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Theorem lautset 38595
Description: The set of lattice automorphisms. (Contributed by NM, 11-May-2012.)
Hypotheses
Ref Expression
lautset.b 𝐡 = (Baseβ€˜πΎ)
lautset.l ≀ = (leβ€˜πΎ)
lautset.i 𝐼 = (LAutβ€˜πΎ)
Assertion
Ref Expression
lautset (𝐾 ∈ 𝐴 β†’ 𝐼 = {𝑓 ∣ (𝑓:𝐡–1-1-onto→𝐡 ∧ βˆ€π‘₯ ∈ 𝐡 βˆ€π‘¦ ∈ 𝐡 (π‘₯ ≀ 𝑦 ↔ (π‘“β€˜π‘₯) ≀ (π‘“β€˜π‘¦)))})
Distinct variable groups:   π‘₯,𝑓,𝑦,𝐡   𝑓,𝐾,π‘₯,𝑦   ≀ ,𝑓
Allowed substitution hints:   𝐴(π‘₯,𝑦,𝑓)   𝐼(π‘₯,𝑦,𝑓)   ≀ (π‘₯,𝑦)

Proof of Theorem lautset
Dummy variable π‘˜ is distinct from all other variables.
StepHypRef Expression
1 elex 3465 . 2 (𝐾 ∈ 𝐴 β†’ 𝐾 ∈ V)
2 lautset.i . . 3 𝐼 = (LAutβ€˜πΎ)
3 fveq2 6846 . . . . . . . . 9 (π‘˜ = 𝐾 β†’ (Baseβ€˜π‘˜) = (Baseβ€˜πΎ))
4 lautset.b . . . . . . . . 9 𝐡 = (Baseβ€˜πΎ)
53, 4eqtr4di 2791 . . . . . . . 8 (π‘˜ = 𝐾 β†’ (Baseβ€˜π‘˜) = 𝐡)
65f1oeq2d 6784 . . . . . . 7 (π‘˜ = 𝐾 β†’ (𝑓:(Baseβ€˜π‘˜)–1-1-ontoβ†’(Baseβ€˜π‘˜) ↔ 𝑓:𝐡–1-1-ontoβ†’(Baseβ€˜π‘˜)))
7 f1oeq3 6778 . . . . . . . 8 ((Baseβ€˜π‘˜) = 𝐡 β†’ (𝑓:𝐡–1-1-ontoβ†’(Baseβ€˜π‘˜) ↔ 𝑓:𝐡–1-1-onto→𝐡))
85, 7syl 17 . . . . . . 7 (π‘˜ = 𝐾 β†’ (𝑓:𝐡–1-1-ontoβ†’(Baseβ€˜π‘˜) ↔ 𝑓:𝐡–1-1-onto→𝐡))
96, 8bitrd 279 . . . . . 6 (π‘˜ = 𝐾 β†’ (𝑓:(Baseβ€˜π‘˜)–1-1-ontoβ†’(Baseβ€˜π‘˜) ↔ 𝑓:𝐡–1-1-onto→𝐡))
10 fveq2 6846 . . . . . . . . . . 11 (π‘˜ = 𝐾 β†’ (leβ€˜π‘˜) = (leβ€˜πΎ))
11 lautset.l . . . . . . . . . . 11 ≀ = (leβ€˜πΎ)
1210, 11eqtr4di 2791 . . . . . . . . . 10 (π‘˜ = 𝐾 β†’ (leβ€˜π‘˜) = ≀ )
1312breqd 5120 . . . . . . . . 9 (π‘˜ = 𝐾 β†’ (π‘₯(leβ€˜π‘˜)𝑦 ↔ π‘₯ ≀ 𝑦))
