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Theorem lautset 37212
 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 3513 . 2 (𝐾𝐴𝐾 ∈ V)
2 lautset.i . . 3 𝐼 = (LAut‘𝐾)
3 fveq2 6665 . . . . . . . . 9 (𝑘 = 𝐾 → (Base‘𝑘) = (Base‘𝐾))
4 lautset.b . . . . . . . . 9 𝐵 = (Base‘𝐾)
53, 4syl6eqr 2874 . . . . . . . 8 (𝑘 = 𝐾 → (Base‘𝑘) = 𝐵)
65f1oeq2d 6606 . . . . . . 7 (𝑘 = 𝐾 → (𝑓:(Base‘𝑘)–1-1-onto→(Base‘𝑘) ↔ 𝑓:𝐵1-1-onto→(Base‘𝑘)))
7 f1oeq3 6601 . . . . . . . 8 ((Base‘𝑘) = 𝐵 → (𝑓:𝐵1-1-onto→(Base‘𝑘) ↔ 𝑓:𝐵1-1-onto𝐵))
85, 7syl 17 . . . . . . 7 (𝑘 = 𝐾 → (𝑓:𝐵1-1-onto→(Base‘𝑘) ↔ 𝑓:𝐵1-1-onto𝐵))
96, 8bitrd 281 . . . . . 6 (𝑘 = 𝐾 → (𝑓:(Base‘𝑘)–1-1-onto→(Base‘𝑘) ↔ 𝑓:𝐵1-1-onto𝐵))
10 fveq2 6665 . . . . . . . . . . 11 (𝑘 = 𝐾 → (le‘𝑘) = (le‘𝐾))
11 lautset.l . . . . . . . . . . 11 = (le‘𝐾)
1210, 11syl6eqr 2874 . . . . . . . . . 10 (𝑘 = 𝐾 → (le‘𝑘) = )
1312breqd 5070 . . . . . . . . 9 (𝑘 = 𝐾 → (𝑥(le‘𝑘)𝑦𝑥 𝑦))
1412breqd 5070 . . . . . . . . 9 (𝑘 = 𝐾 → ((𝑓𝑥)(le‘𝑘)(𝑓𝑦) ↔ (𝑓𝑥) (𝑓𝑦)))
1513, 14bibi12d 348 . . . . . . . 8 (𝑘 = 𝐾 → ((𝑥(le‘𝑘)𝑦 ↔ (𝑓𝑥)(le‘𝑘)(𝑓𝑦)) ↔ (𝑥 𝑦 ↔ (𝑓𝑥) (𝑓𝑦))))
165, 15raleqbidv 3402 . . . . . . 7 (𝑘 = 𝐾 → (∀𝑦 ∈ (Base‘𝑘)(𝑥(le‘𝑘)𝑦 ↔ (𝑓𝑥)(le‘𝑘)(𝑓𝑦)) ↔ ∀𝑦𝐵 (𝑥 𝑦 ↔ (𝑓𝑥) (𝑓𝑦))))
175, 16raleqbidv 3402 . . . . . 6 (𝑘 = 𝐾 → (∀𝑥 ∈ (Base‘𝑘)∀𝑦 ∈ (Base‘𝑘)(𝑥(le‘𝑘)𝑦 ↔ (𝑓𝑥)(le‘𝑘)(𝑓𝑦)) ↔ ∀𝑥𝐵𝑦𝐵 (𝑥 𝑦 ↔ (𝑓𝑥) (𝑓𝑦))))
189, 17anbi12d 632 . . . . 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)))))
1918abbidv 2885 . . . 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)))})
20 df-laut 37119 . . . 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)))})
214fvexi 6679 . . . . . . . 8 𝐵 ∈ V
2221, 21mapval 8412 . . . . . . 7 (𝐵m 𝐵) = {𝑓𝑓:𝐵𝐵}
23 ovex 7183 . . . . . . 7 (𝐵m 𝐵) ∈ V
2422, 23eqeltrri 2910 . . . . . 6 {𝑓𝑓:𝐵𝐵} ∈ V
25 f1of 6610 . . . . . . 7 (𝑓:𝐵1-1-onto𝐵𝑓:𝐵𝐵)
2625ss2abi 4043 . . . . . 6 {𝑓𝑓:𝐵1-1-onto𝐵} ⊆ {𝑓𝑓:𝐵𝐵}
2724, 26ssexi 5219 . . . . 5 {𝑓𝑓:𝐵1-1-onto𝐵} ∈ V
28 simpl 485 . . . . . 6 ((𝑓:𝐵1-1-onto𝐵 ∧ ∀𝑥𝐵𝑦𝐵 (𝑥 𝑦 ↔ (𝑓𝑥) (𝑓𝑦))) → 𝑓:𝐵1-1-onto𝐵)
2928ss2abi 4043 . . . . 5 {𝑓 ∣ (𝑓:𝐵1-1-onto𝐵 ∧ ∀𝑥𝐵𝑦𝐵 (𝑥 𝑦 ↔ (𝑓𝑥) (𝑓𝑦)))} ⊆ {𝑓𝑓:𝐵1-1-onto𝐵}
3027, 29ssexi 5219 . . . 4 {𝑓 ∣ (𝑓:𝐵1-1-onto𝐵 ∧ ∀𝑥𝐵𝑦𝐵 (𝑥 𝑦 ↔ (𝑓𝑥) (𝑓𝑦)))} ∈ V
3119, 20, 30fvmpt 6763 . . 3 (𝐾 ∈ V → (LAut‘𝐾) = {𝑓 ∣ (𝑓:𝐵1-1-onto𝐵 ∧ ∀𝑥𝐵𝑦𝐵 (𝑥 𝑦 ↔ (𝑓𝑥) (𝑓𝑦)))})
322, 31syl5eq 2868 . 2 (𝐾 ∈ V → 𝐼 = {𝑓 ∣ (𝑓:𝐵1-1-onto𝐵 ∧ ∀𝑥𝐵𝑦𝐵 (𝑥 𝑦 ↔ (𝑓𝑥) (𝑓𝑦)))})
331, 32syl 17 1 (𝐾𝐴𝐼 = {𝑓 ∣ (𝑓:𝐵1-1-onto𝐵 ∧ ∀𝑥𝐵𝑦𝐵 (𝑥 𝑦 ↔ (𝑓𝑥) (𝑓𝑦)))})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   = wceq 1533   ∈ wcel 2110  {cab 2799  ∀wral 3138  Vcvv 3495   class class class wbr 5059  ⟶wf 6346  –1-1-onto→wf1o 6349  ‘cfv 6350  (class class class)co 7150   ↑m cmap 8400  Basecbs 16477  lecple 16566  LAutclaut 37115 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2156  ax-12 2172  ax-ext 2793  ax-sep 5196  ax-nul 5203  ax-pow 5259  ax-pr 5322  ax-un 7455 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3497  df-sbc 3773  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4833  df-br 5060  df-opab 5122  df-mpt 5140  df-id 5455  df-xp 5556  df-rel 5557  df-cnv 5558  df-co 5559  df-dm 5560  df-rn 5561  df-iota 6309  df-fun 6352  df-fn 6353  df-f 6354  df-f1 6355  df-fo 6356  df-f1o 6357  df-fv 6358  df-ov 7153  df-oprab 7154  df-mpo 7155  df-map 8402  df-laut 37119 This theorem is referenced by:  islaut  37213
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