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Theorem ispautN 37239
 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 37238 . . 3 (𝐾𝐵𝑀 = {𝑓 ∣ (𝑓:𝑆1-1-onto𝑆 ∧ ∀𝑥𝑆𝑦𝑆 (𝑥𝑦 ↔ (𝑓𝑥) ⊆ (𝑓𝑦)))})
43eleq2d 2901 . 2 (𝐾𝐵 → (𝐹𝑀𝐹 ∈ {𝑓 ∣ (𝑓:𝑆1-1-onto𝑆 ∧ ∀𝑥𝑆𝑦𝑆 (𝑥𝑦 ↔ (𝑓𝑥) ⊆ (𝑓𝑦)))}))
5 f1of 6618 . . . . 5 (𝐹:𝑆1-1-onto𝑆𝐹:𝑆𝑆)
61fvexi 6687 . . . . 5 𝑆 ∈ V
7 fex 6992 . . . . 5 ((𝐹:𝑆𝑆𝑆 ∈ V) → 𝐹 ∈ V)
85, 6, 7sylancl 588 . . . 4 (𝐹:𝑆1-1-onto𝑆𝐹 ∈ V)
98adantr 483 . . 3 ((𝐹:𝑆1-1-onto𝑆 ∧ ∀𝑥𝑆𝑦𝑆 (𝑥𝑦 ↔ (𝐹𝑥) ⊆ (𝐹𝑦))) → 𝐹 ∈ V)
10 f1oeq1 6607 . . . 4 (𝑓 = 𝐹 → (𝑓:𝑆1-1-onto𝑆𝐹:𝑆1-1-onto𝑆))
11 fveq1 6672 . . . . . . 7 (𝑓 = 𝐹 → (𝑓𝑥) = (𝐹𝑥))
12 fveq1 6672 . . . . . . 7 (𝑓 = 𝐹 → (𝑓𝑦) = (𝐹𝑦))
1311, 12sseq12d 4003 . . . . . 6 (𝑓 = 𝐹 → ((𝑓𝑥) ⊆ (𝑓𝑦) ↔ (𝐹𝑥) ⊆ (𝐹𝑦)))
1413bibi2d 345 . . . . 5 (𝑓 = 𝐹 → ((𝑥𝑦 ↔ (𝑓𝑥) ⊆ (𝑓𝑦)) ↔ (𝑥𝑦 ↔ (𝐹𝑥) ⊆ (𝐹𝑦))))
15142ralbidv 3202 . . . 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 3677 . 2 (𝐹 ∈ {𝑓 ∣ (𝑓:𝑆1-1-onto𝑆 ∧ ∀𝑥𝑆𝑦𝑆 (𝑥𝑦 ↔ (𝑓𝑥) ⊆ (𝑓𝑦)))} ↔ (𝐹:𝑆1-1-onto𝑆 ∧ ∀𝑥𝑆𝑦𝑆 (𝑥𝑦 ↔ (𝐹𝑥) ⊆ (𝐹𝑦))))
184, 17syl6bb 289 1 (𝐾𝐵 → (𝐹𝑀 ↔ (𝐹:𝑆1-1-onto𝑆 ∧ ∀𝑥𝑆𝑦𝑆 (𝑥𝑦 ↔ (𝐹𝑥) ⊆ (𝐹𝑦)))))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   = wceq 1536   ∈ wcel 2113  {cab 2802  ∀wral 3141  Vcvv 3497   ⊆ wss 3939  ⟶wf 6354  –1-1-onto→wf1o 6357  ‘cfv 6358  PSubSpcpsubsp 36636  PAutcpautN 37127 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2796  ax-rep 5193  ax-sep 5206  ax-nul 5213  ax-pow 5269  ax-pr 5333  ax-un 7464 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-mo 2621  df-eu 2653  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2966  df-ne 3020  df-ral 3146  df-rex 3147  df-reu 3148  df-rab 3150  df-v 3499  df-sbc 3776  df-csb 3887  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4471  df-pw 4544  df-sn 4571  df-pr 4573  df-op 4577  df-uni 4842  df-iun 4924  df-br 5070  df-opab 5132  df-mpt 5150  df-id 5463  df-xp 5564  df-rel 5565  df-cnv 5566  df-co 5567  df-dm 5568  df-rn 5569  df-res 5570  df-ima 5571  df-iota 6317  df-fun 6360  df-fn 6361  df-f 6362  df-f1 6363  df-fo 6364  df-f1o 6365  df-fv 6366  df-ov 7162  df-oprab 7163  df-mpo 7164  df-map 8411  df-pautN 37131 This theorem is referenced by: (None)
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