1412breqd 5120 . . . . . . . . 9 (π‘˜ = 𝐾 β†’ ((π‘“β€˜π‘₯)(leβ€˜π‘˜)(π‘“β€˜π‘¦) ↔ (π‘“β€˜π‘₯) ≀ (π‘“β€˜π‘¦)))
1513, 14bibi12d 346 . . . . . . . 8 (π‘˜ = 𝐾 β†’ ((π‘₯(leβ€˜π‘˜)𝑦 ↔ (π‘“β€˜π‘₯)(leβ€˜π‘˜)(π‘“β€˜π‘¦)) ↔ (π‘₯ ≀ 𝑦 ↔ (π‘“β€˜π‘₯) ≀ (π‘“β€˜π‘¦))))
165, 15raleqbidv 3318 . . . . . . 7 (π‘˜ = 𝐾 β†’ (βˆ€π‘¦ ∈ (Baseβ€˜π‘˜)(π‘₯(leβ€˜π‘˜)𝑦 ↔ (π‘“β€˜π‘₯)(leβ€˜π‘˜)(π‘“β€˜π‘¦)) ↔ βˆ€π‘¦ ∈ 𝐡 (π‘₯ ≀ 𝑦 ↔ (π‘“β€˜π‘₯) ≀ (π‘“β€˜π‘¦))))
175, 16raleqbidv 3318 . . . . . 6 (π‘˜ = 𝐾 β†’ (βˆ€π‘₯ ∈ (Baseβ€˜π‘˜)βˆ€π‘¦ ∈ (Baseβ€˜π‘˜)(π‘₯(leβ€˜π‘˜)𝑦 ↔ (π‘“β€˜π‘₯)(leβ€˜π‘˜)(π‘“β€˜π‘¦)) ↔ βˆ€π‘₯ ∈ 𝐡 βˆ€π‘¦ ∈ 𝐡 (π‘₯ ≀ 𝑦 ↔ (π‘“β€˜π‘₯) ≀ (π‘“β€˜π‘¦))))
189, 17anbi12d 632 . . . . 5 (π‘˜ = 𝐾 β†’ ((𝑓:(Baseβ€˜π‘˜)–1-1-ontoβ†’(Baseβ€˜π‘˜) ∧ βˆ€π‘₯ ∈ (Baseβ€˜π‘˜)βˆ€π‘¦ ∈ (Baseβ€˜π‘˜)(π‘₯(leβ€˜π‘˜)𝑦 ↔ (π‘“β€˜π‘₯)(leβ€˜π‘˜)(π‘“β€˜π‘¦))) ↔ (𝑓:𝐡–1-1-onto→𝐡 ∧ βˆ€π‘₯ ∈ 𝐡 βˆ€π‘¦ ∈ 𝐡 (π‘₯ ≀ 𝑦 ↔ (π‘“β€˜π‘₯) ≀ (π‘“β€˜π‘¦)))))
1918abbidv 2802 . . . 4 (π‘˜ = 𝐾 β†’ {𝑓 ∣ (𝑓:(Baseβ€˜π‘˜)–1-1-ontoβ†’(Baseβ€˜π‘˜) ∧ βˆ€π‘₯ ∈ (Baseβ€˜π‘˜)βˆ€π‘¦ ∈ (Baseβ€˜π‘˜)(π‘₯(leβ€˜π‘˜)𝑦 ↔ (π‘“β€˜π‘₯)(leβ€˜π‘˜)(π‘“β€˜π‘¦)))} = {𝑓 ∣ (𝑓:𝐡–1-1-onto→𝐡 ∧ βˆ€π‘₯ ∈ 𝐡 βˆ€π‘¦ ∈ 𝐡 (π‘₯ ≀ 𝑦 ↔ (π‘“β€˜π‘₯) ≀ (π‘“β€˜π‘¦)))})
20 df-laut 38502 . . . 4 LAut = (π‘˜ ∈ V ↦ {𝑓 ∣ (𝑓:(Baseβ€˜π‘˜)–1-1-ontoβ†’(Baseβ€˜π‘˜) ∧ βˆ€π‘₯ ∈ (Baseβ€˜π‘˜)βˆ€π‘¦ ∈ (Baseβ€˜π‘˜)(π‘₯(leβ€˜π‘˜)𝑦 ↔ (π‘“β€˜π‘₯)(leβ€˜π‘˜)(π‘“β€˜π‘¦)))})
214fvexi 6860 . . . . . . . 8 𝐡 ∈ V
2221, 21mapval 8783 . . . . . . 7 (𝐡 ↑m 𝐡) = {𝑓 ∣ 𝑓:𝐡⟢𝐡}
23 ovex 7394 . . . . . . 7 (𝐡 ↑m 𝐡) ∈ V
2422, 23eqeltrri 2831 . . . . . 6 {𝑓 ∣ 𝑓:𝐡⟢𝐡} ∈ V
25 f1of 6788 . . . . . . 7 (𝑓:𝐡–1-1-onto→𝐡 β†’ 𝑓:𝐡⟢𝐡)
