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Theorem pautsetN 37306
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 3498 . 2 (𝐾𝐵𝐾 ∈ V)
2 pautset.m . . 3 𝑀 = (PAut‘𝐾)
3 fveq2 6659 . . . . . . . . 9 (𝑘 = 𝐾 → (PSubSp‘𝑘) = (PSubSp‘𝐾))
4 pautset.s . . . . . . . . 9 𝑆 = (PSubSp‘𝐾)
53, 4syl6eqr 2877 . . . . . . . 8 (𝑘 = 𝐾 → (PSubSp‘𝑘) = 𝑆)
65f1oeq2d 6600 . . . . . . 7 (𝑘 = 𝐾 → (𝑓:(PSubSp‘𝑘)–1-1-onto→(PSubSp‘𝑘) ↔ 𝑓:𝑆1-1-onto→(PSubSp‘𝑘)))
7 f1oeq3 6595 . . . . . . . 8 ((PSubSp‘𝑘) = 𝑆 → (𝑓:𝑆1-1-onto→(PSubSp‘𝑘) ↔ 𝑓:𝑆1-1-onto𝑆))
85, 7syl 17 . . . . . . 7 (𝑘 = 𝐾 → (𝑓:𝑆1-1-onto→(PSubSp‘𝑘) ↔ 𝑓:𝑆1-1-onto𝑆))
96, 8bitrd 282 . . . . . 6 (𝑘 = 𝐾 → (𝑓:(PSubSp‘𝑘)–1-1-onto→(PSubSp‘𝑘) ↔ 𝑓:𝑆1-1-onto𝑆))
105raleqdv 3403 . . . . . . 7 (𝑘 = 𝐾 → (∀𝑦 ∈ (PSubSp‘𝑘)(𝑥𝑦 ↔ (𝑓𝑥) ⊆ (𝑓𝑦)) ↔ ∀𝑦𝑆 (𝑥𝑦 ↔ (𝑓𝑥) ⊆ (𝑓𝑦))))
115, 10raleqbidv 3393 . . . . . 6 (𝑘 = 𝐾 → (∀𝑥 ∈ (PSubSp‘𝑘)∀𝑦 ∈ (PSubSp‘𝑘)(𝑥𝑦 ↔ (𝑓𝑥) ⊆ (𝑓𝑦)) ↔ ∀𝑥𝑆𝑦𝑆 (𝑥𝑦 ↔ (𝑓𝑥) ⊆ (𝑓𝑦))))
129, 11anbi12d 633 . . . . 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)))))
1312abbidv 2888 . . . 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)))})
14 df-pautN 37199 . . . 4 PAut = (𝑘 ∈ V ↦ {𝑓 ∣ (𝑓:(PSubSp‘𝑘)–1-1-onto→(PSubSp‘𝑘) ∧ ∀𝑥 ∈ (PSubSp‘𝑘)∀𝑦 ∈ (PSubSp‘𝑘)(𝑥𝑦 ↔ (𝑓𝑥) ⊆ (𝑓𝑦)))})
154fvexi 6673 . . . . . . . 8 𝑆 ∈ V
1615, 15mapval 8410 . . . . . . 7 (𝑆m 𝑆) = {𝑓𝑓:𝑆𝑆}
17 ovex 7179 . . . . . . 7 (𝑆m 𝑆) ∈ V
1816, 17eqeltrri 2913 . . . . . 6 {𝑓𝑓:𝑆𝑆} ∈ V
19 f1of 6604 . . . . . . 7 (𝑓:𝑆1-1-onto𝑆𝑓:𝑆𝑆)
2019ss2abi 4029 . . . . . 6 {𝑓𝑓:𝑆1-1-onto𝑆} ⊆ {𝑓𝑓:𝑆𝑆}
2118, 20ssexi 5213 . . . . 5 {𝑓𝑓:𝑆1-1-onto𝑆} ∈ V
22 simpl 486 . . . . . 6 ((𝑓:𝑆1-1-onto𝑆 ∧ ∀𝑥𝑆𝑦𝑆 (𝑥𝑦 ↔ (𝑓𝑥) ⊆ (𝑓𝑦))) → 𝑓:𝑆1-1-onto𝑆)
2322ss2abi 4029 . . . . 5 {𝑓 ∣ (𝑓:𝑆1-1-onto𝑆 ∧ ∀𝑥𝑆𝑦𝑆 (𝑥𝑦 ↔ (𝑓𝑥) ⊆ (𝑓𝑦)))} ⊆ {𝑓𝑓:𝑆1-1-onto𝑆}
2421, 23ssexi 5213 . . . 4 {𝑓 ∣ (𝑓:𝑆1-1-onto𝑆 ∧ ∀𝑥𝑆𝑦𝑆 (𝑥𝑦 ↔ (𝑓𝑥) ⊆ (𝑓𝑦)))} ∈ V
2513, 14, 24fvmpt 6757 . . 3 (𝐾 ∈ V → (PAut‘𝐾) = {𝑓 ∣ (𝑓:𝑆1-1-onto𝑆 ∧ ∀𝑥𝑆𝑦𝑆 (𝑥𝑦 ↔ (𝑓𝑥) ⊆ (𝑓𝑦)))})
262, 25syl5eq 2871 . 2 (𝐾 ∈ V → 𝑀 = {𝑓 ∣ (𝑓:𝑆1-1-onto𝑆 ∧ ∀𝑥𝑆𝑦𝑆 (𝑥𝑦 ↔ (𝑓𝑥) ⊆ (𝑓𝑦)))})
271, 26syl 17 1 (𝐾𝐵𝑀 = {𝑓 ∣ (𝑓:𝑆1-1-onto𝑆 ∧ ∀𝑥𝑆𝑦𝑆 (𝑥𝑦 ↔ (𝑓𝑥) ⊆ (𝑓𝑦)))})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1538  wcel 2115  {cab 2802  wral 3133  Vcvv 3480  wss 3919  wf 6340  1-1-ontowf1o 6343  cfv 6344  (class class class)co 7146  m cmap 8398  PSubSpcpsubsp 36704  PAutcpautN 37195
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5190  ax-nul 5197  ax-pow 5254  ax-pr 5318  ax-un 7452
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-sbc 3759  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4826  df-br 5054  df-opab 5116  df-mpt 5134  df-id 5448  df-xp 5549  df-rel 5550  df-cnv 5551  df-co 5552  df-dm 5553  df-rn 5554  df-iota 6303  df-fun 6346  df-fn 6347  df-f 6348  df-f1 6349  df-fo 6350  df-f1o 6351  df-fv 6352  df-ov 7149  df-oprab 7150  df-mpo 7151  df-map 8400  df-pautN 37199
This theorem is referenced by:  ispautN  37307
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