Theorem mirf1o 26457

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 26448	. . 3 ⊢ (𝜑 → 𝑀:𝑃⟶𝑃)
10	9	ffnd 6517	. 2 ⊢ (𝜑 → 𝑀 Fn 𝑃)
11	6	adantr 483	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑃) → 𝐺 ∈ TarskiG)
12	7	adantr 483	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑃) → 𝐴 ∈ 𝑃)
13		simpr 487	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑃) → 𝑎 ∈ 𝑃)
14	1, 2, 3, 4, 5, 11, 12, 8, 13	mirmir 26450	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑃) → (𝑀‘(𝑀‘𝑎)) = 𝑎)
15	14	ralrimiva 3184	. . 3 ⊢ (𝜑 → ∀𝑎 ∈ 𝑃 (𝑀‘(𝑀‘𝑎)) = 𝑎)
16		nvocnv 7040	. . 3 ⊢ ((𝑀:𝑃⟶𝑃 ∧ ∀𝑎 ∈ 𝑃 (𝑀‘(𝑀‘𝑎)) = 𝑎) → ◡𝑀 = 𝑀)
17	9, 15, 16	syl2anc 586	. 2 ⊢ (𝜑 → ◡𝑀 = 𝑀)
18		nvof1o 7039	. 2 ⊢ ((𝑀 Fn 𝑃 ∧ ◡𝑀 = 𝑀) → 𝑀:𝑃–1-1-onto→𝑃)
19	10, 17, 18	syl2anc 586	1 ⊢ (𝜑 → 𝑀:𝑃–1-1-onto→𝑃)

Syntax hints:   → wi 4   ∧ wa 398   = wceq 1537   ∈ wcel 2114  ∀wral 3140  ^◡ccnv 5556   Fn wfn 6352  ⟶wf 6353  –1-1-onto→wf1o 6356  ‘cfv 6357  Basecbs 16485  distcds 16576  TarskiGcstrkg 26218  Itvcitv 26224  LineGclng 26225  pInvGcmir 26440

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 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-rep 5192  ax-sep 5205  ax-nul 5212  ax-pr 5332

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-ral 3145  df-rex 3146  df-reu 3147  df-rmo 3148  df-rab 3149  df-v 3498  df-sbc 3775  df-csb 3886  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-pw 4543  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4841  df-iun 4923  df-br 5069  df-opab 5131  df-mpt 5149  df-id 5462  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-res 5569  df-ima 5570  df-iota 6316  df-fun 6359  df-fn 6360  df-f 6361  df-f1 6362  df-fo 6363  df-f1o 6364  df-fv 6365  df-riota 7116  df-ov 7161  df-trkgc 26236  df-trkgb 26237  df-trkgcb 26238  df-trkg 26241  df-mir 26441

This theorem is referenced by:  mirmot  26463