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Theorem pautsetN 35702
Description: The set of projective automorphisms. (Contributed by NM, 26-Jan-2012.) (New usage is discouraged.)
Hypotheses
Ref Expression
pautset.s 𝑆 = (PSubSp‘𝐾)
pautset.m 𝑀 = (PAut‘𝐾)
Assertion
Ref Expression
pautsetN (𝐾𝐵𝑀 = {𝑓 ∣ (𝑓:𝑆1-1-onto𝑆 ∧ ∀𝑥𝑆𝑦𝑆 (𝑥𝑦 ↔ (𝑓𝑥) ⊆ (𝑓𝑦)))})
Distinct variable groups:   𝑥,𝑓,𝑦   𝑓,𝐾,𝑥   𝑆,𝑓,𝑥,𝑦
Allowed substitution hints:   𝐵(𝑥,𝑦,𝑓)   𝐾(𝑦)   𝑀(𝑥,𝑦,𝑓)

Proof of Theorem pautsetN
Dummy variable 𝑘 is distinct from all other variables.
StepHypRef Expression
1 elex 3243 . 2 (𝐾𝐵𝐾 ∈ V)
2 pautset.m . . 3 𝑀 = (PAut‘𝐾)
3 fveq2 6229 . . . . . . . . 9 (𝑘 = 𝐾 → (PSubSp‘𝑘) = (PSubSp‘𝐾))
4 pautset.s . . . . . . . . 9 𝑆 = (PSubSp‘𝐾)
53, 4syl6eqr 2703 . . . . . . . 8 (𝑘 = 𝐾 → (PSubSp‘𝑘) = 𝑆)
6 f1oeq2 6166 . . . . . . . 8 ((PSubSp‘𝑘) = 𝑆 → (𝑓:(PSubSp‘𝑘)–1-1-onto→(PSubSp‘𝑘) ↔ 𝑓:𝑆1-1-onto→(PSubSp‘𝑘)))
75, 6syl 17 . . . . . . 7 (𝑘 = 𝐾 → (𝑓:(PSubSp‘𝑘)–1-1-onto→(PSubSp‘𝑘) ↔ 𝑓:𝑆1-1-onto→(PSubSp‘𝑘)))
8 f1oeq3 6167 . . . . . . . 8 ((PSubSp‘𝑘) = 𝑆 → (𝑓:𝑆1-1-onto→(PSubSp‘𝑘) ↔ 𝑓:𝑆1-1-onto𝑆))
95, 8syl 17 . . . . . . 7 (𝑘 = 𝐾 → (𝑓:𝑆1-1-onto→(PSubSp‘𝑘) ↔ 𝑓:𝑆1-1-onto𝑆))
107, 9bitrd 268 . . . . . 6 (𝑘 = 𝐾 → (𝑓:(PSubSp‘𝑘)–1-1-onto→(PSubSp‘𝑘) ↔ 𝑓:𝑆1-1-onto𝑆))
115raleqdv 3174 . . . . . . 7 (𝑘 = 𝐾 → (∀𝑦 ∈ (PSubSp‘𝑘)(𝑥𝑦 ↔ (𝑓𝑥) ⊆ (𝑓𝑦)) ↔ ∀𝑦𝑆 (𝑥𝑦 ↔ (𝑓𝑥) ⊆ (𝑓𝑦))))
125, 11raleqbidv 3182 . . . . . 6 (𝑘 = 𝐾 → (∀𝑥 ∈ (PSubSp‘𝑘)∀𝑦 ∈ (PSubSp‘𝑘)(𝑥𝑦 ↔ (𝑓𝑥) ⊆ (𝑓𝑦)) ↔ ∀𝑥𝑆𝑦𝑆 (𝑥𝑦 ↔ (𝑓𝑥) ⊆ (𝑓𝑦))))
1310, 12anbi12d 747 . . . . 5 (𝑘 = 𝐾 → ((𝑓:(PSubSp‘𝑘)–1-1-onto→(PSubSp‘𝑘) ∧ ∀𝑥 ∈ (PSubSp‘𝑘)∀𝑦 ∈ (PSubSp‘𝑘)(𝑥𝑦 ↔ (𝑓𝑥) ⊆ (𝑓𝑦))) ↔ (𝑓:𝑆1-1-onto𝑆 ∧ ∀𝑥𝑆𝑦𝑆 (𝑥𝑦 ↔ (𝑓𝑥) ⊆ (𝑓𝑦)))))
