Theorem mirf 25904

Description: Point inversion as function. (Contributed by Thierry Arnoux, 30-May-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
mirf	⊢ (𝜑 → 𝑀:𝑃⟶𝑃)

Proof of Theorem mirf

Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		riotaex 6841	. . 3 ⊢ (℩𝑧 ∈ 𝑃 ((𝐴 − 𝑧) = (𝐴 − 𝑦) ∧ 𝐴 ∈ (𝑧𝐼𝑦))) ∈ V
2	1	a1i 11	. 2 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑃) → (℩𝑧 ∈ 𝑃 ((𝐴 − 𝑧) = (𝐴 − 𝑦) ∧ 𝐴 ∈ (𝑧𝐼𝑦))) ∈ V)
3		mirfv.m	. . 3 ⊢ 𝑀 = (𝑆‘𝐴)
4		mirval.p	. . . 4 ⊢ 𝑃 = (Base‘𝐺)
5		mirval.d	. . . 4 ⊢ − = (dist‘𝐺)
6		mirval.i	. . . 4 ⊢ 𝐼 = (Itv‘𝐺)
7		mirval.l	. . . 4 ⊢ 𝐿 = (LineG‘𝐺)
8		mirval.s	. . . 4 ⊢ 𝑆 = (pInvG‘𝐺)
9		mirval.g	. . . 4 ⊢ (𝜑 → 𝐺 ∈ TarskiG)
10		mirval.a	. . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑃)
11	4, 5, 6, 7, 8, 9, 10	mirval 25899	. . 3 ⊢ (𝜑 → (𝑆‘𝐴) = (𝑦 ∈ 𝑃 ↦ (℩𝑧 ∈ 𝑃 ((𝐴 − 𝑧) = (𝐴 − 𝑦) ∧ 𝐴 ∈ (𝑧𝐼𝑦)))))
12	3, 11	syl5eq 2843	. 2 ⊢ (𝜑 → 𝑀 = (𝑦 ∈ 𝑃 ↦ (℩𝑧 ∈ 𝑃 ((𝐴 − 𝑧) = (𝐴 − 𝑦) ∧ 𝐴 ∈ (𝑧𝐼𝑦)))))
13	9	adantr 473	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑃) → 𝐺 ∈ TarskiG)
14	10	adantr 473	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑃) → 𝐴 ∈ 𝑃)
15		simpr 478	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑃) → 𝑥 ∈ 𝑃)
16	4, 5, 6, 7, 8, 13, 14, 3, 15	mirfv 25900	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑃) → (𝑀‘𝑥) = (℩𝑧 ∈ 𝑃 ((𝐴 − 𝑧) = (𝐴 − 𝑥) ∧ 𝐴 ∈ (𝑧𝐼𝑥))))
17	4, 5, 6, 13, 15, 14	mirreu3 25898	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑃) → ∃!𝑧 ∈ 𝑃 ((𝐴 − 𝑧) = (𝐴 − 𝑥) ∧ 𝐴 ∈ (𝑧𝐼𝑥)))
18		riotacl 6851	. . . 4 ⊢ (∃!𝑧 ∈ 𝑃 ((𝐴 − 𝑧) = (𝐴 − 𝑥) ∧ 𝐴 ∈ (𝑧𝐼𝑥)) → (℩𝑧 ∈ 𝑃 ((𝐴 − 𝑧) = (𝐴 − 𝑥) ∧ 𝐴 ∈ (𝑧𝐼𝑥))) ∈ 𝑃)
19	17, 18	syl 17	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑃) → (℩𝑧 ∈ 𝑃 ((𝐴 − 𝑧) = (𝐴 − 𝑥) ∧ 𝐴 ∈ (𝑧𝐼𝑥))) ∈ 𝑃)
20	16, 19	eqeltrd 2876	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑃) → (𝑀‘𝑥) ∈ 𝑃)
21	2, 12, 20	fmpt2d 6617	1 ⊢ (𝜑 → 𝑀:𝑃⟶𝑃)

Syntax hints:   → wi 4   ∧ wa 385   = wceq 1653   ∈ wcel 2157  ∃!wreu 3089  Vcvv 3383   ↦ cmpt 4920  ⟶wf 6095  ‘cfv 6099  ℩crio 6836  (class class class)co 6876  Basecbs 16181  distcds 16273  TarskiGcstrkg 25678  Itvcitv 25684  LineGclng 25685  pInvGcmir 25896

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2354  ax-ext 2775  ax-rep 4962  ax-sep 4973  ax-nul 4981  ax-pr 5095

This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2590  df-eu 2607  df-clab 2784  df-cleq 2790  df-clel 2793  df-nfc 2928  df-ne 2970  df-ral 3092  df-rex 3093  df-reu 3094  df-rmo 3095  df-rab 3096  df-v 3385  df-sbc 3632  df-csb 3727  df-dif 3770  df-un 3772  df-in 3774  df-ss 3781  df-nul 4114  df-if 4276  df-pw 4349  df-sn 4367  df-pr 4369  df-op 4373  df-uni 4627  df-iun 4710  df-br 4842  df-opab 4904  df-mpt 4921  df-id 5218  df-xp 5316  df-rel 5317  df-cnv 5318  df-co 5319  df-dm 5320  df-rn 5321  df-res 5322  df-ima 5323  df-iota 6062  df-fun 6101  df-fn 6102  df-f 6103  df-f1 6104  df-fo 6105  df-f1o 6106  df-fv 6107  df-riota 6837  df-ov 6879  df-trkgc 25696  df-trkgb 25697  df-trkgcb 25698  df-trkg 25701  df-mir 25897

This theorem is referenced by:  mircl  25905  mirf1o  25913  mirbtwni  25915  mirbtwnb  25916  mirauto  25928  miduniq2  25931  krippenlem  25934