Theorem mirf1o

Description: The point inversion function M is a bijection. Theorem 7.11 of Schwabhauser p. 50. (Contributed by Thierry Arnoux, 6-Jun-2019)

	Ref	Expression
Hypotheses	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	mirf1o	|- ( ph -> M : P -1-1-onto-> P )

Proof

Step	Hyp	Ref	Expression
1		mirval.p	|- P = ( Base ` G )
2		mirval.d	|- .- = ( dist ` G )
3		mirval.i	|- I = ( Itv ` G )
4		mirval.l	|- L = ( LineG ` G )
5		mirval.s	|- S = ( pInvG ` G )
6		mirval.g	|- ( ph -> G e. TarskiG )
7		mirval.a	|- ( ph -> A e. P )
8		mirfv.m	|- M = ( S ` A )
9	1 2 3 4 5 6 7 8	mirf	|- ( ph -> M : P --> P )
10	9	ffnd	|- ( ph -> M Fn P )
11	6	adantr	|- ( ( ph /\ a e. P ) -> G e. TarskiG )
12	7	adantr	|- ( ( ph /\ a e. P ) -> A e. P )
13		simpr	|- ( ( ph /\ a e. P ) -> a e. P )
14	1 2 3 4 5 11 12 8 13	mirmir	|- ( ( ph /\ a e. P ) -> ( M ` ( M ` a ) ) = a )
15	14	ralrimiva	|- ( ph -> A. a e. P ( M ` ( M ` a ) ) = a )
16		nvocnv	|- ( ( M : P --> P /\ A. a e. P ( M ` ( M ` a ) ) = a ) -> `' M = M )
17	9 15 16	syl2anc	|- ( ph -> `' M = M )
18		nvof1o	|- ( ( M Fn P /\ `' M = M ) -> M : P -1-1-onto-> P )
19	10 17 18	syl2anc	|- ( ph -> M : P -1-1-onto-> P )