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Theorem mirf1o 25558
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 25549 . . 3 (𝜑𝑀:𝑃𝑃)
10 ffn 6043 . . 3 (𝑀:𝑃𝑃𝑀 Fn 𝑃)
119, 10syl 17 . 2 (𝜑𝑀 Fn 𝑃)
126adantr 481 . . . . 5 ((𝜑𝑎𝑃) → 𝐺 ∈ TarskiG)
137adantr 481 . . . . 5 ((𝜑𝑎𝑃) → 𝐴𝑃)
14 simpr 477 . . . . 5 ((𝜑𝑎𝑃) → 𝑎𝑃)
151, 2, 3, 4, 5, 12, 13, 8, 14mirmir 25551 . . . 4 ((𝜑𝑎𝑃) → (𝑀‘(𝑀𝑎)) = 𝑎)
1615ralrimiva 2965 . . 3 (𝜑 → ∀𝑎𝑃 (𝑀‘(𝑀𝑎)) = 𝑎)
17 nvocnv 6534 . . 3 ((𝑀:𝑃𝑃 ∧ ∀𝑎𝑃 (𝑀‘(𝑀𝑎)) = 𝑎) → 𝑀 = 𝑀)
189, 16, 17syl2anc 693 . 2 (𝜑𝑀 = 𝑀)
19 nvof1o 6533 . 2 ((𝑀 Fn 𝑃𝑀 = 𝑀) → 𝑀:𝑃1-1-onto𝑃)
2011, 18, 19syl2anc 693 1 (𝜑𝑀:𝑃1-1-onto𝑃)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1482  wcel 1989  wral 2911  ccnv 5111   Fn wfn 5881  wf 5882  1-1-ontowf1o 5885  cfv 5886  Basecbs 15851  distcds 15944  TarskiGcstrkg 25323  Itvcitv 25329  LineGclng 25330  pInvGcmir 25541
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-rep 4769  ax-sep 4779  ax-nul 4787  ax-pr 4904
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ne 2794  df-ral 2916  df-rex 2917  df-reu 2918  df-rmo 2919  df-rab 2920  df-v 3200  df-sbc 3434  df-csb 3532  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-nul 3914  df-if 4085  df-pw 4158  df-sn 4176  df-pr 4178  df-op 4182  df-uni 4435  df-iun 4520  df-br 4652  df-opab 4711  df-mpt 4728  df-id 5022  df-xp 5118  df-rel 5119  df-cnv 5120  df-co 5121  df-dm 5122  df-rn 5123  df-res 5124  df-ima 5125  df-iota 5849  df-fun 5888  df-fn 5889  df-f 5890  df-f1 5891  df-fo 5892  df-f1o 5893  df-fv 5894  df-riota 6608  df-ov 6650  df-trkgc 25341  df-trkgb 25342  df-trkgcb 25343  df-trkg 25346  df-mir 25542
This theorem is referenced by:  mirmot  25564
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