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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 3938 | . . . . . 6 ⊢ (𝑋 ∈ V → {{〈𝑋, 𝑋〉}} ⊆ {{〈𝑋, 𝑋〉}}) | |
2 | eqid 2798 | . . . . . . 7 ⊢ (EndoFMnd‘{𝑋}) = (EndoFMnd‘{𝑋}) | |
3 | eqid 2798 | . . . . . . 7 ⊢ (Base‘(EndoFMnd‘{𝑋})) = (Base‘(EndoFMnd‘{𝑋})) | |
4 | eqid 2798 | . . . . . . 7 ⊢ {𝑋} = {𝑋} | |
5 | 2, 3, 4 | efmnd1bas 18050 | . . . . . 6 ⊢ (𝑋 ∈ V → (Base‘(EndoFMnd‘{𝑋})) = {{〈𝑋, 𝑋〉}}) |
6 | eqid 2798 | . . . . . . 7 ⊢ (SymGrp‘{𝑋}) = (SymGrp‘{𝑋}) | |
7 | eqid 2798 | . . . . . . 7 ⊢ (Base‘(SymGrp‘{𝑋})) = (Base‘(SymGrp‘{𝑋})) | |
8 | 6, 7, 4 | symg1bas 18511 | . . . . . 6 ⊢ (𝑋 ∈ V → (Base‘(SymGrp‘{𝑋})) = {{〈𝑋, 𝑋〉}}) |
9 | 1, 5, 8 | 3sstr4d 3962 | . . . . 5 ⊢ (𝑋 ∈ V → (Base‘(EndoFMnd‘{𝑋})) ⊆ (Base‘(SymGrp‘{𝑋}))) |
10 | fvexd 6660 | . . . . 5 ⊢ (𝑋 ∈ V → (EndoFMnd‘{𝑋}) ∈ V) | |
11 | fvexd 6660 | . . . . 5 ⊢ (𝑋 ∈ V → (Base‘(SymGrp‘{𝑋})) ∈ V) | |
12 | 6, 7, 2 | symgressbas 18502 | . . . . . 6 ⊢ (SymGrp‘{𝑋}) = ((EndoFMnd‘{𝑋}) ↾s (Base‘(SymGrp‘{𝑋}))) |
13 | 12, 3 | ressid2 16544 | . . . . 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 2804 | . . 3 ⊢ (𝑋 ∈ V → (EndoFMnd‘{𝑋}) = (SymGrp‘{𝑋})) |
16 | fveq2 6645 | . . . 4 ⊢ (𝐴 = {𝑋} → (EndoFMnd‘𝐴) = (EndoFMnd‘{𝑋})) | |
17 | fveq2 6645 | . . . 4 ⊢ (𝐴 = {𝑋} → (SymGrp‘𝐴) = (SymGrp‘{𝑋})) | |
18 | 16, 17 | eqeq12d 2814 | . . 3 ⊢ (𝐴 = {𝑋} → ((EndoFMnd‘𝐴) = (SymGrp‘𝐴) ↔ (EndoFMnd‘{𝑋}) = (SymGrp‘{𝑋}))) |
19 | 15, 18 | syl5ibrcom 250 | . 2 ⊢ (𝑋 ∈ V → (𝐴 = {𝑋} → (EndoFMnd‘𝐴) = (SymGrp‘𝐴))) |
20 | snprc 4613 | . . . . 5 ⊢ (¬ 𝑋 ∈ V ↔ {𝑋} = ∅) | |
21 | 20 | biimpi 219 | . . . 4 ⊢ (¬ 𝑋 ∈ V → {𝑋} = ∅) |
22 | 21 | eqeq2d 2809 | . . 3 ⊢ (¬ 𝑋 ∈ V → (𝐴 = {𝑋} ↔ 𝐴 = ∅)) |
23 | 0symgefmndeq 18514 | . . . 4 ⊢ (EndoFMnd‘∅) = (SymGrp‘∅) | |
24 | fveq2 6645 | . . . 4 ⊢ (𝐴 = ∅ → (EndoFMnd‘𝐴) = (EndoFMnd‘∅)) | |
25 | fveq2 6645 | . . . 4 ⊢ (𝐴 = ∅ → (SymGrp‘𝐴) = (SymGrp‘∅)) | |
26 | 23, 24, 25 | 3eqtr4a 2859 | . . 3 ⊢ (𝐴 = ∅ → (EndoFMnd‘𝐴) = (SymGrp‘𝐴)) |
27 | 22, 26 | syl6bi 256 | . 2 ⊢ (¬ 𝑋 ∈ V → (𝐴 = {𝑋} → (EndoFMnd‘𝐴) = (SymGrp‘𝐴))) |
28 | 19, 27 | pm2.61i 185 | 1 ⊢ (𝐴 = {𝑋} → (EndoFMnd‘𝐴) = (SymGrp‘𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1538 ∈ wcel 2111 Vcvv 3441 ⊆ wss 3881 ∅c0 4243 {csn 4525 〈cop 4531 ‘cfv 6324 Basecbs 16475 EndoFMndcefmnd 18025 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: symgvalstruct 18517 |
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