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Theorem islaut 39556
Description: The predicate "is a lattice automorphism". (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
islaut (𝐾 ∈ 𝐴 β†’ (𝐹 ∈ 𝐼 ↔ (𝐹:𝐡–1-1-onto→𝐡 ∧ βˆ€π‘₯ ∈ 𝐡 βˆ€π‘¦ ∈ 𝐡 (π‘₯ ≀ 𝑦 ↔ (πΉβ€˜π‘₯) ≀ (πΉβ€˜π‘¦)))))
Distinct variable groups:   π‘₯,𝑦,𝐡   π‘₯,𝐹,𝑦   π‘₯,𝐾,𝑦
Allowed substitution hints:   𝐴(π‘₯,𝑦)   𝐼(π‘₯,𝑦)   ≀ (π‘₯,𝑦)

Proof of Theorem islaut
Dummy variable 𝑓 is distinct from all other variables.
StepHypRef Expression
1 lautset.b . . . 4 𝐡 = (Baseβ€˜πΎ)
2 lautset.l . . . 4 ≀ = (leβ€˜πΎ)
3 lautset.i . . . 4 𝐼 = (LAutβ€˜πΎ)
41, 2, 3lautset 39555 . . 3 (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‘¦)))})
54eleq2d 2815 . 2 (𝐾 ∈ 𝐴 β†’ (𝐹 ∈ 𝐼 ↔ 𝐹 ∈ {𝑓 ∣ (𝑓:𝐡–1-1-onto→𝐡 ∧ βˆ€π‘₯ ∈ 𝐡 βˆ€π‘¦ ∈ 𝐡 (π‘₯ ≀ 𝑦 ↔ (π‘“β€˜π‘₯) ≀ (π‘“β€˜π‘¦)))}))
6 f1of 6839 . . . . 5 (𝐹:𝐡–1-1-onto→𝐡 β†’ 𝐹:𝐡⟢𝐡)
71fvexi 6911 . . . . 5 𝐡 ∈ V
8 fex 7238 . . . . 5 ((𝐹:𝐡⟢𝐡 ∧ 𝐡 ∈ V) β†’ 𝐹 ∈ V)
96, 7, 8sylancl 585 . . . 4 (𝐹:𝐡–1-1-onto→𝐡 β†’ 𝐹 ∈ V)
109adantr 480 . . 3 ((𝐹:𝐡–1-1-onto→𝐡 ∧ βˆ€π‘₯ ∈ 𝐡 βˆ€π‘¦ ∈ 𝐡 (π‘₯ ≀ 𝑦 ↔ (πΉβ€˜π‘₯) ≀ (πΉβ€˜π‘¦))) β†’ 𝐹 ∈ V)
11 f1oeq1 6827 . . . 4 (𝑓 = 𝐹 β†’ (𝑓:𝐡–1-1-onto→𝐡 ↔ 𝐹:𝐡–1-1-onto→𝐡))
12 fveq1 6896 . . . . . . 7 (𝑓 = 𝐹 β†’ (π‘“β€˜π‘₯) = (πΉβ€˜π‘₯))
13 fveq1 6896 . . . . . . 7 (𝑓 = 𝐹 β†’ (π‘“β€˜π‘¦) = (πΉβ€˜π‘¦))
1412, 13breq12d 5161 . . . . . 6 (𝑓 = 𝐹 β†’ ((π‘“β€˜π‘₯) ≀ (π‘“β€˜π‘¦) ↔ (πΉβ€˜π‘₯) ≀ (πΉβ€˜π‘¦)))
1514bibi2d 342 . . . . 5 (𝑓 = 𝐹 β†’ ((π‘₯ ≀ 𝑦 ↔ (π‘“β€˜π‘₯) ≀ (π‘“β€˜π‘¦)) ↔ (π‘₯ ≀ 𝑦 ↔ (πΉβ€˜π‘₯) ≀ (πΉβ€˜π‘¦))))
16152ralbidv 3215 . . . 4 (𝑓 = 𝐹 β†’ (βˆ€π‘₯ ∈ 𝐡 βˆ€π‘¦ ∈ 𝐡 (π‘₯ ≀ 𝑦 ↔ (π‘“β€˜π‘₯) ≀ (π‘“β€˜π‘¦)) ↔ βˆ€π‘₯ ∈ 𝐡 βˆ€π‘¦ ∈ 𝐡 (π‘₯ ≀ 𝑦 ↔ (πΉβ€˜π‘₯) ≀ (πΉβ€˜π‘¦))))
1711, 16anbi12d 631 . . 3 (𝑓 = 𝐹 β†’ ((𝑓:𝐡–1-1-onto→𝐡 ∧ βˆ€π‘₯ ∈ 𝐡 βˆ€π‘¦ ∈ 𝐡 (π‘₯ ≀ 𝑦 ↔ (π‘“β€˜π‘₯) ≀ (π‘“β€˜π‘¦))) ↔ (𝐹:𝐡–1-1-onto→𝐡 ∧ βˆ€π‘₯ ∈ 𝐡 βˆ€π‘¦ ∈ 𝐡 (π‘₯ ≀ 𝑦 ↔ (πΉβ€˜π‘₯) ≀ (πΉβ€˜π‘¦)))))
1810, 17elab3 3675 . 2 (𝐹 ∈ {𝑓 ∣ (𝑓:𝐡–1-1-onto→𝐡 ∧ βˆ€π‘₯ ∈ 𝐡 βˆ€π‘¦ ∈ 𝐡 (π‘₯ ≀ 𝑦 ↔ (π‘“β€˜π‘₯) ≀ (π‘“β€˜π‘¦)))} ↔ (𝐹:𝐡–1-1-onto→𝐡 ∧ βˆ€π‘₯ ∈ 𝐡 βˆ€π‘¦ ∈ 𝐡 (π‘₯ ≀ 𝑦 ↔ (πΉβ€˜π‘₯) ≀ (πΉβ€˜π‘¦))))
195, 18bitrdi 287 1 (𝐾 ∈ 𝐴 β†’ (𝐹 ∈ 𝐼 ↔ (𝐹:𝐡–1-1-onto→𝐡 ∧ βˆ€π‘₯ ∈ 𝐡 βˆ€π‘¦ ∈ 𝐡 (π‘₯ ≀ 𝑦 ↔ (πΉβ€˜π‘₯) ≀ (πΉβ€˜π‘¦)))))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 395   = wceq 1534   ∈ wcel 2099  {cab 2705  βˆ€wral 3058  Vcvv 3471   class class class wbr 5148  βŸΆwf 6544  β€“1-1-ontoβ†’wf1o 6547  β€˜cfv 6548  Basecbs 17180  lecple 17240  LAutclaut 39458
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5365  ax-pr 5429  ax-un 7740
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3374  df-rab 3430  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6500  df-fun 6550  df-fn 6551  df-f 6552  df-f1 6553  df-fo 6554  df-f1o 6555  df-fv 6556  df-ov 7423  df-oprab 7424  df-mpo 7425  df-map 8847  df-laut 39462
This theorem is referenced by:  lautle  39557  laut1o  39558  lautcnv  39563  idlaut  39569  lautco  39570  cdleme50laut  40020
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