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Theorem ispautN 40082
Description: The predicate "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 40081 . . 3 (𝐾𝐵𝑀 = {𝑓 ∣ (𝑓:𝑆1-1-onto𝑆 ∧ ∀𝑥𝑆𝑦𝑆 (𝑥𝑦 ↔ (𝑓𝑥) ⊆ (𝑓𝑦)))})
43eleq2d 2825 . 2 (𝐾𝐵 → (𝐹𝑀𝐹 ∈ {𝑓 ∣ (𝑓:𝑆1-1-onto𝑆 ∧ ∀𝑥𝑆𝑦𝑆 (𝑥𝑦 ↔ (𝑓𝑥) ⊆ (𝑓𝑦)))}))
5 f1of 6849 . . . . 5 (𝐹:𝑆1-1-onto𝑆𝐹:𝑆𝑆)
61fvexi 6921 . . . . 5 𝑆 ∈ V
7 fex 7246 . . . . 5 ((𝐹:𝑆𝑆𝑆 ∈ V) → 𝐹 ∈ V)
85, 6, 7sylancl 586 . . . 4 (𝐹:𝑆1-1-onto𝑆𝐹 ∈ V)
98adantr 480 . . 3 ((𝐹:𝑆1-1-onto𝑆 ∧ ∀𝑥𝑆𝑦𝑆 (𝑥𝑦 ↔ (𝐹𝑥) ⊆ (𝐹𝑦))) → 𝐹 ∈ V)
10 f1oeq1 6837 . . . 4 (𝑓 = 𝐹 → (𝑓:𝑆1-1-onto𝑆𝐹:𝑆1-1-onto𝑆))
11 fveq1 6906 . . . . . . 7 (𝑓 = 𝐹 → (𝑓𝑥) = (𝐹𝑥))
12 fveq1 6906 . . . . . . 7 (𝑓 = 𝐹 → (𝑓𝑦) = (𝐹𝑦))
1311, 12sseq12d 4029 . . . . . 6 (𝑓 = 𝐹 → ((𝑓𝑥) ⊆ (𝑓𝑦) ↔ (𝐹𝑥) ⊆ (𝐹𝑦)))
1413bibi2d 342 . . . . 5 (𝑓 = 𝐹 → ((𝑥𝑦 ↔ (𝑓𝑥) ⊆ (𝑓𝑦)) ↔ (𝑥𝑦 ↔ (𝐹𝑥) ⊆ (𝐹𝑦))))
15142ralbidv 3219 . . . 4 (𝑓 = 𝐹 → (∀𝑥𝑆𝑦𝑆 (𝑥𝑦 ↔ (𝑓𝑥) ⊆ (𝑓𝑦)) ↔ ∀𝑥𝑆𝑦𝑆 (𝑥𝑦 ↔ (𝐹𝑥) ⊆ (𝐹𝑦))))
1610, 15anbi12d 632 . . 3 (𝑓 = 𝐹 → ((𝑓:𝑆1-1-onto𝑆 ∧ ∀𝑥𝑆𝑦𝑆 (𝑥𝑦 ↔ (𝑓𝑥) ⊆ (𝑓𝑦))) ↔ (𝐹:𝑆1-1-onto𝑆 ∧ ∀𝑥𝑆𝑦𝑆 (𝑥𝑦 ↔ (𝐹𝑥) ⊆ (𝐹𝑦)))))
179, 16elab3 3689 . 2 (𝐹 ∈ {𝑓 ∣ (𝑓:𝑆1-1-onto𝑆 ∧ ∀𝑥𝑆𝑦𝑆 (𝑥𝑦 ↔ (𝑓𝑥) ⊆ (𝑓𝑦)))} ↔ (𝐹:𝑆1-1-onto𝑆 ∧ ∀𝑥𝑆𝑦𝑆 (𝑥𝑦 ↔ (𝐹𝑥) ⊆ (𝐹𝑦))))
184, 17bitrdi 287 1 (𝐾𝐵 → (𝐹𝑀 ↔ (𝐹:𝑆1-1-onto𝑆 ∧ ∀𝑥𝑆𝑦𝑆 (𝑥𝑦 ↔ (𝐹𝑥) ⊆ (𝐹𝑦)))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1537  wcel 2106  {cab 2712  wral 3059  Vcvv 3478  wss 3963  wf 6559  1-1-ontowf1o 6562  cfv 6563  PSubSpcpsubsp 39479  PAutcpautN 39970
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-map 8867  df-pautN 39974
This theorem is referenced by: (None)
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