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Theorem lautset 35788
Description: The set of lattice automorphisms. (Contributed by NM, 11-May-2012.)
Hypotheses
Ref Expression
lautset.b 𝐵 = (Base‘𝐾)
lautset.l = (le‘𝐾)
lautset.i 𝐼 = (LAut‘𝐾)
Assertion
Ref Expression
lautset (𝐾𝐴𝐼 = {𝑓 ∣ (𝑓:𝐵1-1-onto𝐵 ∧ ∀𝑥𝐵𝑦𝐵 (𝑥 𝑦 ↔ (𝑓𝑥) (𝑓𝑦)))})
Distinct variable groups:   𝑥,𝑓,𝑦,𝐵   𝑓,𝐾,𝑥,𝑦   ,𝑓
Allowed substitution hints:   𝐴(𝑥,𝑦,𝑓)   𝐼(𝑥,𝑦,𝑓)   (𝑥,𝑦)

Proof of Theorem lautset
Dummy variable 𝑘 is distinct from all other variables.
StepHypRef Expression
1 elex 3316 . 2 (𝐾𝐴𝐾 ∈ V)
2 lautset.i . . 3 𝐼 = (LAut‘𝐾)
3 fveq2 6304 . . . . . . . . 9 (𝑘 = 𝐾 → (Base‘𝑘) = (Base‘𝐾))
4 lautset.b . . . . . . . . 9 𝐵 = (Base‘𝐾)
53, 4syl6eqr 2776 . . . . . . . 8 (𝑘 = 𝐾 → (Base‘𝑘) = 𝐵)
6 f1oeq2 6241 . . . . . . . 8 ((Base‘𝑘) = 𝐵 → (𝑓:(Base‘𝑘)–1-1-onto→(Base‘𝑘) ↔ 𝑓:𝐵1-1-onto→(Base‘𝑘)))
75, 6syl 17 . . . . . . 7 (𝑘 = 𝐾 → (𝑓:(Base‘𝑘)–1-1-onto→(Base‘𝑘) ↔ 𝑓:𝐵1-1-onto→(Base‘𝑘)))
8 f1oeq3 6242 . . . . . . . 8 ((Base‘𝑘) = 𝐵 → (𝑓:𝐵1-1-onto→(Base‘𝑘) ↔ 𝑓:𝐵1-1-onto𝐵))
95, 8syl 17 . . . . . . 7 (𝑘 = 𝐾 → (𝑓:𝐵1-1-onto→(Base‘𝑘) ↔ 𝑓:𝐵1-1-onto𝐵))
107, 9bitrd 268 . . . . . 6 (𝑘 = 𝐾 → (𝑓:(Base‘𝑘)–1-1-onto→(Base‘𝑘) ↔ 𝑓:𝐵1-1-onto𝐵))
11 fveq2 6304 . . . . . . . . . . 11 (𝑘 = 𝐾 → (le‘𝑘) = (le‘𝐾))
12 lautset.l . . . . . . . . . . 11 = (le‘𝐾)
1311, 12syl6eqr 2776 . . . . . . . . . 10 (𝑘 = 𝐾 → (le‘𝑘) = )
1413breqd 4771 . . . . . . . . 9 (𝑘 = 𝐾 → (𝑥(le‘𝑘)𝑦𝑥 𝑦))
1513breqd 4771 . . . . . . . . 9 (𝑘 = 𝐾 → ((𝑓𝑥)(le‘𝑘)(𝑓𝑦) ↔ (𝑓𝑥) (𝑓𝑦)))
1614, 15bibi12d 334 . . . . . . . 8 (𝑘 = 𝐾 → ((𝑥(le‘𝑘)𝑦 ↔ (𝑓𝑥)(le‘𝑘)(𝑓𝑦)) ↔ (𝑥 𝑦 ↔ (𝑓𝑥) (𝑓𝑦))))
175, 16raleqbidv 3255 . . . . . . 7 (𝑘 = 𝐾 → (∀𝑦 ∈ (Base‘𝑘)(𝑥(le‘𝑘)𝑦 ↔ (𝑓𝑥)(le‘𝑘)(𝑓𝑦)) ↔ ∀𝑦𝐵 (𝑥 𝑦 ↔ (𝑓𝑥) (𝑓𝑦))))
185, 17raleqbidv 3255 . . . . . 6 (𝑘 = 𝐾 → (∀𝑥 ∈ (Base‘𝑘)∀𝑦 ∈ (Base‘𝑘)(𝑥(le‘𝑘)𝑦 ↔ (𝑓𝑥)(le‘𝑘)(𝑓𝑦)) ↔ ∀𝑥𝐵𝑦𝐵 (𝑥 𝑦 ↔ (𝑓𝑥) (𝑓𝑦))))
1910, 18anbi12d 749 . . . . 5 (𝑘 = 𝐾 → ((𝑓:(Base‘𝑘)–1-1-onto→(Base‘𝑘) ∧ ∀𝑥 ∈ (Base‘𝑘)∀𝑦 ∈ (Base‘𝑘)(𝑥(le‘𝑘)𝑦 ↔ (𝑓𝑥)(le‘𝑘)(𝑓𝑦))) ↔ (𝑓:𝐵1-1-onto𝐵 ∧ ∀𝑥𝐵𝑦𝐵 (𝑥 𝑦 ↔ (𝑓𝑥) (𝑓𝑦)))))
2019abbidv 2843 . . . 4 (𝑘 = 𝐾 → {𝑓 ∣ (𝑓:(Base‘𝑘)–1-1-onto→(Base‘𝑘) ∧ ∀𝑥 ∈ (Base‘𝑘)∀𝑦 ∈ (Base‘𝑘)(𝑥(le‘𝑘)𝑦 ↔ (𝑓𝑥)(le‘𝑘)(𝑓𝑦)))} = {𝑓 ∣ (𝑓:𝐵1-1-onto𝐵 ∧ ∀𝑥𝐵𝑦𝐵 (𝑥 𝑦 ↔ (𝑓𝑥) (𝑓𝑦)))})
