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Theorem mirf 26433
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 7095 . . 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 26428 . . 3 (𝜑 → (𝑆𝐴) = (𝑦𝑃 ↦ (𝑧𝑃 ((𝐴 𝑧) = (𝐴 𝑦) ∧ 𝐴 ∈ (𝑧𝐼𝑦)))))
123, 11syl5eq 2867 . 2 (𝜑𝑀 = (𝑦𝑃 ↦ (𝑧𝑃 ((𝐴 𝑧) = (𝐴 𝑦) ∧ 𝐴 ∈ (𝑧𝐼𝑦)))))
139adantr 483 . . . 4 ((𝜑𝑥𝑃) → 𝐺 ∈ TarskiG)
1410adantr 483 . . . 4 ((𝜑𝑥𝑃) → 𝐴𝑃)
15 simpr 487 . . . 4 ((𝜑𝑥𝑃) → 𝑥𝑃)
164, 5, 6, 7, 8, 13, 14, 3, 15mirfv 26429 . . 3 ((𝜑𝑥𝑃) → (𝑀𝑥) = (𝑧𝑃 ((𝐴 𝑧) = (𝐴 𝑥) ∧ 𝐴 ∈ (𝑧𝐼𝑥))))
174, 5, 6, 13, 15, 14mirreu3 26427 . . . 4 ((𝜑𝑥𝑃) → ∃!𝑧𝑃 ((𝐴 𝑧) = (𝐴 𝑥) ∧ 𝐴 ∈ (𝑧𝐼𝑥)))
18 riotacl 7108 . . . 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 2911 . 2 ((𝜑𝑥𝑃) → (𝑀𝑥) ∈ 𝑃)
212, 12, 20fmpt2d 6863 1 (𝜑𝑀:𝑃𝑃)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 398   = wceq 1537  wcel 2114  ∃!wreu 3127  Vcvv 3473  cmpt 5122  wf 6327  cfv 6331  crio 7090  (class class class)co 7133  Basecbs 16462  distcds 16553  TarskiGcstrkg 26203  Itvcitv 26209  LineGclng 26210  pInvGcmir 26425
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2792  ax-rep 5166  ax-sep 5179  ax-nul 5186  ax-pr 5306
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2799  df-cleq 2813  df-clel 2891  df-nfc 2959  df-ne 3007  df-ral 3130  df-rex 3131  df-reu 3132  df-rmo 3133  df-rab 3134  df-v 3475  df-sbc 3753  df-csb 3861  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4270  df-if 4444  df-pw 4517  df-sn 4544  df-pr 4546  df-op 4550  df-uni 4815  df-iun 4897  df-br 5043  df-opab 5105  df-mpt 5123  df-id 5436  df-xp 5537  df-rel 5538  df-cnv 5539  df-co 5540  df-dm 5541  df-rn 5542  df-res 5543  df-ima 5544  df-iota 6290  df-fun 6333  df-fn 6334  df-f 6335  df-f1 6336  df-fo 6337  df-f1o 6338  df-fv 6339  df-riota 7091  df-ov 7136  df-trkgc 26221  df-trkgb 26222  df-trkgcb 26223  df-trkg 26226  df-mir 26426
This theorem is referenced by:  mircl  26434  mirf1o  26442  mirbtwni  26444  mirbtwnb  26445  mirauto  26457  miduniq2  26460  krippenlem  26463
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