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Theorem lautset 34834
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 (𝐾𝐴𝐼 = {𝑓 ∣ (𝑓:𝐵1-1-onto𝐵 ∧ ∀𝑥𝐵𝑦𝐵 (𝑥 𝑦 ↔ (𝑓𝑥) (𝑓𝑦)))})
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.
StepHypRef Expression
1 elex 3203 . 2 (𝐾𝐴𝐾 ∈ V)
2 lautset.i . . 3 𝐼 = (LAut‘𝐾)
3 fveq2 6150 . . . . . . . . 9 (𝑘 = 𝐾 → (Base‘𝑘) = (Base‘𝐾))
4 lautset.b . . . . . . . . 9 𝐵 = (Base‘𝐾)
53, 4syl6eqr 2678 . . . . . . . 8 (𝑘 = 𝐾 → (Base‘𝑘) = 𝐵)
6 f1oeq2 6087 . . . . . . . 8 ((Base‘𝑘) = 𝐵 → (𝑓:(Base‘𝑘)–1-1-onto→(Base‘𝑘) ↔ 𝑓:𝐵1-1-onto→(Base‘𝑘)))
75, 6syl 17 . . . . . . 7 (𝑘 = 𝐾 → (𝑓:(Base‘𝑘)–1-1-onto→(Base‘𝑘) ↔ 𝑓:𝐵1-1-onto→(Base‘𝑘)))
8 f1oeq3 6088 . . . . . . . 8 ((Base‘𝑘) = 𝐵 → (𝑓:𝐵1-1-onto→(Base‘𝑘) ↔ 𝑓:𝐵1-1-onto𝐵))
95, 8syl 17 . . . . . . 7 (𝑘 = 𝐾 → (𝑓:𝐵1-1-onto→(Base‘𝑘) ↔ 𝑓:𝐵1-1-onto𝐵))
107, 9bitrd 268 . . . . . 6 (𝑘 = 𝐾 → (𝑓:(Base‘𝑘)–1-1-onto→(Base‘𝑘) ↔ 𝑓:𝐵1-1-onto𝐵))
11 fveq2 6150 . . . . . . . . . . 11 (𝑘 = 𝐾 → (le‘𝑘) = (le‘𝐾))
12 lautset.l . . . . . . . . . . 11 = (le‘𝐾)
1311, 12syl6eqr 2678 . . . . . . . . . 10 (𝑘 = 𝐾 → (le‘𝑘) = )
1413breqd 4629 . . . . . . . . 9 (𝑘 = 𝐾 → (𝑥(le‘𝑘)𝑦𝑥 𝑦))
1513breqd 4629 . . . . . . . . 9 (𝑘 = 𝐾 → ((𝑓𝑥)(le‘𝑘)(𝑓𝑦) ↔ (𝑓𝑥) (𝑓𝑦)))
1614, 15bibi12d 335 . . . . . . . 8 (𝑘 = 𝐾 → ((𝑥(le‘𝑘)𝑦 ↔ (𝑓𝑥)(le‘𝑘)(𝑓𝑦)) ↔ (𝑥 𝑦 ↔ (𝑓𝑥) (𝑓𝑦))))
175, 16raleqbidv 3146 . . . . . . 7 (𝑘 = 𝐾 → (∀𝑦 ∈ (Base‘𝑘)(𝑥(le‘𝑘)𝑦 ↔ (𝑓𝑥)(le‘𝑘)(𝑓𝑦)) ↔ ∀𝑦𝐵 (𝑥 𝑦 ↔ (𝑓𝑥) (𝑓𝑦))))
185, 17raleqbidv 3146 . . . . . 6 (𝑘 = 𝐾 → (∀𝑥 ∈ (Base‘𝑘)∀𝑦 ∈ (Base‘𝑘)(𝑥(le‘𝑘)𝑦 ↔ (𝑓𝑥)(le‘𝑘)(𝑓𝑦)) ↔ ∀𝑥𝐵𝑦𝐵 (𝑥 𝑦 ↔ (𝑓𝑥) (𝑓𝑦))))
1910, 18anbi12d 746 . . . . 5 (𝑘 = 𝐾 → ((𝑓:(Base‘𝑘)–1-1-onto→(Base‘𝑘) ∧ ∀𝑥 ∈ (Base‘𝑘)∀𝑦 ∈ (Base‘𝑘)(𝑥(le‘𝑘)𝑦 ↔ (𝑓𝑥)(le‘𝑘)(𝑓𝑦))) ↔ (𝑓:𝐵1-1-onto𝐵 ∧ ∀𝑥𝐵𝑦𝐵 (𝑥 𝑦 ↔ (𝑓𝑥) (𝑓𝑦)))))
