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Theorem mirf1o 25277
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 25268 . . 3 (𝜑𝑀:𝑃𝑃)
10 ffn 5939 . . 3 (𝑀:𝑃𝑃𝑀 Fn 𝑃)
119, 10syl 17 . 2 (𝜑𝑀 Fn 𝑃)
126adantr 479 . . . . 5 ((𝜑𝑎𝑃) → 𝐺 ∈ TarskiG)
137adantr 479 . . . . 5 ((𝜑𝑎𝑃) → 𝐴𝑃)
14 simpr 475 . . . . 5 ((𝜑𝑎𝑃) → 𝑎𝑃)
151, 2, 3, 4, 5, 12, 13, 8, 14mirmir 25270 . . . 4 ((𝜑𝑎𝑃) → (𝑀‘(𝑀𝑎)) = 𝑎)
1615ralrimiva 2943 . . 3 (𝜑 → ∀𝑎𝑃 (𝑀‘(𝑀𝑎)) = 𝑎)
17 nvocnv 6410 . . 3 ((𝑀:𝑃𝑃 ∧ ∀𝑎𝑃 (𝑀‘(𝑀𝑎)) = 𝑎) → 𝑀 = 𝑀)
189, 16, 17syl2anc 690 . 2 (𝜑𝑀 = 𝑀)
19 nvof1o 6409 . 2 ((𝑀 Fn 𝑃𝑀 = 𝑀) → 𝑀:𝑃1-1-onto𝑃)
2011, 18, 19syl2anc 690 1 (𝜑𝑀:𝑃1-1-onto𝑃)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1474  wcel 1975  wral 2890  ccnv 5022   Fn wfn 5780  wf 5781  1-1-ontowf1o 5784  cfv 5785  Basecbs 15636  distcds 15718  TarskiGcstrkg 25041  Itvcitv 25047  LineGclng 25048  pInvGcmir 25260
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-9 1984  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2227  ax-ext 2584  ax-rep 4688  ax-sep 4698  ax-nul 4707  ax-pr 4823
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1866  df-eu 2456  df-mo 2457  df-clab 2591  df-cleq 2597  df-clel 2600  df-nfc 2734  df-ne 2776  df-ral 2895  df-rex 2896  df-reu 2897  df-rmo 2898  df-rab 2899  df-v 3169  df-sbc 3397  df-csb 3494  df-dif 3537  df-un 3539  df-in 3541  df-ss 3548  df-nul 3869  df-if 4031  df-pw 4104  df-sn 4120  df-pr 4122  df-op 4126  df-uni 4362  df-iun 4446  df-br 4573  df-opab 4633  df-mpt 4634  df-id 4938  df-xp 5029  df-rel 5030  df-cnv 5031  df-co 5032  df-dm 5033  df-rn 5034  df-res 5035  df-ima 5036  df-iota 5749  df-fun 5787  df-fn 5788  df-f 5789  df-f1 5790  df-fo 5791  df-f1o 5792  df-fv 5793  df-riota 6484  df-ov 6525  df-trkgc 25059  df-trkgb 25060  df-trkgcb 25061  df-trkg 25064  df-mir 25261
This theorem is referenced by:  mirmot  25283
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