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Theorem pautsetN 34402
 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 3185 . 2 (𝐾𝐵𝐾 ∈ V)
2 pautset.m . . 3 𝑀 = (PAut‘𝐾)
3 fveq2 6103 . . . . . . . . 9 (𝑘 = 𝐾 → (PSubSp‘𝑘) = (PSubSp‘𝐾))
4 pautset.s . . . . . . . . 9 𝑆 = (PSubSp‘𝐾)
53, 4syl6eqr 2662 . . . . . . . 8 (𝑘 = 𝐾 → (PSubSp‘𝑘) = 𝑆)
6 f1oeq2 6041 . . . . . . . 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 6042 . . . . . . . 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 267 . . . . . 6 (𝑘 = 𝐾 → (𝑓:(PSubSp‘𝑘)–1-1-onto→(PSubSp‘𝑘) ↔ 𝑓:𝑆1-1-onto𝑆))
115raleqdv 3121 . . . . . . 7 (𝑘 = 𝐾 → (∀𝑦 ∈ (PSubSp‘𝑘)(𝑥𝑦 ↔ (𝑓𝑥) ⊆ (𝑓𝑦)) ↔ ∀𝑦𝑆 (𝑥𝑦 ↔ (𝑓𝑥) ⊆ (𝑓𝑦))))
125, 11raleqbidv 3129 . . . . . 6 (𝑘 = 𝐾 → (∀𝑥 ∈ (PSubSp‘𝑘)∀𝑦 ∈ (PSubSp‘𝑘)(𝑥𝑦 ↔ (𝑓𝑥) ⊆ (𝑓𝑦)) ↔ ∀𝑥𝑆𝑦𝑆 (𝑥𝑦 ↔ (𝑓𝑥) ⊆ (𝑓𝑦))))
1310, 12anbi12d 743 . . . . 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 2728 . . . 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 34295 . . . 4 PAut = (𝑘 ∈ V ↦ {𝑓 ∣ (𝑓:(PSubSp‘𝑘)–1-1-onto→(PSubSp‘𝑘) ∧ ∀𝑥 ∈ (PSubSp‘𝑘)∀𝑦 ∈ (PSubSp‘𝑘)(𝑥𝑦 ↔ (𝑓𝑥) ⊆ (𝑓𝑦)))})
16 fvex 6113 . . . . . . . . 9 (PSubSp‘𝐾) ∈ V
174, 16eqeltri 2684 . . . . . . . 8 𝑆 ∈ V
1817, 17mapval 7756 . . . . . . 7 (𝑆𝑚 𝑆) = {𝑓𝑓:𝑆𝑆}
19 ovex 6577 . . . . . . 7 (𝑆𝑚 𝑆) ∈ V
2018, 19eqeltrri 2685 . . . . . 6 {𝑓𝑓:𝑆𝑆} ∈ V
21 f1of 6050 . . . . . . 7 (𝑓:𝑆1-1-onto𝑆𝑓:𝑆𝑆)
2221ss2abi 3637 . . . . . 6 {𝑓𝑓:𝑆1-1-onto𝑆} ⊆ {𝑓𝑓:𝑆𝑆}
2320, 22ssexi 4731 . . . . 5 {𝑓𝑓:𝑆1-1-onto𝑆} ∈ V
24 simpl 472 . . . . . 6 ((𝑓:𝑆1-1-onto𝑆 ∧ ∀𝑥𝑆𝑦𝑆 (𝑥𝑦 ↔ (𝑓𝑥) ⊆ (𝑓𝑦))) → 𝑓:𝑆1-1-onto𝑆)
2524ss2abi 3637 . . . . 5 {𝑓 ∣ (𝑓:𝑆1-1-onto𝑆 ∧ ∀𝑥𝑆𝑦𝑆 (𝑥𝑦 ↔ (𝑓𝑥) ⊆ (𝑓𝑦)))} ⊆ {𝑓𝑓:𝑆1-1-onto𝑆}
2623, 25ssexi 4731 . . . 4 {𝑓 ∣ (𝑓:𝑆1-1-onto𝑆 ∧ ∀𝑥𝑆𝑦𝑆 (𝑥𝑦 ↔ (𝑓𝑥) ⊆ (𝑓𝑦)))} ∈ V
2714, 15, 26fvmpt 6191 . . 3 (𝐾 ∈ V → (PAut‘𝐾) = {𝑓 ∣ (𝑓:𝑆1-1-onto𝑆 ∧ ∀𝑥𝑆𝑦𝑆 (𝑥𝑦 ↔ (𝑓𝑥) ⊆ (𝑓𝑦)))})
282, 27syl5eq 2656 . 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 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  {cab 2596  ∀wral 2896  Vcvv 3173   ⊆ wss 3540  ⟶wf 5800  –1-1-onto→wf1o 5803  ‘cfv 5804  (class class class)co 6549   ↑𝑚 cmap 7744  PSubSpcpsubsp 33800  PAutcpautN 34291 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-map 7746  df-pautN 34295 This theorem is referenced by:  ispautN  34403
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