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Theorem islaut 39465
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 39464 . . 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 2813 . 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 6826 . . . . 5 (𝐹:𝐡–1-1-onto→𝐡 β†’ 𝐹:𝐡⟢𝐡)
71fvexi 6898 . . . . 5 𝐡 ∈ V
8 fex 7222 . . . . 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 6814 . . . 4 (𝑓 = 𝐹 β†’ (𝑓:𝐡–1-1-onto→𝐡 ↔ 𝐹:𝐡–1-1-onto→𝐡))
12 fveq1 6883 . . . . . . 7 (𝑓 = 𝐹 β†’ (π‘“β€˜π‘₯) = (πΉβ€˜π‘₯))
13 fveq1 6883 . . . . . . 7 (𝑓 = 𝐹 β†’ (π‘“β€˜π‘¦) = (πΉβ€˜π‘¦))
1412, 13breq12d 5154 . . . . . 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 3212 . . . 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 630 . . 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 3671 . 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 1533   ∈ wcel 2098  {cab 2703  βˆ€wral 3055  Vcvv 3468   class class class wbr 5141  βŸΆwf 6532  β€“1-1-ontoβ†’wf1o 6535  β€˜cfv 6536  Basecbs 17151  lecple 17211  LAutclaut 39367
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7721
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6488  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7407  df-oprab 7408  df-mpo 7409  df-map 8821  df-laut 39371
This theorem is referenced by:  lautle  39466  laut1o  39467  lautcnv  39472  idlaut  39478  lautco  39479  cdleme50laut  39929
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