Theorem mirf1o 29028

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	⊢ (𝜑 → 𝐺 ∈ TarskiG)
mirval.a	⊢ (𝜑 → 𝐴 ∈ 𝑃)
mirfv.m	⊢ 𝑀 = (𝑆‘𝐴)

Assertion

Ref	Expression
mirf1o	⊢ (𝜑 → 𝑀:𝑃–1-1-onto→𝑃)

Proof of Theorem mirf1o

Dummy variable 𝑎 is distinct from all other variables.

Step	Hyp	Ref	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 ⊢ (𝜑 → 𝐺 ∈ TarskiG)
7		mirval.a	. . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑃)
8		mirfv.m	. . . 4 ⊢ 𝑀 = (𝑆‘𝐴)
9	1, 2, 3, 4, 5, 6, 7, 8	mirf 29019	. . 3 ⊢ (𝜑 → 𝑀:𝑃⟶𝑃)
10	9	ffnd 6707	. 2 ⊢ (𝜑 → 𝑀 Fn 𝑃)
11	6	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑃) → 𝐺 ∈ TarskiG)
12	7	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑃) → 𝐴 ∈ 𝑃)
13		simpr 490	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑃) → 𝑎 ∈ 𝑃)
14	1, 2, 3, 4, 5, 11, 12, 8, 13	mirmir 29021	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑃) → (𝑀‘(𝑀‘𝑎)) = 𝑎)
15	14	ralrimiva 3156	. . 3 ⊢ (𝜑 → ∀𝑎 ∈ 𝑃 (𝑀‘(𝑀‘𝑎)) = 𝑎)
16		nvocnv 7286	. . 3 ⊢ ((𝑀:𝑃⟶𝑃 ∧ ∀𝑎 ∈ 𝑃 (𝑀‘(𝑀‘𝑎)) = 𝑎) → ◡𝑀 = 𝑀)
17	9, 15, 16	syl2anc 596	. 2 ⊢ (𝜑 → ◡𝑀 = 𝑀)
18		nvof1o 7285	. 2 ⊢ ((𝑀 Fn 𝑃 ∧ ◡𝑀 = 𝑀) → 𝑀:𝑃–1-1-onto→𝑃)
19	10, 17, 18	syl2anc 596	1 ⊢ (𝜑 → 𝑀:𝑃–1-1-onto→𝑃)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ∀wral 3078  ^◡ccnv 5658   Fn wfn 6532  ⟶wf 6533  –1-1-onto→wf1o 6536  ‘cfv 6537  Basecbs 17307  distcds 17357  TarskiGcstrkg 28776  Itvcitv 28782  LineGclng 28783  pInvGcmir 29011

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-rep 5236  ax-sep 5255  ax-nul 5267  ax-pr 5402

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-rmo 3367  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-iun 4956  df-br 5108  df-opab 5172  df-mpt 5191  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7374  df-ov 7420  df-trkgc 28797  df-trkgb 28798  df-trkgcb 28799  df-trkg 28802  df-mir 29012

This theorem is used by:  mirmot  29034