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Theorem ispautN 35703
Description: The predictate "is a projective automorphism." (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
ispautN (𝐾𝐵 → (𝐹𝑀 ↔ (𝐹:𝑆1-1-onto𝑆 ∧ ∀𝑥𝑆𝑦𝑆 (𝑥𝑦 ↔ (𝐹𝑥) ⊆ (𝐹𝑦)))))
Distinct variable groups:   𝑥,𝑦,𝐹   𝑥,𝐾   𝑥,𝑆,𝑦
Allowed substitution hints:   𝐵(𝑥,𝑦)   𝐾(𝑦)   𝑀(𝑥,𝑦)

Proof of Theorem ispautN
Dummy variable 𝑓 is distinct from all other variables.
StepHypRef Expression
1 pautset.s . . . 4 𝑆 = (PSubSp‘𝐾)
2 pautset.m . . . 4 𝑀 = (PAut‘𝐾)
31, 2pautsetN 35702 . . 3 (𝐾𝐵𝑀 = {𝑓 ∣ (𝑓:𝑆1-1-onto𝑆 ∧ ∀𝑥𝑆𝑦𝑆 (𝑥𝑦 ↔ (𝑓𝑥) ⊆ (𝑓𝑦)))})
43eleq2d 2716 . 2 (𝐾𝐵 → (𝐹𝑀𝐹 ∈ {𝑓 ∣ (𝑓:𝑆1-1-onto𝑆 ∧ ∀𝑥𝑆𝑦𝑆 (𝑥𝑦 ↔ (𝑓𝑥) ⊆ (𝑓𝑦)))}))
5 f1of 6175 . . . . 5 (𝐹:𝑆1-1-onto𝑆𝐹:𝑆𝑆)
6 fvex 6239 . . . . . 6 (PSubSp‘𝐾) ∈ V
71, 6eqeltri 2726 . . . . 5 𝑆 ∈ V
8 fex 6530 . . . . 5 ((𝐹:𝑆𝑆𝑆 ∈ V) → 𝐹 ∈ V)
95, 7, 8sylancl 695 . . . 4 (𝐹:𝑆1-1-onto𝑆𝐹 ∈ V)
109adantr 480 . . 3 ((𝐹:𝑆1-1-onto𝑆 ∧ ∀𝑥𝑆𝑦𝑆 (𝑥𝑦 ↔ (𝐹𝑥) ⊆ (𝐹𝑦))) → 𝐹 ∈ V)
11 f1oeq1 6165 . . . 4 (𝑓 = 𝐹 → (𝑓:𝑆1-1-onto𝑆𝐹:𝑆1-1-onto𝑆))
12 fveq1 6228 . . . . . . 7 (𝑓 = 𝐹 → (𝑓𝑥) = (𝐹𝑥))
13 fveq1 6228 . . . . . . 7 (𝑓 = 𝐹 → (𝑓𝑦) = (𝐹𝑦))
1412, 13sseq12d 3667 . . . . . 6 (𝑓 = 𝐹 → ((𝑓𝑥) ⊆ (𝑓𝑦) ↔ (𝐹𝑥) ⊆ (𝐹𝑦)))
1514bibi2d 331 . . . . 5 (𝑓 = 𝐹 → ((𝑥𝑦 ↔ (𝑓𝑥) ⊆ (𝑓𝑦)) ↔ (𝑥𝑦 ↔ (𝐹𝑥) ⊆ (𝐹𝑦))))
16152ralbidv 3018 . . . 4 (𝑓 = 𝐹 → (∀𝑥𝑆𝑦𝑆 (𝑥𝑦 ↔ (𝑓𝑥) ⊆ (𝑓𝑦)) ↔ ∀𝑥𝑆𝑦𝑆 (𝑥𝑦 ↔ (𝐹𝑥) ⊆ (𝐹𝑦))))
1711, 16anbi12d 747 . . 3 (𝑓 = 𝐹 → ((𝑓:𝑆1-1-onto𝑆 ∧ ∀𝑥𝑆𝑦𝑆 (𝑥𝑦 ↔ (𝑓𝑥) ⊆ (𝑓𝑦))) ↔ (𝐹:𝑆1-1-onto𝑆 ∧ ∀𝑥𝑆𝑦𝑆 (𝑥𝑦 ↔ (𝐹𝑥) ⊆ (𝐹𝑦)))))
1810, 17elab3 3390 . 2 (𝐹 ∈ {𝑓 ∣ (𝑓:𝑆1-1-onto𝑆 ∧ ∀𝑥𝑆𝑦𝑆 (𝑥𝑦 ↔ (𝑓𝑥) ⊆ (𝑓𝑦)))} ↔ (𝐹:𝑆1-1-onto𝑆 ∧ ∀𝑥𝑆𝑦𝑆 (𝑥𝑦 ↔ (𝐹𝑥) ⊆ (𝐹𝑦))))
194, 18syl6bb 276 1 (𝐾𝐵 → (𝐹𝑀 ↔ (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  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-rep 4804  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-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  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-iun 4554  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-res 5155  df-ima 5156  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: (None)
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