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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 19290 | . 2 ⊢ 𝐵 = {𝑓 ∣ 𝑓:𝐴–1-1-onto→𝐴} |
4 | symg1bas.0 | . . . . . 6 ⊢ 𝐴 = {𝐼} | |
5 | eqidd 2727 | . . . . . . 7 ⊢ (𝐴 = {𝐼} → 𝑝 = 𝑝) | |
6 | id 22 | . . . . . . 7 ⊢ (𝐴 = {𝐼} → 𝐴 = {𝐼}) | |
7 | 5, 6, 6 | f1oeq123d 6821 | . . . . . 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 6827 | . . . . . . 7 ⊢ (𝑝:{𝐼}–1-1-onto→{𝐼} → 𝑝:{𝐼}⟶{𝐼}) | |
10 | fsng 7131 | . . . . . . . 8 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐼 ∈ 𝑉) → (𝑝:{𝐼}⟶{𝐼} ↔ 𝑝 = {⟨𝐼, 𝐼⟩})) | |
11 | 10 | anidms 566 | . . . . . . 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 6868 | . . . . . . . 8 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐼 ∈ 𝑉) → {⟨𝐼, 𝐼⟩}:{𝐼}–1-1-onto→{𝐼}) | |
14 | 13 | anidms 566 | . . . . . . 7 ⊢ (𝐼 ∈ 𝑉 → {⟨𝐼, 𝐼⟩}:{𝐼}–1-1-onto→{𝐼}) |
15 | f1oeq1 6815 | . . . . . . 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 283 | . . . 4 ⊢ (𝐼 ∈ 𝑉 → (𝑝:𝐴–1-1-onto→𝐴 ↔ 𝑝 = {⟨𝐼, 𝐼⟩})) |
19 | vex 3472 | . . . . 5 ⊢ 𝑝 ∈ V | |
20 | f1oeq1 6815 | . . . . 5 ⊢ (𝑓 = 𝑝 → (𝑓:𝐴–1-1-onto→𝐴 ↔ 𝑝:𝐴–1-1-onto→𝐴)) | |
21 | 19, 20 | elab 3663 | . . . 4 ⊢ (𝑝 ∈ {𝑓 ∣ 𝑓:𝐴–1-1-onto→𝐴} ↔ 𝑝:𝐴–1-1-onto→𝐴) |
22 | velsn 4639 | . . . 4 ⊢ (𝑝 ∈ {{⟨𝐼, 𝐼⟩}} ↔ 𝑝 = {⟨𝐼, 𝐼⟩}) | |
23 | 18, 21, 22 | 3bitr4g 314 | . . 3 ⊢ (𝐼 ∈ 𝑉 → (𝑝 ∈ {𝑓 ∣ 𝑓:𝐴–1-1-onto→𝐴} ↔ 𝑝 ∈ {{⟨𝐼, 𝐼⟩}})) |
24 | 23 | eqrdv 2724 | . 2 ⊢ (𝐼 ∈ 𝑉 → {𝑓 ∣ 𝑓:𝐴–1-1-onto→𝐴} = {{⟨𝐼, 𝐼⟩}}) |
25 | 3, 24 | eqtrid 2778 | 1 ⊢ (𝐼 ∈ 𝑉 → 𝐵 = {{⟨𝐼, 𝐼⟩}}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 = wceq 1533 ∈ wcel 2098 {cab 2703 {csn 4623 ⟨cop 4629 ⟶wf 6533 –1-1-onto→wf1o 6536 ‘cfv 6537 Basecbs 17153 SymGrpcsymg 19286 |
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 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-tp 4628 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6294 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7853 df-1st 7974 df-2nd 7975 df-frecs 8267 df-wrecs 8298 df-recs 8372 df-rdg 8411 df-1o 8467 df-er 8705 df-map 8824 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-nn 12217 df-2 12279 df-3 12280 df-4 12281 df-5 12282 df-6 12283 df-7 12284 df-8 12285 df-9 12286 df-n0 12477 df-z 12563 df-uz 12827 df-fz 13491 df-struct 17089 df-sets 17106 df-slot 17124 df-ndx 17136 df-base 17154 df-ress 17183 df-plusg 17219 df-tset 17225 df-efmnd 18794 df-symg 19287 |
This theorem is referenced by: symg2bas 19312 snsymgefmndeq 19314 psgnsn 19440 m1detdiag 22454 |
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