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Theorem mirf 25489
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 6580 . . 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 25484 . . 3 (𝜑 → (𝑆𝐴) = (𝑦𝑃 ↦ (𝑧𝑃 ((𝐴 𝑧) = (𝐴 𝑦) ∧ 𝐴 ∈ (𝑧𝐼𝑦)))))
123, 11syl5eq 2667 . 2 (𝜑𝑀 = (𝑦𝑃 ↦ (𝑧𝑃 ((𝐴 𝑧) = (𝐴 𝑦) ∧ 𝐴 ∈ (𝑧𝐼𝑦)))))
139adantr 481 . . . 4 ((𝜑𝑥𝑃) → 𝐺 ∈ TarskiG)
1410adantr 481 . . . 4 ((𝜑𝑥𝑃) → 𝐴𝑃)
15 simpr 477 . . . 4 ((𝜑𝑥𝑃) → 𝑥𝑃)
164, 5, 6, 7, 8, 13, 14, 3, 15mirfv 25485 . . 3 ((𝜑𝑥𝑃) → (𝑀𝑥) = (𝑧𝑃 ((𝐴 𝑧) = (𝐴 𝑥) ∧ 𝐴 ∈ (𝑧𝐼𝑥))))
174, 5, 6, 13, 15, 14mirreu3 25483 . . . 4 ((𝜑𝑥𝑃) → ∃!𝑧𝑃 ((𝐴 𝑧) = (𝐴 𝑥) ∧ 𝐴 ∈ (𝑧𝐼𝑥)))
18 riotacl 6590 . . . 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 2698 . 2 ((𝜑𝑥𝑃) → (𝑀𝑥) ∈ 𝑃)
212, 12, 20fmpt2d 6359 1 (𝜑𝑀:𝑃𝑃)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wcel 1987  ∃!wreu 2910  Vcvv 3190  cmpt 4683  wf 5853  cfv 5857  crio 6575  (class class class)co 6615  Basecbs 15800  distcds 15890  TarskiGcstrkg 25263  Itvcitv 25269  LineGclng 25270  pInvGcmir 25481
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4741  ax-sep 4751  ax-nul 4759  ax-pr 4877
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2913  df-rex 2914  df-reu 2915  df-rmo 2916  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-iun 4494  df-br 4624  df-opab 4684  df-mpt 4685  df-id 4999  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-fv 5865  df-riota 6576  df-ov 6618  df-trkgc 25281  df-trkgb 25282  df-trkgcb 25283  df-trkg 25286  df-mir 25482
This theorem is referenced by:  mircl  25490  mirf1o  25498  mirbtwni  25500  mirbtwnb  25501  mirauto  25513  miduniq2  25516  krippenlem  25519
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