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Mirrors > Home > MPE Home > Th. List > snsymgefmndeq | Structured version Visualization version GIF version |
Description: The symmetric group on a singleton 𝐴 is identical with the monoid of endofunctions on 𝐴. (Contributed by AV, 31-Mar-2024.) |
Ref | Expression |
---|---|
snsymgefmndeq | ⊢ (𝐴 = {𝑋} → (EndoFMnd‘𝐴) = (SymGrp‘𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssidd 3997 | . . . . . 6 ⊢ (𝑋 ∈ V → {{〈𝑋, 𝑋〉}} ⊆ {{〈𝑋, 𝑋〉}}) | |
2 | eqid 2724 | . . . . . . 7 ⊢ (EndoFMnd‘{𝑋}) = (EndoFMnd‘{𝑋}) | |
3 | eqid 2724 | . . . . . . 7 ⊢ (Base‘(EndoFMnd‘{𝑋})) = (Base‘(EndoFMnd‘{𝑋})) | |
4 | eqid 2724 | . . . . . . 7 ⊢ {𝑋} = {𝑋} | |
5 | 2, 3, 4 | efmnd1bas 18805 | . . . . . 6 ⊢ (𝑋 ∈ V → (Base‘(EndoFMnd‘{𝑋})) = {{〈𝑋, 𝑋〉}}) |
6 | eqid 2724 | . . . . . . 7 ⊢ (SymGrp‘{𝑋}) = (SymGrp‘{𝑋}) | |
7 | eqid 2724 | . . . . . . 7 ⊢ (Base‘(SymGrp‘{𝑋})) = (Base‘(SymGrp‘{𝑋})) | |
8 | 6, 7, 4 | symg1bas 19295 | . . . . . 6 ⊢ (𝑋 ∈ V → (Base‘(SymGrp‘{𝑋})) = {{〈𝑋, 𝑋〉}}) |
9 | 1, 5, 8 | 3sstr4d 4021 | . . . . 5 ⊢ (𝑋 ∈ V → (Base‘(EndoFMnd‘{𝑋})) ⊆ (Base‘(SymGrp‘{𝑋}))) |
10 | fvexd 6896 | . . . . 5 ⊢ (𝑋 ∈ V → (EndoFMnd‘{𝑋}) ∈ V) | |
11 | fvexd 6896 | . . . . 5 ⊢ (𝑋 ∈ V → (Base‘(SymGrp‘{𝑋})) ∈ V) | |
12 | 6, 7, 2 | symgressbas 19286 | . . . . . 6 ⊢ (SymGrp‘{𝑋}) = ((EndoFMnd‘{𝑋}) ↾s (Base‘(SymGrp‘{𝑋}))) |
13 | 12, 3 | ressid2 17173 | . . . . 5 ⊢ (((Base‘(EndoFMnd‘{𝑋})) ⊆ (Base‘(SymGrp‘{𝑋})) ∧ (EndoFMnd‘{𝑋}) ∈ V ∧ (Base‘(SymGrp‘{𝑋})) ∈ V) → (SymGrp‘{𝑋}) = (EndoFMnd‘{𝑋})) |
14 | 9, 10, 11, 13 | syl3anc 1368 | . . . 4 ⊢ (𝑋 ∈ V → (SymGrp‘{𝑋}) = (EndoFMnd‘{𝑋})) |
15 | 14 | eqcomd 2730 | . . 3 ⊢ (𝑋 ∈ V → (EndoFMnd‘{𝑋}) = (SymGrp‘{𝑋})) |
16 | fveq2 6881 | . . . 4 ⊢ (𝐴 = {𝑋} → (EndoFMnd‘𝐴) = (EndoFMnd‘{𝑋})) | |
17 | fveq2 6881 | . . . 4 ⊢ (𝐴 = {𝑋} → (SymGrp‘𝐴) = (SymGrp‘{𝑋})) | |
18 | 16, 17 | eqeq12d 2740 | . . 3 ⊢ (𝐴 = {𝑋} → ((EndoFMnd‘𝐴) = (SymGrp‘𝐴) ↔ (EndoFMnd‘{𝑋}) = (SymGrp‘{𝑋}))) |
19 | 15, 18 | syl5ibrcom 246 | . 2 ⊢ (𝑋 ∈ V → (𝐴 = {𝑋} → (EndoFMnd‘𝐴) = (SymGrp‘𝐴))) |
20 | snprc 4713 | . . . . 5 ⊢ (¬ 𝑋 ∈ V ↔ {𝑋} = ∅) | |
21 | 20 | biimpi 215 | . . . 4 ⊢ (¬ 𝑋 ∈ V → {𝑋} = ∅) |
22 | 21 | eqeq2d 2735 | . . 3 ⊢ (¬ 𝑋 ∈ V → (𝐴 = {𝑋} ↔ 𝐴 = ∅)) |
