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Theorem mirf 25355
 Description: Point inversion as function. (Contributed by Thierry Arnoux, 30-May-2019.)
Hypotheses
Ref Expression
mirval.p 𝑃 = (Base‘𝐺)
mirval.d = (dist‘𝐺)
mirval.i 𝐼 = (Itv‘𝐺)
mirval.l 𝐿 = (LineG‘𝐺)
mirval.s 𝑆 = (pInvG‘𝐺)
mirval.g (𝜑𝐺 ∈ TarskiG)
mirval.a (𝜑𝐴𝑃)
mirfv.m 𝑀 = (𝑆𝐴)
Assertion
Ref Expression
mirf (𝜑𝑀:𝑃𝑃)

Proof of Theorem mirf
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 riotaex 6515 . . 3 (𝑧𝑃 ((𝐴 𝑧) = (𝐴 𝑦) ∧ 𝐴 ∈ (𝑧𝐼𝑦))) ∈ V
21a1i 11 . 2 ((𝜑𝑦𝑃) → (𝑧𝑃 ((𝐴 𝑧) = (𝐴 𝑦) ∧ 𝐴 ∈ (𝑧𝐼𝑦))) ∈ V)
3 mirfv.m . . 3 𝑀 = (𝑆𝐴)
4 mirval.p . . . 4 𝑃 = (Base‘𝐺)
5 mirval.d . . . 4 = (dist‘𝐺)
6 mirval.i . . . 4 𝐼 = (Itv‘𝐺)
7 mirval.l . . . 4 𝐿 = (LineG‘𝐺)
8 mirval.s . . . 4 𝑆 = (pInvG‘𝐺)
9 mirval.g . . . 4 (𝜑𝐺 ∈ TarskiG)
10 mirval.a . . . 4 (𝜑𝐴𝑃)
114, 5, 6, 7, 8, 9, 10mirval 25350 . . 3 (𝜑 → (𝑆𝐴) = (𝑦𝑃 ↦ (𝑧𝑃 ((𝐴 𝑧) = (𝐴 𝑦) ∧ 𝐴 ∈ (𝑧𝐼𝑦)))))
123, 11syl5eq 2656 . 2 (𝜑𝑀 = (𝑦𝑃 ↦ (𝑧𝑃 ((𝐴 𝑧) = (𝐴 𝑦) ∧ 𝐴 ∈ (𝑧𝐼𝑦)))))
139adantr 480 . . . 4 ((𝜑𝑥𝑃) → 𝐺 ∈ TarskiG)
1410adantr 480 . . . 4 ((𝜑𝑥𝑃) → 𝐴𝑃)
15 simpr 476 . . . 4 ((𝜑𝑥𝑃) → 𝑥𝑃)
164, 5, 6, 7, 8, 13, 14, 3, 15mirfv 25351 . . 3 ((𝜑𝑥𝑃) → (𝑀𝑥) = (𝑧𝑃 ((𝐴 𝑧) = (𝐴 𝑥) ∧ 𝐴 ∈ (𝑧𝐼𝑥))))
174, 5, 6, 13, 15, 14mirreu3 25349 . . . 4 ((𝜑𝑥𝑃) → ∃!𝑧𝑃 ((𝐴 𝑧) = (𝐴 𝑥) ∧ 𝐴 ∈ (𝑧𝐼𝑥)))
18 riotacl 6525 . . . 4 (∃!𝑧𝑃 ((𝐴 𝑧) = (𝐴 𝑥) ∧ 𝐴 ∈ (𝑧𝐼𝑥)) → (𝑧𝑃 ((𝐴 𝑧) = (𝐴 𝑥) ∧ 𝐴 ∈ (𝑧𝐼𝑥))) ∈ 𝑃)
1917, 18syl 17 . . 3 ((𝜑𝑥𝑃) → (𝑧𝑃 ((𝐴 𝑧) = (𝐴 𝑥) ∧ 𝐴 ∈ (𝑧𝐼𝑥))) ∈ 𝑃)
2016, 19eqeltrd 2688 . 2 ((𝜑𝑥𝑃) → (𝑀𝑥) ∈ 𝑃)
212, 12, 20fmpt2d 6300 1 (𝜑𝑀:𝑃𝑃)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∃!wreu 2898  Vcvv 3173   ↦ cmpt 4643  ⟶wf 5800  ‘cfv 5804  ℩crio 6510  (class class class)co 6549  Basecbs 15695  distcds 15777  TarskiGcstrkg 25129  Itvcitv 25135  LineGclng 25136  pInvGcmir 25347 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-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-pr 4833 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-rmo 2904  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-riota 6511  df-ov 6552  df-trkgc 25147  df-trkgb 25148  df-trkgcb 25149  df-trkg 25152  df-mir 25348 This theorem is referenced by:  mircl  25356  mirf1o  25364  mirbtwni  25366  mirbtwnb  25367  mirauto  25379  miduniq2  25382  krippenlem  25385
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