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Theorem mirf1o 24250
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  |-  ( ph  ->  G  e. TarskiG )
mirval.a  |-  ( ph  ->  A  e.  P )
mirfv.m  |-  M  =  ( S `  A
)
Assertion
Ref Expression
mirf1o  |-  ( ph  ->  M : P -1-1-onto-> P )

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  |-  ( ph  ->  G  e. TarskiG )
7 mirval.a . . . 4  |-  ( ph  ->  A  e.  P )
8 mirfv.m . . . 4  |-  M  =  ( S `  A
)
91, 2, 3, 4, 5, 6, 7, 8mirf 24242 . . 3  |-  ( ph  ->  M : P --> P )
10 ffn 5713 . . 3  |-  ( M : P --> P  ->  M  Fn  P )
119, 10syl 16 . 2  |-  ( ph  ->  M  Fn  P )
126adantr 463 . . . . 5  |-  ( (
ph  /\  a  e.  P )  ->  G  e. TarskiG )
137adantr 463 . . . . 5  |-  ( (
ph  /\  a  e.  P )  ->  A  e.  P )
14 simpr 459 . . . . 5  |-  ( (
ph  /\  a  e.  P )  ->  a  e.  P )
151, 2, 3, 4, 5, 12, 13, 8, 14mirmir 24244 . . . 4  |-  ( (
ph  /\  a  e.  P )  ->  ( M `  ( M `  a ) )  =  a )
1615ralrimiva 2868 . . 3  |-  ( ph  ->  A. a  e.  P  ( M `  ( M `
 a ) )  =  a )
17 nvocnv 6162 . . 3  |-  ( ( M : P --> P  /\  A. a  e.  P  ( M `  ( M `
 a ) )  =  a )  ->  `' M  =  M
)
189, 16, 17syl2anc 659 . 2  |-  ( ph  ->  `' M  =  M
)
19 nvof1o 6161 . 2  |-  ( ( M  Fn  P  /\  `' M  =  M
)  ->  M : P
-1-1-onto-> P )
2011, 18, 19syl2anc 659 1  |-  ( ph  ->  M : P -1-1-onto-> P )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1398    e. wcel 1823   A.wral 2804   `'ccnv 4987    Fn wfn 5565   -->wf 5566   -1-1-onto->wf1o 5569   ` cfv 5570   Basecbs 14716   distcds 14793  TarskiGcstrkg 24023  Itvcitv 24030  LineGclng 24031  pInvGcmir 24234
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-trkgc 24042  df-trkgb 24043  df-trkgcb 24044  df-trkg 24048  df-mir 24235
This theorem is referenced by:  mirmot  24256
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