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Theorem mirf1o 25785
Description: The point inversion function 𝑀 is a bijection. Theorem 7.11 of [Schwabhauser] p. 50. (Contributed by Thierry Arnoux, 6-Jun-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
mirf1o (𝜑𝑀:𝑃1-1-onto𝑃)

Proof of Theorem mirf1o
Dummy variable 𝑎 is distinct from all other variables.
StepHypRef Expression
1 mirval.p . . . 4 𝑃 = (Base‘𝐺)
2 mirval.d . . . 4 = (dist‘𝐺)
3 mirval.i . . . 4 𝐼 = (Itv‘𝐺)
4 mirval.l . . . 4 𝐿 = (LineG‘𝐺)
5 mirval.s . . . 4 𝑆 = (pInvG‘𝐺)
6 mirval.g . . . 4 (𝜑𝐺 ∈ TarskiG)
7 mirval.a . . . 4 (𝜑𝐴𝑃)
8 mirfv.m . . . 4 𝑀 = (𝑆𝐴)
91, 2, 3, 4, 5, 6, 7, 8mirf 25776 . . 3 (𝜑𝑀:𝑃𝑃)
10 ffn 6185 . . 3 (𝑀:𝑃𝑃𝑀 Fn 𝑃)
119, 10syl 17 . 2 (𝜑𝑀 Fn 𝑃)
126adantr 466 . . . . 5 ((𝜑𝑎𝑃) → 𝐺 ∈ TarskiG)
137adantr 466 . . . . 5 ((𝜑𝑎𝑃) → 𝐴𝑃)
14 simpr 471 . . . . 5 ((𝜑𝑎𝑃) → 𝑎𝑃)
151, 2, 3, 4, 5, 12, 13, 8, 14mirmir 25778 . . . 4 ((𝜑𝑎𝑃) → (𝑀‘(𝑀𝑎)) = 𝑎)
1615ralrimiva 3115 . . 3 (𝜑 → ∀𝑎𝑃 (𝑀‘(𝑀𝑎)) = 𝑎)
17 nvocnv 6680 . . 3 ((𝑀:𝑃𝑃 ∧ ∀𝑎𝑃 (𝑀‘(𝑀𝑎)) = 𝑎) → 𝑀 = 𝑀)
189, 16, 17syl2anc 573 . 2 (𝜑𝑀 = 𝑀)
19 nvof1o 6679 . 2 ((𝑀 Fn 𝑃𝑀 = 𝑀) → 𝑀:𝑃1-1-onto𝑃)
2011, 18, 19syl2anc 573 1 (𝜑𝑀:𝑃1-1-onto𝑃)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1631  wcel 2145  wral 3061  ccnv 5248   Fn wfn 6026  wf 6027  1-1-ontowf1o 6030  cfv 6031  Basecbs 16064  distcds 16158  TarskiGcstrkg 25550  Itvcitv 25556  LineGclng 25557  pInvGcmir 25768
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-riota 6754  df-ov 6796  df-trkgc 25568  df-trkgb 25569  df-trkgcb 25570  df-trkg 25573  df-mir 25769
This theorem is referenced by:  mirmot  25791
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