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	⊢ ( 𝜑 → 𝐺 ∈ TarskiG )
	mirval.a	⊢ ( 𝜑 → 𝐴 ∈ 𝑃 )
	mirfv.m	⊢ 𝑀 = ( 𝑆 ‘ 𝐴 )
Assertion	mirf1o	⊢ ( 𝜑 → 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	⊢ ( 𝜑 → 𝐺 ∈ TarskiG )
7		mirval.a	⊢ ( 𝜑 → 𝐴 ∈ 𝑃 )
8		mirfv.m	⊢ 𝑀 = ( 𝑆 ‘ 𝐴 )
9	1 2 3 4 5 6 7 8	mirf	⊢ ( 𝜑 → 𝑀 : 𝑃 ⟶ 𝑃 )
10	9	ffnd	⊢ ( 𝜑 → 𝑀 Fn 𝑃 )
11	6	adantr	⊢ ( ( 𝜑 ∧ 𝑎 ∈ 𝑃 ) → 𝐺 ∈ TarskiG )
12	7	adantr	⊢ ( ( 𝜑 ∧ 𝑎 ∈ 𝑃 ) → 𝐴 ∈ 𝑃 )
13		simpr	⊢ ( ( 𝜑 ∧ 𝑎 ∈ 𝑃 ) → 𝑎 ∈ 𝑃 )
14	1 2 3 4 5 11 12 8 13	mirmir	⊢ ( ( 𝜑 ∧ 𝑎 ∈ 𝑃 ) → ( 𝑀 ‘ ( 𝑀 ‘ 𝑎 ) ) = 𝑎 )
15	14	ralrimiva	⊢ ( 𝜑 → ∀ 𝑎 ∈ 𝑃 ( 𝑀 ‘ ( 𝑀 ‘ 𝑎 ) ) = 𝑎 )
16		nvocnv	⊢ ( ( 𝑀 : 𝑃 ⟶ 𝑃 ∧ ∀ 𝑎 ∈ 𝑃 ( 𝑀 ‘ ( 𝑀 ‘ 𝑎 ) ) = 𝑎 ) → ◡ 𝑀 = 𝑀 )
17	9 15 16	syl2anc	⊢ ( 𝜑 → ◡ 𝑀 = 𝑀 )
18		nvof1o	⊢ ( ( 𝑀 Fn 𝑃 ∧ ◡ 𝑀 = 𝑀 ) → 𝑀 : 𝑃 –1-1-onto→ 𝑃 )
19	10 17 18	syl2anc	⊢ ( 𝜑 → 𝑀 : 𝑃 –1-1-onto→ 𝑃 )