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Theorem ispautN 34403
 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 34402 . . 3 (𝐾𝐵𝑀 = {𝑓 ∣ (𝑓:𝑆1-1-onto𝑆 ∧ ∀𝑥𝑆𝑦𝑆 (𝑥𝑦 ↔ (𝑓𝑥) ⊆ (𝑓𝑦)))})
43eleq2d 2673 . 2 (𝐾𝐵 → (𝐹𝑀𝐹 ∈ {𝑓 ∣ (𝑓:𝑆1-1-onto𝑆 ∧ ∀𝑥𝑆𝑦𝑆 (𝑥𝑦 ↔ (𝑓𝑥) ⊆ (𝑓𝑦)))}))
5 f1of 6050 . . . . 5 (𝐹:𝑆1-1-onto𝑆𝐹:𝑆𝑆)
6 fvex 6113 . . . . . 6 (PSubSp‘𝐾) ∈ V
71, 6eqeltri 2684 . . . . 5 𝑆 ∈ V
8 fex 6394 . . . . 5 ((𝐹:𝑆𝑆𝑆 ∈ V) → 𝐹 ∈ V)
95, 7, 8sylancl 693 . . . 4 (𝐹:𝑆1-1-onto𝑆𝐹 ∈ V)
109adantr 480 . . 3 ((𝐹:𝑆1-1-onto𝑆 ∧ ∀𝑥𝑆𝑦𝑆 (𝑥𝑦 ↔ (𝐹𝑥) ⊆ (𝐹𝑦))) → 𝐹 ∈ V)
11 f1oeq1 6040 . . . 4 (𝑓 = 𝐹 → (𝑓:𝑆1-1-onto𝑆𝐹:𝑆1-1-onto𝑆))
12 fveq1 6102 . . . . . . 7 (𝑓 = 𝐹 → (𝑓𝑥) = (𝐹𝑥))
13 fveq1 6102 . . . . . . 7 (𝑓 = 𝐹 → (𝑓𝑦) = (𝐹𝑦))
1412, 13sseq12d 3597 . . . . . 6 (𝑓 = 𝐹 → ((𝑓𝑥) ⊆ (𝑓𝑦) ↔ (𝐹𝑥) ⊆ (𝐹𝑦)))
1514bibi2d 331 . . . . 5 (𝑓 = 𝐹 → ((𝑥𝑦 ↔ (𝑓𝑥) ⊆ (𝑓𝑦)) ↔ (𝑥𝑦 ↔ (𝐹𝑥) ⊆ (𝐹𝑦))))
16152ralbidv 2972 . . . 4 (𝑓 = 𝐹 → (∀𝑥𝑆𝑦𝑆 (𝑥𝑦 ↔ (𝑓𝑥) ⊆ (𝑓𝑦)) ↔ ∀𝑥𝑆𝑦𝑆 (𝑥𝑦 ↔ (𝐹𝑥) ⊆ (𝐹𝑦))))
1711, 16anbi12d 743 . . 3 (𝑓 = 𝐹 → ((𝑓:𝑆1-1-onto𝑆 ∧ ∀𝑥𝑆𝑦𝑆 (𝑥𝑦 ↔ (𝑓𝑥) ⊆ (𝑓𝑦))) ↔ (𝐹:𝑆1-1-onto𝑆 ∧ ∀𝑥𝑆𝑦𝑆 (𝑥𝑦 ↔ (𝐹𝑥) ⊆ (𝐹𝑦)))))
1810, 17elab3 3327 . 2 (𝐹 ∈ {𝑓 ∣ (𝑓:𝑆1-1-onto𝑆 ∧ ∀𝑥𝑆𝑦𝑆 (𝑥𝑦 ↔ (𝑓𝑥) ⊆ (𝑓𝑦)))} ↔ (𝐹:𝑆1-1-onto𝑆 ∧ ∀𝑥𝑆𝑦𝑆 (𝑥𝑦 ↔ (𝐹𝑥) ⊆ (𝐹𝑦))))
194, 18syl6bb 275 1 (𝐾𝐵 → (𝐹𝑀 ↔ (𝐹:𝑆1-1-onto𝑆 ∧ ∀𝑥𝑆𝑦𝑆 (𝑥𝑦 ↔ (𝐹𝑥) ⊆ (𝐹𝑦)))))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  {cab 2596  ∀wral 2896  Vcvv 3173   ⊆ wss 3540  ⟶wf 5800  –1-1-onto→wf1o 5803  ‘cfv 5804  PSubSpcpsubsp 33800  PAutcpautN 34291 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-map 7746  df-pautN 34295 This theorem is referenced by: (None)
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