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Mirrors > Home > MPE Home > Th. List > motf1o | Structured version Visualization version GIF version |
Description: Motions are bijections. (Contributed by Thierry Arnoux, 15-Dec-2019.) |
Ref | Expression |
---|---|
ismot.p | ⊢ 𝑃 = (Base‘𝐺) |
ismot.m | ⊢ − = (dist‘𝐺) |
motgrp.1 | ⊢ (𝜑 → 𝐺 ∈ 𝑉) |
motco.2 | ⊢ (𝜑 → 𝐹 ∈ (𝐺Ismt𝐺)) |
Ref | Expression |
---|---|
motf1o | ⊢ (𝜑 → 𝐹:𝑃–1-1-onto→𝑃) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | motco.2 | . . 3 ⊢ (𝜑 → 𝐹 ∈ (𝐺Ismt𝐺)) | |
2 | motgrp.1 | . . . 4 ⊢ (𝜑 → 𝐺 ∈ 𝑉) | |
3 | ismot.p | . . . . 5 ⊢ 𝑃 = (Base‘𝐺) | |
4 | ismot.m | . . . . 5 ⊢ − = (dist‘𝐺) | |
5 | 3, 4 | ismot 26321 | . . . 4 ⊢ (𝐺 ∈ 𝑉 → (𝐹 ∈ (𝐺Ismt𝐺) ↔ (𝐹:𝑃–1-1-onto→𝑃 ∧ ∀𝑎 ∈ 𝑃 ∀𝑏 ∈ 𝑃 ((𝐹‘𝑎) − (𝐹‘𝑏)) = (𝑎 − 𝑏)))) |
6 | 2, 5 | syl 17 | . . 3 ⊢ (𝜑 → (𝐹 ∈ (𝐺Ismt𝐺) ↔ (𝐹:𝑃–1-1-onto→𝑃 ∧ ∀𝑎 ∈ 𝑃 ∀𝑏 ∈ 𝑃 ((𝐹‘𝑎) − (𝐹‘𝑏)) = (𝑎 − 𝑏)))) |
7 | 1, 6 | mpbid 234 | . 2 ⊢ (𝜑 → (𝐹:𝑃–1-1-onto→𝑃 ∧ ∀𝑎 ∈ 𝑃 ∀𝑏 ∈ 𝑃 ((𝐹‘𝑎) − (𝐹‘𝑏)) = (𝑎 − 𝑏))) |
8 | 7 | simpld 497 | 1 ⊢ (𝜑 → 𝐹:𝑃–1-1-onto→𝑃) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∀wral 3138 –1-1-onto→wf1o 6354 ‘cfv 6355 (class class class)co 7156 Basecbs 16483 distcds 16574 Ismtcismt 26318 |
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 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 |
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 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-ov 7159 df-oprab 7160 df-mpo 7161 df-map 8408 df-ismt 26319 |
This theorem is referenced by: motcl 26325 motco 26326 cnvmot 26327 motcgrg 26330 |
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