Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 18501 | . 2 ⊢ 𝐵 = {𝑓 ∣ 𝑓:𝐴–1-1-onto→𝐴} |
4 | symg1bas.0 | . . . . . 6 ⊢ 𝐴 = {𝐼} | |
5 | eqidd 2824 | . . . . . . 7 ⊢ (𝐴 = {𝐼} → 𝑝 = 𝑝) | |
6 | id 22 | . . . . . . 7 ⊢ (𝐴 = {𝐼} → 𝐴 = {𝐼}) | |
7 | 5, 6, 6 | f1oeq123d 6612 | . . . . . 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 6617 | . . . . . . 7 ⊢ (𝑝:{𝐼}–1-1-onto→{𝐼} → 𝑝:{𝐼}⟶{𝐼}) | |
10 | fsng 6901 | . . . . . . . 8 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐼 ∈ 𝑉) → (𝑝:{𝐼}⟶{𝐼} ↔ 𝑝 = {〈𝐼, 𝐼〉})) | |
11 | 10 | anidms 569 | . . . . . . 7 ⊢ (𝐼 ∈ 𝑉 → (𝑝:{𝐼}⟶{𝐼} ↔ 𝑝 = {〈𝐼, 𝐼〉})) |
12 | 9, 11 | syl5ib 246 | . . . . . 6 ⊢ (𝐼 ∈ 𝑉 → (𝑝:{𝐼}–1-1-onto→{𝐼} → 𝑝 = {〈𝐼, 𝐼〉})) |
13 | f1osng 6657 | . . . . . . . 8 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐼 ∈ 𝑉) → {〈𝐼, 𝐼〉}:{𝐼}–1-1-onto→{𝐼}) | |
14 | 13 | anidms 569 | . . . . . . 7 ⊢ (𝐼 ∈ 𝑉 → {〈𝐼, 𝐼〉}:{𝐼}–1-1-onto→{𝐼}) |
15 | f1oeq1 6606 | . . . . . . 7 ⊢ (𝑝 = {〈𝐼, 𝐼〉} → (𝑝:{𝐼}–1-1-onto→{𝐼} ↔ {〈𝐼, 𝐼〉}:{𝐼}–1-1-onto→{𝐼})) | |
16 | 14, 15 | syl5ibrcom 249 | . . . . . 6 ⊢ (𝐼 ∈ 𝑉 → (𝑝 = {〈𝐼, 𝐼〉} → 𝑝:{𝐼}–1-1-onto→{𝐼})) |
17 | 12, 16 | impbid 214 | . . . . 5 ⊢ (𝐼 ∈ 𝑉 → (𝑝:{𝐼}–1-1-onto→{𝐼} ↔ 𝑝 = {〈𝐼, 𝐼〉})) |
18 | 8, 17 | syl5bb 285 | . . . 4 ⊢ (𝐼 ∈ 𝑉 → (𝑝:𝐴–1-1-onto→𝐴 ↔ 𝑝 = {〈𝐼, 𝐼〉})) |
19 | vex 3499 | . . . . 5 ⊢ 𝑝 ∈ V | |
20 | f1oeq1 6606 | . . . . 5 ⊢ (𝑓 = 𝑝 → (𝑓:𝐴–1-1-onto→𝐴 ↔ 𝑝:𝐴–1-1-onto→𝐴)) | |
21 | 19, 20 | elab 3669 | . . . 4 ⊢ (𝑝 ∈ {𝑓 ∣ 𝑓:𝐴–1-1-onto→𝐴} ↔ 𝑝:𝐴–1-1-onto→𝐴) |
22 | velsn 4585 | . . . 4 ⊢ (𝑝 ∈ {{〈𝐼, 𝐼〉}} ↔ 𝑝 = {〈𝐼, 𝐼〉}) | |
23 | 18, 21, 22 | 3bitr4g 316 | . . 3 ⊢ (𝐼 ∈ 𝑉 → (𝑝 ∈ {𝑓 ∣ 𝑓:𝐴–1-1-onto→𝐴} ↔ 𝑝 ∈ {{〈𝐼, 𝐼〉}})) |
24 | 23 | eqrdv 2821 | . 2 ⊢ (𝐼 ∈ 𝑉 → {𝑓 ∣ 𝑓:𝐴–1-1-onto→𝐴} = {{〈𝐼, 𝐼〉}}) |
25 | 3, 24 | syl5eq 2870 | 1 ⊢ (𝐼 ∈ 𝑉 → 𝐵 = {{〈𝐼, 𝐼〉}}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 = wceq 1537 ∈ wcel 2114 {cab 2801 {csn 4569 〈cop 4575 ⟶wf 6353 –1-1-onto→wf1o 6356 ‘cfv 6357 Basecbs 16485 SymGrpcsymg 18497 |
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-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 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-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 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-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 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-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 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-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-oadd 8108 df-er 8291 df-map 8410 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-nn 11641 df-2 11703 df-3 11704 df-4 11705 df-5 11706 df-6 11707 df-7 11708 df-8 11709 df-9 11710 df-n0 11901 df-z 11985 df-uz 12247 df-fz 12896 df-struct 16487 df-ndx 16488 df-slot 16489 df-base 16491 df-sets 16492 df-ress 16493 df-plusg 16580 df-tset 16586 df-efmnd 18036 df-symg 18498 |
This theorem is referenced by: symg2bas 18523 snsymgefmndeq 18525 psgnsn 18650 m1detdiag 21208 |
Copyright terms: Public domain | W3C validator |