1413abbidv 2770 . . . 4 (𝑘 = 𝐾 → {𝑓 ∣ (𝑓:(PSubSp‘𝑘)–1-1-onto→(PSubSp‘𝑘) ∧ ∀𝑥 ∈ (PSubSp‘𝑘)∀𝑦 ∈ (PSubSp‘𝑘)(𝑥𝑦 ↔ (𝑓𝑥) ⊆ (𝑓𝑦)))} = {𝑓 ∣ (𝑓:𝑆1-1-onto𝑆 ∧ ∀𝑥𝑆𝑦𝑆 (𝑥𝑦 ↔ (𝑓𝑥) ⊆ (𝑓𝑦)))})
15 df-pautN 35595 . . . 4 PAut = (𝑘 ∈ V ↦ {𝑓 ∣ (𝑓:(PSubSp‘𝑘)–1-1-onto→(PSubSp‘𝑘) ∧ ∀𝑥 ∈ (PSubSp‘𝑘)∀𝑦 ∈ (PSubSp‘𝑘)(𝑥𝑦 ↔ (𝑓𝑥) ⊆ (𝑓𝑦)))})
16 fvex 6239 . . . . . . . . 9 (PSubSp‘𝐾) ∈ V
174, 16eqeltri 2726 . . . . . . . 8 𝑆 ∈ V
1817, 17mapval 7911 . . . . . . 7 (𝑆𝑚 𝑆) = {𝑓𝑓:𝑆𝑆}
19 ovex 6718 . . . . . . 7 (𝑆𝑚 𝑆) ∈ V
2018, 19eqeltrri 2727 . . . . . 6 {𝑓𝑓:𝑆𝑆} ∈ V
21 f1of 6175 . . . . . . 7 (𝑓:𝑆1-1-onto𝑆𝑓:𝑆𝑆)
2221ss2abi 3707 . . . . . 6 {𝑓𝑓:𝑆1-1-onto𝑆} ⊆ {𝑓𝑓:𝑆𝑆}
2320, 22ssexi 4836 . . . . 5 {𝑓𝑓:𝑆1-1-onto𝑆} ∈ V
24 simpl 472 . . . . . 6 ((𝑓:𝑆1-1-onto𝑆 ∧ ∀𝑥𝑆𝑦𝑆 (𝑥𝑦 ↔ (𝑓𝑥) ⊆ (𝑓𝑦))) → 𝑓:𝑆1-1-onto𝑆)
2524ss2abi 3707 . . . . 5 {𝑓 ∣ (𝑓:𝑆1-1-onto𝑆 ∧ ∀𝑥𝑆𝑦𝑆 (𝑥𝑦 ↔ (𝑓𝑥) ⊆ (𝑓𝑦)))} ⊆ {𝑓𝑓:𝑆1-1-onto𝑆}
2623, 25ssexi 4836 . . . 4 {𝑓 ∣ (𝑓:𝑆1-1-onto𝑆 ∧ ∀𝑥𝑆𝑦𝑆 (𝑥𝑦 ↔ (𝑓𝑥) ⊆ (𝑓𝑦)))} ∈ V
2714, 15, 26fvmpt 6321 . . 3 (𝐾 ∈ V → (PAut‘𝐾) = {𝑓 ∣ (𝑓:𝑆1-1-onto𝑆 ∧ ∀𝑥𝑆𝑦𝑆 (𝑥𝑦 ↔ (𝑓𝑥) ⊆ (𝑓𝑦)))})
282, 27syl5eq 2697 . 2 (𝐾 ∈ V → 𝑀 = {𝑓 ∣ (𝑓:𝑆1-1-onto𝑆 ∧ ∀𝑥𝑆𝑦𝑆 (𝑥𝑦 ↔ (𝑓𝑥) ⊆ (𝑓𝑦)))})
291, 28syl 17 1 (𝐾𝐵𝑀 = {𝑓 ∣ (𝑓:𝑆1-1-onto𝑆 ∧ ∀𝑥𝑆𝑦𝑆 (𝑥𝑦 ↔ (𝑓𝑥) ⊆ (𝑓𝑦)))})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1523  wcel 2030  {cab 2637  wral 2941  Vcvv 3231  wss 3607  wf 5922  1-1-ontowf1o 5925  cfv 5926  (class class class)co 6690  𝑚 cmap 7899  PSubSpcpsubsp 35100  PAutcpautN 35591
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-map 7901  df-pautN 35595
This theorem is referenced by:  ispautN  35703
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