝐵–1-1-onto→𝐵 ↔ 𝐹:𝐵–1-1-onto→𝐵))
13		fveq1 6190	. . . . . . 7 ⊢ (𝑓 = 𝐹 → (𝑓‘𝑥) = (𝐹‘𝑥))
14		fveq1 6190	. . . . . . 7 ⊢ (𝑓 = 𝐹 → (𝑓‘𝑦) = (𝐹‘𝑦))
15	13, 14	breq12d 4666	. . . . . 6 ⊢ (𝑓 = 𝐹 → ((𝑓‘𝑥) ≤ (𝑓‘𝑦) ↔ (𝐹‘𝑥) ≤ (𝐹‘𝑦)))
16	15	bibi2d 332	. . . . 5 ⊢ (𝑓 = 𝐹 → ((𝑥 ≤ 𝑦 ↔ (𝑓‘𝑥) ≤ (𝑓‘𝑦)) ↔ (𝑥 ≤ 𝑦 ↔ (𝐹‘𝑥) ≤ (𝐹‘𝑦))))
17	16	2ralbidv 2989	. . . 4 ⊢ (𝑓 = 𝐹 → (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 ≤ 𝑦 ↔ (𝑓‘𝑥) ≤ (𝑓‘𝑦)) ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 ≤ 𝑦 ↔ (𝐹‘𝑥) ≤ (𝐹‘𝑦))))
18	12, 17	anbi12d 747	. . 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)))))
19	11, 18	elab3 3358	. 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))))
20	5, 19	syl6bb 276	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 196   ∧ wa 384   = wceq 1483   ∈ wcel 1990  {cab 2608  ∀wral 2912  Vcvv 3200   class class class wbr 4653  ⟶wf 5884  –1-1-onto→wf1o 5887  ‘cfv 5888  Basecbs 15857  lecple 15948  LAutclaut 35271

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-map 7859  df-laut 35275

This theorem is referenced by:  lautle  35370  laut1o  35371  lautcnv  35376  idlaut  35382  lautco  35383  cdleme50laut  35835