Theorem mirf1o 28645

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 28636	. . 3 ⊢ (𝜑 → 𝑀:𝑃⟶𝑃)
10	9	ffnd 6652	. 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 28638	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑃) → (𝑀‘(𝑀‘𝑎)) = 𝑎)
15	14	ralrimiva 3124	. . 3 ⊢ (𝜑 → ∀𝑎 ∈ 𝑃 (𝑀‘(𝑀‘𝑎)) = 𝑎)
16		nvocnv 7215	. . 3 ⊢ ((𝑀:𝑃⟶𝑃 ∧ ∀𝑎 ∈ 𝑃 (𝑀‘(𝑀‘𝑎)) = 𝑎) → ◡𝑀 = 𝑀)
17	9, 15, 16	syl2anc 584	. 2 ⊢ (𝜑 → ◡𝑀 = 𝑀)
18		nvof1o 7214	. 2 ⊢ ((𝑀 Fn 𝑃 ∧ ◡𝑀 = 𝑀) → 𝑀:𝑃–1-1-onto→𝑃)
19	10, 17, 18	syl2anc 584	1 ⊢ (𝜑 → 𝑀:𝑃–1-1-onto→𝑃)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2111  ∀wral 3047  ^◡ccnv 5615   Fn wfn 6476  ⟶wf 6477  –1-1-onto→wf1o 6480  ‘cfv 6481  Basecbs 17117  distcds 17167  TarskiGcstrkg 28403  Itvcitv 28409  LineGclng 28410  pInvGcmir 28628

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5217  ax-sep 5234  ax-nul 5244  ax-pr 5370

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rmo 3346  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4476  df-pw 4552  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-iun 4943  df-br 5092  df-opab 5154  df-mpt 5173  df-id 5511  df-xp 5622  df-rel 5623  df-cnv 5624  df-co 5625  df-dm 5626  df-rn 5627  df-res 5628  df-ima 5629  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-riota 7303  df-ov 7349  df-trkgc 28424  df-trkgb 28425  df-trkgcb 28426  df-trkg 28429  df-mir 28629

This theorem is referenced by:  mirmot  28651