2625ss2abi 4027 . . . . . 6 {𝑓 ∣ 𝑓:𝐡–1-1-onto→𝐡} βŠ† {𝑓 ∣ 𝑓:𝐡⟢𝐡}
2724, 26ssexi 5283 . . . . 5 {𝑓 ∣ 𝑓:𝐡–1-1-onto→𝐡} ∈ V
28 simpl 484 . . . . . 6 ((𝑓:𝐡–1-1-onto→𝐡 ∧ βˆ€π‘₯ ∈ 𝐡 βˆ€π‘¦ ∈ 𝐡 (π‘₯ ≀ 𝑦 ↔ (π‘“β€˜π‘₯) ≀ (π‘“β€˜π‘¦))) β†’ 𝑓:𝐡–1-1-onto→𝐡)
2928ss2abi 4027 . . . . 5 {𝑓 ∣ (𝑓:𝐡–1-1-onto→𝐡 ∧ βˆ€π‘₯ ∈ 𝐡 βˆ€π‘¦ ∈ 𝐡 (π‘₯ ≀ 𝑦 ↔ (π‘“β€˜π‘₯) ≀ (π‘“β€˜π‘¦)))} βŠ† {𝑓 ∣ 𝑓:𝐡–1-1-onto→𝐡}
3027, 29ssexi 5283 . . . 4 {𝑓 ∣ (𝑓:𝐡–1-1-onto→𝐡 ∧ βˆ€π‘₯ ∈ 𝐡 βˆ€π‘¦ ∈ 𝐡 (π‘₯ ≀ 𝑦 ↔ (π‘“β€˜π‘₯) ≀ (π‘“β€˜π‘¦)))} ∈ V
3119, 20, 30fvmpt 6952 . . 3 (𝐾 ∈ V β†’ (LAutβ€˜πΎ) = {𝑓 ∣ (𝑓:𝐡–1-1-onto→𝐡 ∧ βˆ€π‘₯ ∈ 𝐡 βˆ€π‘¦ ∈ 𝐡 (π‘₯ ≀ 𝑦 ↔ (π‘“β€˜π‘₯) ≀ (π‘“β€˜π‘¦)))})
322, 31eqtrid 2785 . 2 (𝐾 ∈ V β†’ 𝐼 = {𝑓 ∣ (𝑓:𝐡–1-1-onto→𝐡 ∧ βˆ€π‘₯ ∈ 𝐡 βˆ€π‘¦ ∈ 𝐡 (π‘₯ ≀ 𝑦 ↔ (π‘“β€˜π‘₯) ≀ (π‘“β€˜π‘¦)))})
331, 32syl 17 1 (𝐾 ∈ 𝐴 β†’ 𝐼 = {𝑓 ∣ (𝑓:𝐡–1-1-onto→𝐡 ∧ βˆ€π‘₯ ∈ 𝐡 βˆ€π‘¦ ∈ 𝐡 (π‘₯ ≀ 𝑦 ↔ (π‘“β€˜π‘₯) ≀ (π‘“β€˜π‘¦)))})
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 397   = wceq 1542   ∈ wcel 2107  {cab 2710  βˆ€wral 3061  Vcvv 3447   class class class wbr 5109  βŸΆwf 6496  β€“1-1-ontoβ†’wf1o 6499  β€˜cfv 6500  (class class class)co 7361   ↑m cmap 8771  Basecbs 17091  lecple 17148  LAutclaut 38498
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3449  df-sbc 3744  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-br 5110  df-opab 5172  df-mpt 5193  df-id 5535  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-ov 7364  df-oprab 7365  df-mpo 7366  df-map 8773  df-laut 38502
This theorem is referenced by:  islaut  38596
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