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Theorem islaut 38592
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 38591 . . 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 2820 . 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 6785 . . . . 5 (𝐹:𝐡–1-1-onto→𝐡 β†’ 𝐹:𝐡⟢𝐡)
71fvexi 6857 . . . . 5 𝐡 ∈ V
8 fex 7177 . . . . 5 ((𝐹:𝐡⟢𝐡 ∧ 𝐡 ∈ V) β†’ 𝐹 ∈ V)
96, 7, 8sylancl 587 . . . 4 (𝐹:𝐡–1-1-onto→𝐡 β†’ 𝐹 ∈ V)
109adantr 482 . . 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 6773 . . . 4 (𝑓 = 𝐹 β†’ (𝑓:𝐡–1-1-onto→𝐡 ↔ 𝐹:𝐡–1-1-onto→𝐡))
12 fveq1 6842 . . . . . . 7 (𝑓 = 𝐹 β†’ (π‘“β€˜π‘₯) = (πΉβ€˜π‘₯))
13 fveq1 6842 . . . . . . 7 (𝑓 = 𝐹 β†’ (π‘“β€˜π‘¦) = (πΉβ€˜π‘¦))
1412, 13breq12d 5119 . . . . . 6 (𝑓 = 𝐹 β†’ ((π‘“β€˜π‘₯) ≀ (π‘“β€˜π‘¦) ↔ (πΉβ€˜π‘₯) ≀ (πΉβ€˜π‘¦)))
1514bibi2d 343 . . . . 5 (𝑓 = 𝐹 β†’ ((π‘₯ ≀ 𝑦 ↔ (π‘“β€˜π‘₯) ≀ (π‘“β€˜π‘¦)) ↔ (π‘₯ ≀ 𝑦 ↔ (πΉβ€˜π‘₯) ≀ (πΉβ€˜π‘¦))))
16152ralbidv 3209 . . . 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 632 . . 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 3639 . 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 397   = wceq 1542   ∈ wcel 2107  {cab 2710  βˆ€wral 3061  Vcvv 3444   class class class wbr 5106  βŸΆwf 6493  β€“1-1-ontoβ†’wf1o 6496  β€˜cfv 6497  Basecbs 17088  lecple 17145  LAutclaut 38494
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-rep 5243  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673
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-reu 3353  df-rab 3407  df-v 3446  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-ov 7361  df-oprab 7362  df-mpo 7363  df-map 8770  df-laut 38498
This theorem is referenced by:  lautle  38593  laut1o  38594  lautcnv  38599  idlaut  38605  lautco  38606  cdleme50laut  39056
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