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Theorem mirf1o 26461
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 26452 . . 3 (𝜑𝑀:𝑃𝑃)
109ffnd 6495 . 2 (𝜑𝑀 Fn 𝑃)
116adantr 484 . . . . 5 ((𝜑𝑎𝑃) → 𝐺 ∈ TarskiG)
127adantr 484 . . . . 5 ((𝜑𝑎𝑃) → 𝐴𝑃)
13 simpr 488 . . . . 5 ((𝜑𝑎𝑃) → 𝑎𝑃)
141, 2, 3, 4, 5, 11, 12, 8, 13mirmir 26454 . . . 4 ((𝜑𝑎𝑃) → (𝑀‘(𝑀𝑎)) = 𝑎)
1514ralrimiva 3174 . . 3 (𝜑 → ∀𝑎𝑃 (𝑀‘(𝑀𝑎)) = 𝑎)
16 nvocnv 7021 . . 3 ((𝑀:𝑃𝑃 ∧ ∀𝑎𝑃 (𝑀‘(𝑀𝑎)) = 𝑎) → 𝑀 = 𝑀)
179, 15, 16syl2anc 587 . 2 (𝜑𝑀 = 𝑀)
18 nvof1o 7020 . 2 ((𝑀 Fn 𝑃𝑀 = 𝑀) → 𝑀:𝑃1-1-onto𝑃)
1910, 17, 18syl2anc 587 1 (𝜑𝑀:𝑃1-1-onto𝑃)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wcel 2114  wral 3130  ccnv 5531   Fn wfn 6329  wf 6330  1-1-ontowf1o 6333  cfv 6334  Basecbs 16474  distcds 16565  TarskiGcstrkg 26222  Itvcitv 26228  LineGclng 26229  pInvGcmir 26444
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-rep 5166  ax-sep 5179  ax-nul 5186  ax-pr 5307
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-ral 3135  df-rex 3136  df-reu 3137  df-rmo 3138  df-rab 3139  df-v 3471  df-sbc 3748  df-csb 3856  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-pw 4513  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4814  df-iun 4896  df-br 5043  df-opab 5105  df-mpt 5123  df-id 5437  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-rn 5543  df-res 5544  df-ima 5545  df-iota 6293  df-fun 6336  df-fn 6337  df-f 6338  df-f1 6339  df-fo 6340  df-f1o 6341  df-fv 6342  df-riota 7098  df-ov 7143  df-trkgc 26240  df-trkgb 26241  df-trkgcb 26242  df-trkg 26245  df-mir 26445
This theorem is referenced by:  mirmot  26467
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