23 | 0symgefmndeq 19298 | . . . 4 ⊢ (EndoFMnd‘∅) = (SymGrp‘∅) | |
24 | fveq2 6881 | . . . 4 ⊢ (𝐴 = ∅ → (EndoFMnd‘𝐴) = (EndoFMnd‘∅)) | |
25 | fveq2 6881 | . . . 4 ⊢ (𝐴 = ∅ → (SymGrp‘𝐴) = (SymGrp‘∅)) | |
26 | 23, 24, 25 | 3eqtr4a 2790 | . . 3 ⊢ (𝐴 = ∅ → (EndoFMnd‘𝐴) = (SymGrp‘𝐴)) |
27 | 22, 26 | syl6bi 253 | . 2 ⊢ (¬ 𝑋 ∈ V → (𝐴 = {𝑋} → (EndoFMnd‘𝐴) = (SymGrp‘𝐴))) |
28 | 19, 27 | pm2.61i 182 | 1 ⊢ (𝐴 = {𝑋} → (EndoFMnd‘𝐴) = (SymGrp‘𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1533 ∈ wcel 2098 Vcvv 3466 ⊆ wss 3940 ∅c0 4314 {csn 4620 〈cop 4626 ‘cfv 6533 Basecbs 17140 EndoFMndcefmnd 18780 SymGrpcsymg 19271 |
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 2695 ax-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7718 ax-cnex 11161 ax-resscn 11162 ax-1cn 11163 ax-icn 11164 ax-addcl 11165 ax-addrcl 11166 ax-mulcl 11167 ax-mulrcl 11168 ax-mulcom 11169 ax-addass 11170 ax-mulass 11171 ax-distr 11172 ax-i2m1 11173 ax-1ne0 11174 ax-1rid 11175 ax-rnegex 11176 ax-rrecex 11177 ax-cnre 11178 ax-pre-lttri 11179 ax-pre-lttrn 11180 ax-pre-ltadd 11181 ax-pre-mulgt0 11182 |
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 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3959 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-tp 4625 df-op 4627 df-uni 4900 df-iun 4989 df-br 5139 df-opab 5201 df-mpt 5222 df-tr 5256 df-id 5564 df-eprel 5570 df-po 5578 df-so 5579 df-fr 5621 df-we 5623 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-pred 6290 df-ord 6357 df-on 6358 df-lim 6359 df-suc 6360 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-riota 7357 df-ov 7404 df-oprab 7405 df-mpo 7406 df-om 7849 df-1st 7968 df-2nd 7969 df-frecs 8261 df-wrecs 8292 df-recs 8366 df-rdg 8405 df-1o 8461 df-er 8698 df-map 8817 df-en 8935 df-dom 8936 df-sdom 8937 df-fin 8938 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-nn 12209 df-2 12271 df-3 12272 df-4 12273 df-5 12274 df-6 12275 df-7 12276 df-8 12277 df-9 12278 df-n0 12469 df-z 12555 df-uz 12819 df-fz 13481 df-struct 17076 df-sets 17093 df-slot 17111 df-ndx 17123 df-base 17141 df-ress 17170 df-plusg 17206 df-tset 17212 df-efmnd 18781 df-symg 19272 |
This theorem is referenced by: symgvalstruct 19301 symgvalstructOLD 19302 |
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