2019abbidv 2744 . . . 4 (𝑘 = 𝐾 → {𝑓 ∣ (𝑓:(Base‘𝑘)–1-1-onto→(Base‘𝑘) ∧ ∀𝑥 ∈ (Base‘𝑘)∀𝑦 ∈ (Base‘𝑘)(𝑥(le‘𝑘)𝑦 ↔ (𝑓𝑥)(le‘𝑘)(𝑓𝑦)))} = {𝑓 ∣ (𝑓:𝐵1-1-onto𝐵 ∧ ∀𝑥𝐵𝑦𝐵 (𝑥 𝑦 ↔ (𝑓𝑥) (𝑓𝑦)))})
21 df-laut 34741 . . . 4 LAut = (𝑘 ∈ V ↦ {𝑓 ∣ (𝑓:(Base‘𝑘)–1-1-onto→(Base‘𝑘) ∧ ∀𝑥 ∈ (Base‘𝑘)∀𝑦 ∈ (Base‘𝑘)(𝑥(le‘𝑘)𝑦 ↔ (𝑓𝑥)(le‘𝑘)(𝑓𝑦)))})
22 fvex 6160 . . . . . . . . 9 (Base‘𝐾) ∈ V
234, 22eqeltri 2700 . . . . . . . 8 𝐵 ∈ V
2423, 23mapval 7815 . . . . . . 7 (𝐵𝑚 𝐵) = {𝑓𝑓:𝐵𝐵}
25 ovex 6633 . . . . . . 7 (𝐵𝑚 𝐵) ∈ V
2624, 25eqeltrri 2701 . . . . . 6 {𝑓𝑓:𝐵𝐵} ∈ V
27 f1of 6096 . . . . . . 7 (𝑓:𝐵1-1-onto𝐵𝑓:𝐵𝐵)
2827ss2abi 3658 . . . . . 6 {𝑓𝑓:𝐵1-1-onto𝐵} ⊆ {𝑓𝑓:𝐵𝐵}
2926, 28ssexi 4768 . . . . 5 {𝑓𝑓:𝐵1-1-onto𝐵} ∈ V
30 simpl 473 . . . . . 6 ((𝑓:𝐵1-1-onto𝐵 ∧ ∀𝑥𝐵𝑦𝐵 (𝑥 𝑦 ↔ (𝑓𝑥) (𝑓𝑦))) → 𝑓:𝐵1-1-onto𝐵)
3130ss2abi 3658 . . . . 5 {𝑓 ∣ (𝑓:𝐵1-1-onto𝐵 ∧ ∀𝑥𝐵𝑦𝐵 (𝑥 𝑦 ↔ (𝑓𝑥) (𝑓𝑦)))} ⊆ {𝑓𝑓:𝐵1-1-onto𝐵}
3229, 31ssexi 4768 . . . 4 {𝑓 ∣ (𝑓:𝐵1-1-onto𝐵 ∧ ∀𝑥𝐵𝑦𝐵 (𝑥 𝑦 ↔ (𝑓𝑥) (𝑓𝑦)))} ∈ V
3320, 21, 32fvmpt 6240 . . 3 (𝐾 ∈ V → (LAut‘𝐾) = {𝑓 ∣ (𝑓:𝐵1-1-onto𝐵 ∧ ∀𝑥𝐵𝑦𝐵 (𝑥 𝑦 ↔ (𝑓𝑥) (𝑓𝑦)))})
342, 33syl5eq 2672 . 2 (𝐾 ∈ V → 𝐼 = {𝑓 ∣ (𝑓:𝐵1-1-onto𝐵 ∧ ∀𝑥𝐵𝑦𝐵 (𝑥 𝑦 ↔ (𝑓𝑥) (𝑓𝑦)))})
351, 34syl 17 1 (𝐾𝐴𝐼 = {𝑓 ∣ (𝑓:𝐵1-1-onto𝐵 ∧ ∀𝑥𝐵𝑦𝐵 (𝑥 𝑦 ↔ (𝑓𝑥) (𝑓𝑦)))})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1480  wcel 1992  {cab 2612  wral 2912  Vcvv 3191   class class class wbr 4618  wf 5846  1-1-ontowf1o 5849  cfv 5850  (class class class)co 6605  𝑚 cmap 7803  Basecbs 15776  lecple 15864  LAutclaut 34737
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6903
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3193  df-sbc 3423  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-iota 5813  df-fun 5852  df-fn 5853  df-f 5854  df-f1 5855  df-fo 5856  df-f1o 5857  df-fv 5858  df-ov 6608  df-oprab 6609  df-mpt2 6610  df-map 7805  df-laut 34741
This theorem is referenced by:  islaut  34835
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