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Theorem pautsetN 34850
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 3203 . 2 (𝐾𝐵𝐾 ∈ V)
2 pautset.m . . 3 𝑀 = (PAut‘𝐾)
3 fveq2 6150 . . . . . . . . 9 (𝑘 = 𝐾 → (PSubSp‘𝑘) = (PSubSp‘𝐾))
4 pautset.s . . . . . . . . 9 𝑆 = (PSubSp‘𝐾)
53, 4syl6eqr 2678 . . . . . . . 8 (𝑘 = 𝐾 → (PSubSp‘𝑘) = 𝑆)
6 f1oeq2 6087 . . . . . . . 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 6088 . . . . . . . 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 3138 . . . . . . 7 (𝑘 = 𝐾 → (∀𝑦 ∈ (PSubSp‘𝑘)(𝑥𝑦 ↔ (𝑓𝑥) ⊆ (𝑓𝑦)) ↔ ∀𝑦𝑆 (𝑥𝑦 ↔ (𝑓𝑥) ⊆ (𝑓𝑦))))
125, 11raleqbidv 3146 . . . . . 6 (𝑘 = 𝐾 → (∀𝑥 ∈ (PSubSp‘𝑘)∀𝑦 ∈ (PSubSp‘𝑘)(𝑥𝑦 ↔ (𝑓𝑥) ⊆ (𝑓𝑦)) ↔ ∀𝑥𝑆𝑦𝑆 (𝑥𝑦 ↔ (𝑓𝑥) ⊆ (𝑓𝑦))))
1310, 12anbi12d 746 . . . . 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 2744 . . . 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 34743 . . . 4 PAut = (𝑘 ∈ V ↦ {𝑓 ∣ (𝑓:(PSubSp‘𝑘)–1-1-onto→(PSubSp‘𝑘) ∧ ∀𝑥 ∈ (PSubSp‘𝑘)∀𝑦 ∈ (PSubSp‘𝑘)(𝑥𝑦 ↔ (𝑓𝑥) ⊆ (𝑓𝑦)))})
16 fvex 6160 . . . . . . . . 9 (PSubSp‘𝐾) ∈ V
174, 16eqeltri 2700 . . . . . . . 8 𝑆 ∈ V
1817, 17mapval 7815 . . . . . . 7 (𝑆𝑚 𝑆) = {𝑓𝑓:𝑆𝑆}
19 ovex 6633 . . . . . . 7 (𝑆𝑚 𝑆) ∈ V
2018, 19eqeltrri 2701 . . . . . 6 {𝑓𝑓:𝑆𝑆} ∈ V
21 f1of 6096 . . . . . . 7 (𝑓:𝑆1-1-onto𝑆𝑓:𝑆𝑆)
2221ss2abi 3658 . . . . . 6 {𝑓𝑓:𝑆1-1-onto𝑆} ⊆ {𝑓𝑓:𝑆𝑆}
2320, 22ssexi 4768 . . . . 5 {𝑓𝑓:𝑆1-1-onto𝑆} ∈ V
24 simpl 473 . . . . . 6 ((𝑓:𝑆1-1-onto𝑆 ∧ ∀𝑥𝑆𝑦𝑆 (𝑥𝑦 ↔ (𝑓𝑥) ⊆ (𝑓𝑦))) → 𝑓:𝑆1-1-onto𝑆)
2524ss2abi 3658 . . . . 5 {𝑓 ∣ (𝑓:𝑆1-1-onto𝑆 ∧ ∀𝑥𝑆𝑦𝑆 (𝑥𝑦 ↔ (𝑓𝑥) ⊆ (𝑓𝑦)))} ⊆ {𝑓𝑓:𝑆1-1-onto𝑆}
2623, 25ssexi 4768 . . . 4 {𝑓 ∣ (𝑓:𝑆1-1-onto𝑆 ∧ ∀𝑥𝑆𝑦𝑆 (𝑥𝑦 ↔ (𝑓𝑥) ⊆ (𝑓𝑦)))} ∈ V
2714, 15, 26fvmpt 6240 . . 3 (𝐾 ∈ V → (PAut‘𝐾) = {𝑓 ∣ (𝑓:𝑆1-1-onto𝑆 ∧ ∀𝑥𝑆𝑦𝑆 (𝑥𝑦 ↔ (𝑓𝑥) ⊆ (𝑓𝑦)))})
282, 27syl5eq 2672 . 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 384   = wceq 1480  wcel 1992  {cab 2612  wral 2912  Vcvv 3191  wss 3560  wf 5846  1-1-ontowf1o 5849  cfv 5850  (class class class)co 6605  𝑚 cmap 7803  PSubSpcpsubsp 34248  PAutcpautN 34739
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6903
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3193  df-sbc 3423  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-iota 5813  df-fun 5852  df-fn 5853  df-f 5854  df-f1 5855  df-fo 5856  df-f1o 5857  df-fv 5858  df-ov 6608  df-oprab 6609  df-mpt2 6610  df-map 7805  df-pautN 34743
This theorem is referenced by:  ispautN  34851
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