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Theorem lautcnvclN 38089
Description: Reverse closure of a lattice automorphism. (Contributed by NM, 25-May-2012.) (New usage is discouraged.)
Hypotheses
Ref Expression
laut1o.b 𝐵 = (Base‘𝐾)
laut1o.i 𝐼 = (LAut‘𝐾)
Assertion
Ref Expression
lautcnvclN (((𝐾𝑉𝐹𝐼) ∧ 𝑋𝐵) → (𝐹𝑋) ∈ 𝐵)

Proof of Theorem lautcnvclN
StepHypRef Expression
1 laut1o.b . . 3 𝐵 = (Base‘𝐾)
2 laut1o.i . . 3 𝐼 = (LAut‘𝐾)
31, 2laut1o 38086 . 2 ((𝐾𝑉𝐹𝐼) → 𝐹:𝐵1-1-onto𝐵)
4 f1ocnvdm 7151 . 2 ((𝐹:𝐵1-1-onto𝐵𝑋𝐵) → (𝐹𝑋) ∈ 𝐵)
53, 4sylan 580 1 (((𝐾𝑉𝐹𝐼) ∧ 𝑋𝐵) → (𝐹𝑋) ∈ 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1539  wcel 2106  ccnv 5585  1-1-ontowf1o 6427  cfv 6428  Basecbs 16901  LAutclaut 37986
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5210  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7580
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3433  df-sbc 3718  df-csb 3834  df-dif 3891  df-un 3893  df-in 3895  df-ss 3905  df-nul 4259  df-if 4462  df-pw 4537  df-sn 4564  df-pr 4566  df-op 4570  df-uni 4842  df-iun 4928  df-br 5076  df-opab 5138  df-mpt 5159  df-id 5486  df-xp 5592  df-rel 5593  df-cnv 5594  df-co 5595  df-dm 5596  df-rn 5597  df-res 5598  df-ima 5599  df-iota 6386  df-fun 6430  df-fn 6431  df-f 6432  df-f1 6433  df-fo 6434  df-f1o 6435  df-fv 6436  df-ov 7272  df-oprab 7273  df-mpo 7274  df-map 8606  df-laut 37990
This theorem is referenced by: (None)
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