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Theorem mirf1o 23790
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 23782 . . 3  |-  ( ph  ->  M : P --> P )
10 ffn 5731 . . 3  |-  ( M : P --> P  ->  M  Fn  P )
119, 10syl 16 . 2  |-  ( ph  ->  M  Fn  P )
126adantr 465 . . . . 5  |-  ( (
ph  /\  a  e.  P )  ->  G  e. TarskiG )
137adantr 465 . . . . 5  |-  ( (
ph  /\  a  e.  P )  ->  A  e.  P )
14 simpr 461 . . . . 5  |-  ( (
ph  /\  a  e.  P )  ->  a  e.  P )
151, 2, 3, 4, 5, 12, 13, 8, 14mirmir 23784 . . . 4  |-  ( (
ph  /\  a  e.  P )  ->  ( M `  ( M `  a ) )  =  a )
1615ralrimiva 2878 . . 3  |-  ( ph  ->  A. a  e.  P  ( M `  ( M `
 a ) )  =  a )
17 nvocnv 6175 . . 3  |-  ( ( M : P --> P  /\  A. a  e.  P  ( M `  ( M `
 a ) )  =  a )  ->  `' M  =  M
)
189, 16, 17syl2anc 661 . 2  |-  ( ph  ->  `' M  =  M
)
19 nvof1o 6174 . 2  |-  ( ( M  Fn  P  /\  `' M  =  M
)  ->  M : P
-1-1-onto-> P )
2011, 18, 19syl2anc 661 1  |-  ( ph  ->  M : P -1-1-onto-> P )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767   A.wral 2814   `'ccnv 4998    Fn wfn 5583   -->wf 5584   -1-1-onto->wf1o 5587   ` cfv 5588   Basecbs 14490   distcds 14564  TarskiGcstrkg 23581  Itvcitv 23588  LineGclng 23589  pInvGcmir 23774
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-riota 6245  df-ov 6287  df-trkgc 23600  df-trkgb 23601  df-trkgcb 23602  df-trkg 23606  df-mir 23775
This theorem is referenced by:  mirmot  23796
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