21 df-laut 35695 . . . 4 LAut = (𝑘 ∈ V ↦ {𝑓 ∣ (𝑓:(Base‘𝑘)–1-1-onto→(Base‘𝑘) ∧ ∀𝑥 ∈ (Base‘𝑘)∀𝑦 ∈ (Base‘𝑘)(𝑥(le‘𝑘)𝑦 ↔ (𝑓𝑥)(le‘𝑘)(𝑓𝑦)))})
22 fvex 6314 . . . . . . . . 9 (Base‘𝐾) ∈ V
234, 22eqeltri 2799 . . . . . . . 8 𝐵 ∈ V
2423, 23mapval 7986 . . . . . . 7 (𝐵𝑚 𝐵) = {𝑓𝑓:𝐵𝐵}
25 ovex 6793 . . . . . . 7 (𝐵𝑚 𝐵) ∈ V
2624, 25eqeltrri 2800 . . . . . 6 {𝑓𝑓:𝐵𝐵} ∈ V
27 f1of 6250 . . . . . . 7 (𝑓:𝐵1-1-onto𝐵𝑓:𝐵𝐵)
2827ss2abi 3780 . . . . . 6 {𝑓𝑓:𝐵1-1-onto𝐵} ⊆ {𝑓𝑓:𝐵𝐵}
2926, 28ssexi 4911 . . . . 5 {𝑓𝑓:𝐵1-1-onto𝐵} ∈ V
30 simpl 474 . . . . . 6 ((𝑓:𝐵1-1-onto𝐵 ∧ ∀𝑥𝐵𝑦𝐵 (𝑥 𝑦 ↔ (𝑓𝑥) (𝑓𝑦))) → 𝑓:𝐵1-1-onto𝐵)
3130ss2abi 3780 . . . . 5 {𝑓 ∣ (𝑓:𝐵1-1-onto𝐵 ∧ ∀𝑥𝐵𝑦𝐵 (𝑥 𝑦 ↔ (𝑓𝑥) (𝑓𝑦)))} ⊆ {𝑓𝑓:𝐵1-1-onto𝐵}
3229, 31ssexi 4911 . . . 4 {𝑓 ∣ (𝑓:𝐵1-1-onto𝐵 ∧ ∀𝑥𝐵𝑦𝐵 (𝑥 𝑦 ↔ (𝑓𝑥) (𝑓𝑦)))} ∈ V
3320, 21, 32fvmpt 6396 . . 3 (𝐾 ∈ V → (LAut‘𝐾) = {𝑓 ∣ (𝑓:𝐵1-1-onto𝐵 ∧ ∀𝑥𝐵𝑦𝐵 (𝑥 𝑦 ↔ (𝑓𝑥) (𝑓𝑦)))})
342, 33syl5eq 2770 . 2 (𝐾 ∈ V → 𝐼 = {𝑓 ∣ (𝑓:𝐵1-1-onto𝐵 ∧ ∀𝑥𝐵𝑦𝐵 (𝑥 𝑦 ↔ (𝑓𝑥) (𝑓𝑦)))})
351, 34syl 17 1 (𝐾𝐴𝐼 = {𝑓 ∣ (𝑓:𝐵1-1-onto𝐵 ∧ ∀𝑥𝐵𝑦𝐵 (𝑥 𝑦 ↔ (𝑓𝑥) (𝑓𝑦)))})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1596  wcel 2103  {cab 2710  wral 3014  Vcvv 3304   class class class wbr 4760  wf 5997  1-1-ontowf1o 6000  cfv 6001  (class class class)co 6765  𝑚 cmap 7974  Basecbs 15980  lecple 16071  LAutclaut 35691
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-8 2105  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704  ax-sep 4889  ax-nul 4897  ax-pow 4948  ax-pr 5011  ax-un 7066
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1599  df-ex 1818  df-nf 1823  df-sb 2011  df-eu 2575  df-mo 2576  df-clab 2711  df-cleq 2717  df-clel 2720  df-nfc 2855  df-ral 3019  df-rex 3020  df-rab 3023  df-v 3306  df-sbc 3542  df-dif 3683  df-un 3685  df-in 3687  df-ss 3694  df-nul 4024  df-if 4195  df-pw 4268  df-sn 4286  df-pr 4288  df-op 4292  df-uni 4545  df-br 4761  df-opab 4821  df-mpt 4838  df-id 5128  df-xp 5224  df-rel 5225  df-cnv 5226  df-co 5227  df-dm 5228  df-rn 5229  df-iota 5964  df-fun 6003  df-fn 6004  df-f 6005  df-f1 6006  df-fo 6007  df-f1o 6008  df-fv 6009  df-ov 6768  df-oprab 6769  df-mpt2 6770  df-map 7976  df-laut 35695
This theorem is referenced by:  islaut  35789
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