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Mirrors > Home > MPE Home > Th. List > symg1bas | Structured version Visualization version GIF version |
Description: The symmetric group on a singleton is the symmetric group S1 consisting of the identity only. (Contributed by AV, 9-Dec-2018.) |
Ref | Expression |
---|---|
symg1bas.1 | ⊢ 𝐺 = (SymGrp‘𝐴) |
symg1bas.2 | ⊢ 𝐵 = (Base‘𝐺) |
symg1bas.0 | ⊢ 𝐴 = {𝐼} |
Ref | Expression |
---|---|
symg1bas | ⊢ (𝐼 ∈ 𝑉 → 𝐵 = {{〈𝐼, 𝐼〉}}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | symg1bas.1 | . . 3 ⊢ 𝐺 = (SymGrp‘𝐴) | |
2 | symg1bas.2 | . . 3 ⊢ 𝐵 = (Base‘𝐺) | |
3 | 1, 2 | symgbas 18491 | . 2 ⊢ 𝐵 = {𝑓 ∣ 𝑓:𝐴–1-1-onto→𝐴} |
4 | symg1bas.0 | . . . . . 6 ⊢ 𝐴 = {𝐼} | |
5 | eqidd 2799 | . . . . . . 7 ⊢ (𝐴 = {𝐼} → 𝑝 = 𝑝) | |
6 | id 22 | . . . . . . 7 ⊢ (𝐴 = {𝐼} → 𝐴 = {𝐼}) | |
7 | 5, 6, 6 | f1oeq123d 6585 | . . . . . 6 ⊢ (𝐴 = {𝐼} → (𝑝:𝐴–1-1-onto→𝐴 ↔ 𝑝:{𝐼}–1-1-onto→{𝐼})) |
8 | 4, 7 | ax-mp 5 | . . . . 5 ⊢ (𝑝:𝐴–1-1-onto→𝐴 ↔ 𝑝:{𝐼}–1-1-onto→{𝐼}) |
9 | f1of 6590 | . . . . . . 7 ⊢ (𝑝:{𝐼}–1-1-onto→{𝐼} → 𝑝:{𝐼}⟶{𝐼}) | |
10 | fsng 6876 | . . . . . . . 8 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐼 ∈ 𝑉) → (𝑝:{𝐼}⟶{𝐼} ↔ 𝑝 = {〈𝐼, 𝐼〉})) | |
11 | 10 | anidms 570 | . . . . . . 7 ⊢ (𝐼 ∈ 𝑉 → (𝑝:{𝐼}⟶{𝐼} ↔ 𝑝 = {〈𝐼, 𝐼〉})) |
12 | 9, 11 | syl5ib 247 | . . . . . 6 ⊢ (𝐼 ∈ 𝑉 → (𝑝:{𝐼}–1-1-onto→{𝐼} → 𝑝 = {〈𝐼, 𝐼〉})) |
13 | f1osng 6630 | . . . . . . . 8 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐼 ∈ 𝑉) → {〈𝐼, 𝐼〉}:{𝐼}–1-1-onto→{𝐼}) | |
14 | 13 | anidms 570 | . . . . . . 7 ⊢ (𝐼 ∈ 𝑉 → {〈𝐼, 𝐼〉}:{𝐼}–1-1-onto→{𝐼}) |
15 | f1oeq1 6579 | . . . . . . 7 ⊢ (𝑝 = {〈𝐼, 𝐼〉} → (𝑝:{𝐼}–1-1-onto→{𝐼} ↔ {〈𝐼, 𝐼〉}:{𝐼}–1-1-onto→{𝐼})) | |
16 | 14, 15 | syl5ibrcom 250 | . . . . . 6 ⊢ (𝐼 ∈ 𝑉 → (𝑝 = {〈𝐼, 𝐼〉} → 𝑝:{𝐼}–1-1-onto→{𝐼})) |
17 | 12, 16 | impbid 215 | . . . . 5 ⊢ (𝐼 ∈ 𝑉 → (𝑝:{𝐼}–1-1-onto→{𝐼} ↔ 𝑝 = {〈𝐼, 𝐼〉})) |
18 | 8, 17 | syl5bb 286 | . . . 4 ⊢ (𝐼 ∈ 𝑉 → (𝑝:𝐴–1-1-onto→𝐴 ↔ 𝑝 = {〈𝐼, 𝐼〉})) |
19 | vex 3444 | . . . . 5 ⊢ 𝑝 ∈ V | |
20 | f1oeq1 6579 | . . . . 5 ⊢ (𝑓 = 𝑝 → (𝑓:𝐴–1-1-onto→𝐴 ↔ 𝑝:𝐴–1-1-onto→𝐴)) | |
21 | 19, 20 | elab 3615 | . . . 4 ⊢ (𝑝 ∈ {𝑓 ∣ 𝑓:𝐴–1-1-onto→𝐴} ↔ 𝑝:𝐴–1-1-onto→𝐴) |
22 | velsn 4541 | . . . 4 ⊢ (𝑝 ∈ {{〈𝐼, 𝐼〉}} ↔ 𝑝 = {〈𝐼, 𝐼〉}) | |
23 | 18, 21, 22 | 3bitr4g 317 | . . 3 ⊢ (𝐼 ∈ 𝑉 → (𝑝 ∈ {𝑓 ∣ 𝑓:𝐴–1-1-onto→𝐴} ↔ 𝑝 ∈ {{〈𝐼, 𝐼〉}})) |
24 | 23 | eqrdv 2796 | . 2 ⊢ (𝐼 ∈ 𝑉 → {𝑓 ∣ 𝑓:𝐴–1-1-onto→𝐴} = {{〈𝐼, 𝐼〉}}) |
25 | 3, 24 | syl5eq 2845 | 1 ⊢ (𝐼 ∈ 𝑉 → 𝐵 = {{〈𝐼, 𝐼〉}}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 = wceq 1538 ∈ wcel 2111 {cab 2776 {csn 4525 〈cop 4531 ⟶wf 6320 –1-1-onto→wf1o 6323 ‘cfv 6324 Basecbs 16475 SymGrpcsymg 18487 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-oadd 8089 df-er 8272 df-map 8391 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11626 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-z 11970 df-uz 12232 df-fz 12886 df-struct 16477 df-ndx 16478 df-slot 16479 df-base 16481 df-sets 16482 df-ress 16483 df-plusg 16570 df-tset 16576 df-efmnd 18026 df-symg 18488 |
This theorem is referenced by: symg2bas 18513 snsymgefmndeq 18515 psgnsn 18640 m1detdiag 21202 |
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