Theorem mirf1o 28695

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 28686	. . 3 ⊢ (𝜑 → 𝑀:𝑃⟶𝑃)
10	9	ffnd 6748	. 2 ⊢ (𝜑 → 𝑀 Fn 𝑃)
11	6	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑃) → 𝐺 ∈ TarskiG)
12	7	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑃) → 𝐴 ∈ 𝑃)
13		simpr 484	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑃) → 𝑎 ∈ 𝑃)
14	1, 2, 3, 4, 5, 11, 12, 8, 13	mirmir 28688	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑃) → (𝑀‘(𝑀‘𝑎)) = 𝑎)
15	14	ralrimiva 3152	. . 3 ⊢ (𝜑 → ∀𝑎 ∈ 𝑃 (𝑀‘(𝑀‘𝑎)) = 𝑎)
16		nvocnv 7317	. . 3 ⊢ ((𝑀:𝑃⟶𝑃 ∧ ∀𝑎 ∈ 𝑃 (𝑀‘(𝑀‘𝑎)) = 𝑎) → ◡𝑀 = 𝑀)
17	9, 15, 16	syl2anc 583	. 2 ⊢ (𝜑 → ◡𝑀 = 𝑀)
18		nvof1o 7316	. 2 ⊢ ((𝑀 Fn 𝑃 ∧ ◡𝑀 = 𝑀) → 𝑀:𝑃–1-1-onto→𝑃)
19	10, 17, 18	syl2anc 583	1 ⊢ (𝜑 → 𝑀:𝑃–1-1-onto→𝑃)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1537   ∈ wcel 2108  ∀wral 3067  ^◡ccnv 5699   Fn wfn 6568  ⟶wf 6569  –1-1-onto→wf1o 6572  ‘cfv 6573  Basecbs 17258  distcds 17320  TarskiGcstrkg 28453  Itvcitv 28459  LineGclng 28460  pInvGcmir 28678

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-riota 7404  df-ov 7451  df-trkgc 28474  df-trkgb 28475  df-trkgcb 28476  df-trkg 28479  df-mir 28679

This theorem is referenced by:  mirmot  28701