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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 19332 | . 2 ⊢ 𝐵 = {𝑓 ∣ 𝑓:𝐴–1-1-onto→𝐴} |
4 | symg1bas.0 | . . . . . 6 ⊢ 𝐴 = {𝐼} | |
5 | eqidd 2729 | . . . . . . 7 ⊢ (𝐴 = {𝐼} → 𝑝 = 𝑝) | |
6 | id 22 | . . . . . . 7 ⊢ (𝐴 = {𝐼} → 𝐴 = {𝐼}) | |
7 | 5, 6, 6 | f1oeq123d 6838 | . . . . . 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 6844 | . . . . . . 7 ⊢ (𝑝:{𝐼}–1-1-onto→{𝐼} → 𝑝:{𝐼}⟶{𝐼}) | |
10 | fsng 7152 | . . . . . . . 8 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐼 ∈ 𝑉) → (𝑝:{𝐼}⟶{𝐼} ↔ 𝑝 = {⟨𝐼, 𝐼⟩})) | |
11 | 10 | anidms 565 | . . . . . . 7 ⊢ (𝐼 ∈ 𝑉 → (𝑝:{𝐼}⟶{𝐼} ↔ 𝑝 = {⟨𝐼, 𝐼⟩})) |
12 | 9, 11 | imbitrid 243 | . . . . . 6 ⊢ (𝐼 ∈ 𝑉 → (𝑝:{𝐼}–1-1-onto→{𝐼} → 𝑝 = {⟨𝐼, 𝐼⟩})) |
13 | f1osng 6885 | . . . . . . . 8 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐼 ∈ 𝑉) → {⟨𝐼, 𝐼⟩}:{𝐼}–1-1-onto→{𝐼}) | |
14 | 13 | anidms 565 | . . . . . . 7 ⊢ (𝐼 ∈ 𝑉 → {⟨𝐼, 𝐼⟩}:{𝐼}–1-1-onto→{𝐼}) |
15 | f1oeq1 6832 | . . . . . . 7 ⊢ (𝑝 = {⟨𝐼, 𝐼⟩} → (𝑝:{𝐼}–1-1-onto→{𝐼} ↔ {⟨𝐼, 𝐼⟩}:{𝐼}–1-1-onto→{𝐼})) | |
16 | 14, 15 | syl5ibrcom 246 | . . . . . 6 ⊢ (𝐼 ∈ 𝑉 → (𝑝 = {⟨𝐼, 𝐼⟩} → 𝑝:{𝐼}–1-1-onto→{𝐼})) |
17 | 12, 16 | impbid 211 | . . . . 5 ⊢ (𝐼 ∈ 𝑉 → (𝑝:{𝐼}–1-1-onto→{𝐼} ↔ 𝑝 = {⟨𝐼, 𝐼⟩})) |
18 | 8, 17 | bitrid 282 | . . . 4 ⊢ (𝐼 ∈ 𝑉 → (𝑝:𝐴–1-1-onto→𝐴 ↔ 𝑝 = {⟨𝐼, 𝐼⟩})) |
19 | vex 3477 | . . . . 5 ⊢ 𝑝 ∈ V | |
20 | f1oeq1 6832 | . . . . 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 4648 | . . . 4 ⊢ (𝑝 ∈ {{⟨𝐼, 𝐼⟩}} ↔ 𝑝 = {⟨𝐼, 𝐼⟩}) | |
23 | 18, 21, 22 | 3bitr4g 313 | . . 3 ⊢ (𝐼 ∈ 𝑉 → (𝑝 ∈ {𝑓 ∣ 𝑓:𝐴–1-1-onto→𝐴} ↔ 𝑝 ∈ {{⟨𝐼, 𝐼⟩}})) |
24 | 23 | eqrdv 2726 | . 2 ⊢ (𝐼 ∈ 𝑉 → {𝑓 ∣ 𝑓:𝐴–1-1-onto→𝐴} = {{⟨𝐼, 𝐼⟩}}) |
25 | 3, 24 | eqtrid 2780 | 1 ⊢ (𝐼 ∈ 𝑉 → 𝐵 = {{⟨𝐼, 𝐼⟩}}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 = wceq 1533 ∈ wcel 2098 {cab 2705 {csn 4632 ⟨cop 4638 ⟶wf 6549 –1-1-onto→wf1o 6552 ‘cfv 6553 Basecbs 17187 SymGrpcsymg 19328 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11202 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 ax-pre-mulgt0 11223 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-tp 4637 df-op 4639 df-uni 4913 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-om 7877 df-1st 7999 df-2nd 8000 df-frecs 8293 df-wrecs 8324 df-recs 8398 df-rdg 8437 df-1o 8493 df-er 8731 df-map 8853 df-en 8971 df-dom 8972 df-sdom 8973 df-fin 8974 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-sub 11484 df-neg 11485 df-nn 12251 df-2 12313 df-3 12314 df-4 12315 df-5 12316 df-6 12317 df-7 12318 df-8 12319 df-9 12320 df-n0 12511 df-z 12597 df-uz 12861 df-fz 13525 df-struct 17123 df-sets 17140 df-slot 17158 df-ndx 17170 df-base 17188 df-ress 17217 df-plusg 17253 df-tset 17259 df-efmnd 18828 df-symg 19329 |
This theorem is referenced by: symg2bas 19354 snsymgefmndeq 19356 psgnsn 19482 m1detdiag 22519 |
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