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Theorem mirf1o 25919
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 25910 . . 3 (𝜑𝑀:𝑃𝑃)
109ffnd 6258 . 2 (𝜑𝑀 Fn 𝑃)
116adantr 473 . . . . 5 ((𝜑𝑎𝑃) → 𝐺 ∈ TarskiG)
127adantr 473 . . . . 5 ((𝜑𝑎𝑃) → 𝐴𝑃)
13 simpr 478 . . . . 5 ((𝜑𝑎𝑃) → 𝑎𝑃)
141, 2, 3, 4, 5, 11, 12, 8, 13mirmir 25912 . . . 4 ((𝜑𝑎𝑃) → (𝑀‘(𝑀𝑎)) = 𝑎)
1514ralrimiva 3148 . . 3 (𝜑 → ∀𝑎𝑃 (𝑀‘(𝑀𝑎)) = 𝑎)
16 nvocnv 6766 . . 3 ((𝑀:𝑃𝑃 ∧ ∀𝑎𝑃 (𝑀‘(𝑀𝑎)) = 𝑎) → 𝑀 = 𝑀)
179, 15, 16syl2anc 580 . 2 (𝜑𝑀 = 𝑀)
18 nvof1o 6765 . 2 ((𝑀 Fn 𝑃𝑀 = 𝑀) → 𝑀:𝑃1-1-onto𝑃)
1910, 17, 18syl2anc 580 1 (𝜑𝑀:𝑃1-1-onto𝑃)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 385   = wceq 1653  wcel 2157  wral 3090  ccnv 5312   Fn wfn 6097  wf 6098  1-1-ontowf1o 6101  cfv 6102  Basecbs 16183  distcds 16275  TarskiGcstrkg 25680  Itvcitv 25686  LineGclng 25687  pInvGcmir 25902
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2378  ax-ext 2778  ax-rep 4965  ax-sep 4976  ax-nul 4984  ax-pr 5098
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2592  df-eu 2610  df-clab 2787  df-cleq 2793  df-clel 2796  df-nfc 2931  df-ne 2973  df-ral 3095  df-rex 3096  df-reu 3097  df-rmo 3098  df-rab 3099  df-v 3388  df-sbc 3635  df-csb 3730  df-dif 3773  df-un 3775  df-in 3777  df-ss 3784  df-nul 4117  df-if 4279  df-pw 4352  df-sn 4370  df-pr 4372  df-op 4376  df-uni 4630  df-iun 4713  df-br 4845  df-opab 4907  df-mpt 4924  df-id 5221  df-xp 5319  df-rel 5320  df-cnv 5321  df-co 5322  df-dm 5323  df-rn 5324  df-res 5325  df-ima 5326  df-iota 6065  df-fun 6104  df-fn 6105  df-f 6106  df-f1 6107  df-fo 6108  df-f1o 6109  df-fv 6110  df-riota 6840  df-ov 6882  df-trkgc 25698  df-trkgb 25699  df-trkgcb 25700  df-trkg 25703  df-mir 25903
This theorem is referenced by:  mirmot  25925
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