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Theorem lautset 35883
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 3361 . 2 (𝐾𝐴𝐾 ∈ V)
2 lautset.i . . 3 𝐼 = (LAut‘𝐾)
3 fveq2 6332 . . . . . . . . 9 (𝑘 = 𝐾 → (Base‘𝑘) = (Base‘𝐾))
4 lautset.b . . . . . . . . 9 𝐵 = (Base‘𝐾)
53, 4syl6eqr 2822 . . . . . . . 8 (𝑘 = 𝐾 → (Base‘𝑘) = 𝐵)
6 f1oeq2 6269 . . . . . . . 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 6270 . . . . . . . 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 6332 . . . . . . . . . . 11 (𝑘 = 𝐾 → (le‘𝑘) = (le‘𝐾))
12 lautset.l . . . . . . . . . . 11 = (le‘𝐾)
1311, 12syl6eqr 2822 . . . . . . . . . 10 (𝑘 = 𝐾 → (le‘𝑘) = )
1413breqd 4795 . . . . . . . . 9 (𝑘 = 𝐾 → (𝑥(le‘𝑘)𝑦𝑥 𝑦))
1513breqd 4795 . . . . . . . . 9 (𝑘 = 𝐾 → ((𝑓𝑥)(le‘𝑘)(𝑓𝑦) ↔ (𝑓𝑥) (𝑓𝑦)))
1614, 15bibi12d 334 . . . . . . . 8 (𝑘 = 𝐾 → ((𝑥(le‘𝑘)𝑦 ↔ (𝑓𝑥)(le‘𝑘)(𝑓𝑦)) ↔ (𝑥 𝑦 ↔ (𝑓𝑥) (𝑓𝑦))))
175, 16raleqbidv 3300 . . . . . . 7 (𝑘 = 𝐾 → (∀𝑦 ∈ (Base‘𝑘)(𝑥(le‘𝑘)𝑦 ↔ (𝑓𝑥)(le‘𝑘)(𝑓𝑦)) ↔ ∀𝑦𝐵 (𝑥 𝑦 ↔ (𝑓𝑥) (𝑓𝑦))))
185, 17raleqbidv 3300 . . . . . 6 (𝑘 = 𝐾 → (∀𝑥 ∈ (Base‘𝑘)∀𝑦 ∈ (Base‘𝑘)(𝑥(le‘𝑘)𝑦 ↔ (𝑓𝑥)(le‘𝑘)(𝑓𝑦)) ↔ ∀𝑥𝐵𝑦𝐵 (𝑥 𝑦 ↔ (𝑓𝑥) (𝑓𝑦))))
1910, 18anbi12d 608 . . . . 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 2889 . . . 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 35790 . . . 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 6342 . . . . . . . . 9 (Base‘𝐾) ∈ V
234, 22eqeltri 2845 . . . . . . . 8 𝐵 ∈ V
2423, 23mapval 8020 . . . . . . 7 (𝐵𝑚 𝐵) = {𝑓𝑓:𝐵𝐵}
25 ovex 6822 . . . . . . 7 (𝐵𝑚 𝐵) ∈ V
2624, 25eqeltrri 2846 . . . . . 6 {𝑓𝑓:𝐵𝐵} ∈ V
27 f1of 6278 . . . . . . 7 (𝑓:𝐵1-1-onto𝐵𝑓:𝐵𝐵)
2827ss2abi 3821 . . . . . 6 {𝑓𝑓:𝐵1-1-onto𝐵} ⊆ {𝑓𝑓:𝐵𝐵}
2926, 28ssexi 4934 . . . . 5 {𝑓𝑓:𝐵1-1-onto𝐵} ∈ V
30 simpl 468 . . . . . 6 ((𝑓:𝐵1-1-onto𝐵 ∧ ∀𝑥𝐵𝑦𝐵 (𝑥 𝑦 ↔ (𝑓𝑥) (𝑓𝑦))) → 𝑓:𝐵1-1-onto𝐵)
3130ss2abi 3821 . . . . 5 {𝑓 ∣ (𝑓:𝐵1-1-onto𝐵 ∧ ∀𝑥𝐵𝑦𝐵 (𝑥 𝑦 ↔ (𝑓𝑥) (𝑓𝑦)))} ⊆ {𝑓𝑓:𝐵1-1-onto𝐵}
3229, 31ssexi 4934 . . . 4 {𝑓 ∣ (𝑓:𝐵1-1-onto𝐵 ∧ ∀𝑥𝐵𝑦𝐵 (𝑥 𝑦 ↔ (𝑓𝑥) (𝑓𝑦)))} ∈ V
3320, 21, 32fvmpt 6424 . . 3 (𝐾 ∈ V → (LAut‘𝐾) = {𝑓 ∣ (𝑓:𝐵1-1-onto𝐵 ∧ ∀𝑥𝐵𝑦𝐵 (𝑥 𝑦 ↔ (𝑓𝑥) (𝑓𝑦)))})
342, 33syl5eq 2816 . 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 382   = wceq 1630  wcel 2144  {cab 2756  wral 3060  Vcvv 3349   class class class wbr 4784  wf 6027  1-1-ontowf1o 6030  cfv 6031  (class class class)co 6792  𝑚 cmap 8008  Basecbs 16063  lecple 16155  LAutclaut 35786
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034  ax-un 7095
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ral 3065  df-rex 3066  df-rab 3069  df-v 3351  df-sbc 3586  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-pw 4297  df-sn 4315  df-pr 4317  df-op 4321  df-uni 4573  df-br 4785  df-opab 4845  df-mpt 4862  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-ov 6795  df-oprab 6796  df-mpt2 6797  df-map 8010  df-laut 35790
This theorem is referenced by:  islaut  35884
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