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Theorem pautsetN 38611
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 3465 . 2 (𝐾𝐵𝐾 ∈ V)
2 pautset.m . . 3 𝑀 = (PAut‘𝐾)
3 fveq2 6846 . . . . . . . . 9 (𝑘 = 𝐾 → (PSubSp‘𝑘) = (PSubSp‘𝐾))
4 pautset.s . . . . . . . . 9 𝑆 = (PSubSp‘𝐾)
53, 4eqtr4di 2791 . . . . . . . 8 (𝑘 = 𝐾 → (PSubSp‘𝑘) = 𝑆)
65f1oeq2d 6784 . . . . . . 7 (𝑘 = 𝐾 → (𝑓:(PSubSp‘𝑘)–1-1-onto→(PSubSp‘𝑘) ↔ 𝑓:𝑆1-1-onto→(PSubSp‘𝑘)))
7 f1oeq3 6778 . . . . . . . 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 279 . . . . . 6 (𝑘 = 𝐾 → (𝑓:(PSubSp‘𝑘)–1-1-onto→(PSubSp‘𝑘) ↔ 𝑓:𝑆1-1-onto𝑆))
105raleqdv 3312 . . . . . . 7 (𝑘 = 𝐾 → (∀𝑦 ∈ (PSubSp‘𝑘)(𝑥𝑦 ↔ (𝑓𝑥) ⊆ (𝑓𝑦)) ↔ ∀𝑦𝑆 (𝑥𝑦 ↔ (𝑓𝑥) ⊆ (𝑓𝑦))))
115, 10raleqbidv 3318 . . . . . 6 (𝑘 = 𝐾 → (∀𝑥 ∈ (PSubSp‘𝑘)∀𝑦 ∈ (PSubSp‘𝑘)(𝑥𝑦 ↔ (𝑓𝑥) ⊆ (𝑓𝑦)) ↔ ∀𝑥𝑆𝑦𝑆 (𝑥𝑦 ↔ (𝑓𝑥) ⊆ (𝑓𝑦))))
129, 11anbi12d 632 . . . . 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 2802 . . . 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 38504 . . . 4 PAut = (𝑘 ∈ V ↦ {𝑓 ∣ (𝑓:(PSubSp‘𝑘)–1-1-onto→(PSubSp‘𝑘) ∧ ∀𝑥 ∈ (PSubSp‘𝑘)∀𝑦 ∈ (PSubSp‘𝑘)(𝑥𝑦 ↔ (𝑓𝑥) ⊆ (𝑓𝑦)))})
154fvexi 6860 . . . . . . . 8 𝑆 ∈ V
1615, 15mapval 8783 . . . . . . 7 (𝑆m 𝑆) = {𝑓𝑓:𝑆𝑆}
17 ovex 7394 . . . . . . 7 (𝑆m 𝑆) ∈ V
1816, 17eqeltrri 2831 . . . . . 6 {𝑓𝑓:𝑆𝑆} ∈ V
19 f1of 6788 . . . . . . 7 (𝑓:𝑆1-1-onto𝑆𝑓:𝑆𝑆)
2019ss2abi 4027 . . . . . 6 {𝑓𝑓:𝑆1-1-onto𝑆} ⊆ {𝑓𝑓:𝑆𝑆}
2118, 20ssexi 5283 . . . . 5 {𝑓𝑓:𝑆1-1-onto𝑆} ∈ V
22 simpl 484 . . . . . 6 ((𝑓:𝑆1-1-onto𝑆 ∧ ∀𝑥𝑆𝑦𝑆 (𝑥𝑦 ↔ (𝑓𝑥) ⊆ (𝑓𝑦))) → 𝑓:𝑆1-1-onto𝑆)
2322ss2abi 4027 . . . . 5 {𝑓 ∣ (𝑓:𝑆1-1-onto𝑆 ∧ ∀𝑥𝑆𝑦𝑆 (𝑥𝑦 ↔ (𝑓𝑥) ⊆ (𝑓𝑦)))} ⊆ {𝑓𝑓:𝑆1-1-onto𝑆}
2421, 23ssexi 5283 . . . 4 {𝑓 ∣ (𝑓:𝑆1-1-onto𝑆 ∧ ∀𝑥𝑆𝑦𝑆 (𝑥𝑦 ↔ (𝑓𝑥) ⊆ (𝑓𝑦)))} ∈ V
2513, 14, 24fvmpt 6952 . . 3 (𝐾 ∈ V → (PAut‘𝐾) = {𝑓 ∣ (𝑓:𝑆1-1-onto𝑆 ∧ ∀𝑥𝑆𝑦𝑆 (𝑥𝑦 ↔ (𝑓𝑥) ⊆ (𝑓𝑦)))})
262, 25eqtrid 2785 . 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 205  wa 397   = wceq 1542  wcel 2107  {cab 2710  wral 3061  Vcvv 3447  wss 3914  wf 6496  1-1-ontowf1o 6499  cfv 6500  (class class class)co 7361  m cmap 8771  PSubSpcpsubsp 38009  PAutcpautN 38500
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3449  df-sbc 3744  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-br 5110  df-opab 5172  df-mpt 5193  df-id 5535  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-ov 7364  df-oprab 7365  df-mpo 7366  df-map 8773  df-pautN 38504
This theorem is referenced by:  ispautN  38612
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