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Theorem laut1o 40068
Description: A lattice automorphism is one-to-one and onto. (Contributed by NM, 19-May-2012.)
Hypotheses
Ref Expression
laut1o.b 𝐵 = (Base‘𝐾)
laut1o.i 𝐼 = (LAut‘𝐾)
Assertion
Ref Expression
laut1o ((𝐾𝐴𝐹𝐼) → 𝐹:𝐵1-1-onto𝐵)

Proof of Theorem laut1o
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 laut1o.b . . 3 𝐵 = (Base‘𝐾)
2 eqid 2735 . . 3 (le‘𝐾) = (le‘𝐾)
3 laut1o.i . . 3 𝐼 = (LAut‘𝐾)
41, 2, 3islaut 40066 . 2 (𝐾𝐴 → (𝐹𝐼 ↔ (𝐹:𝐵1-1-onto𝐵 ∧ ∀𝑥𝐵𝑦𝐵 (𝑥(le‘𝐾)𝑦 ↔ (𝐹𝑥)(le‘𝐾)(𝐹𝑦)))))
54simprbda 498 1 ((𝐾𝐴𝐹𝐼) → 𝐹:𝐵1-1-onto𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1537  wcel 2106  wral 3059   class class class wbr 5148  1-1-ontowf1o 6562  cfv 6563  Basecbs 17245  lecple 17305  LAutclaut 39968
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 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-map 8867  df-laut 39972
This theorem is referenced by:  laut11  40069  lautcl  40070  lautcnvclN  40071  lautcnvle  40072  lautcnv  40073  lautcvr  40075  lautj  40076  lautm  40077  lautco  40080  ldil1o  40095  ltrn1o  40107
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