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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 3983 | . . . . . 6 ⊢ (𝑋 ∈ V → {{〈𝑋, 𝑋〉}} ⊆ {{〈𝑋, 𝑋〉}}) | |
2 | eqid 2820 | . . . . . . 7 ⊢ (EndoFMnd‘{𝑋}) = (EndoFMnd‘{𝑋}) | |
3 | eqid 2820 | . . . . . . 7 ⊢ (Base‘(EndoFMnd‘{𝑋})) = (Base‘(EndoFMnd‘{𝑋})) | |
4 | eqid 2820 | . . . . . . 7 ⊢ {𝑋} = {𝑋} | |
5 | 2, 3, 4 | efmnd1bas 18051 | . . . . . 6 ⊢ (𝑋 ∈ V → (Base‘(EndoFMnd‘{𝑋})) = {{〈𝑋, 𝑋〉}}) |
6 | eqid 2820 | . . . . . . 7 ⊢ (SymGrp‘{𝑋}) = (SymGrp‘{𝑋}) | |
7 | eqid 2820 | . . . . . . 7 ⊢ (Base‘(SymGrp‘{𝑋})) = (Base‘(SymGrp‘{𝑋})) | |
8 | 6, 7, 4 | symg1bas 18512 | . . . . . 6 ⊢ (𝑋 ∈ V → (Base‘(SymGrp‘{𝑋})) = {{〈𝑋, 𝑋〉}}) |
9 | 1, 5, 8 | 3sstr4d 4007 | . . . . 5 ⊢ (𝑋 ∈ V → (Base‘(EndoFMnd‘{𝑋})) ⊆ (Base‘(SymGrp‘{𝑋}))) |
10 | fvexd 6678 | . . . . 5 ⊢ (𝑋 ∈ V → (EndoFMnd‘{𝑋}) ∈ V) | |
11 | fvexd 6678 | . . . . 5 ⊢ (𝑋 ∈ V → (Base‘(SymGrp‘{𝑋})) ∈ V) | |
12 | 6, 7, 2 | symgressbas 18503 | . . . . . 6 ⊢ (SymGrp‘{𝑋}) = ((EndoFMnd‘{𝑋}) ↾s (Base‘(SymGrp‘{𝑋}))) |
13 | 12, 3 | ressid2 16545 | . . . . 5 ⊢ (((Base‘(EndoFMnd‘{𝑋})) ⊆ (Base‘(SymGrp‘{𝑋})) ∧ (EndoFMnd‘{𝑋}) ∈ V ∧ (Base‘(SymGrp‘{𝑋})) ∈ V) → (SymGrp‘{𝑋}) = (EndoFMnd‘{𝑋})) |
14 | 9, 10, 11, 13 | syl3anc 1366 | . . . 4 ⊢ (𝑋 ∈ V → (SymGrp‘{𝑋}) = (EndoFMnd‘{𝑋})) |
15 | 14 | eqcomd 2826 | . . 3 ⊢ (𝑋 ∈ V → (EndoFMnd‘{𝑋}) = (SymGrp‘{𝑋})) |
16 | fveq2 6663 | . . . 4 ⊢ (𝐴 = {𝑋} → (EndoFMnd‘𝐴) = (EndoFMnd‘{𝑋})) | |
17 | fveq2 6663 | . . . 4 ⊢ (𝐴 = {𝑋} → (SymGrp‘𝐴) = (SymGrp‘{𝑋})) | |
18 | 16, 17 | eqeq12d 2836 | . . 3 ⊢ (𝐴 = {𝑋} → ((EndoFMnd‘𝐴) = (SymGrp‘𝐴) ↔ (EndoFMnd‘{𝑋}) = (SymGrp‘{𝑋}))) |
19 | 15, 18 | syl5ibrcom 249 | . 2 ⊢ (𝑋 ∈ V → (𝐴 = {𝑋} → (EndoFMnd‘𝐴) = (SymGrp‘𝐴))) |
20 | snprc 4646 | . . . . 5 ⊢ (¬ 𝑋 ∈ V ↔ {𝑋} = ∅) | |
21 | 20 | biimpi 218 | . . . 4 ⊢ (¬ 𝑋 ∈ V → {𝑋} = ∅) |
22 | 21 | eqeq2d 2831 | . . 3 ⊢ (¬ 𝑋 ∈ V → (𝐴 = {𝑋} ↔ 𝐴 = ∅)) |
23 | 0symgefmndeq 18515 | . . . 4 ⊢ (EndoFMnd‘∅) = (SymGrp‘∅) | |
24 | fveq2 6663 | . . . 4 ⊢ (𝐴 = ∅ → (EndoFMnd‘𝐴) = (EndoFMnd‘∅)) | |
25 | fveq2 6663 | . . . 4 ⊢ (𝐴 = ∅ → (SymGrp‘𝐴) = (SymGrp‘∅)) | |
26 | 23, 24, 25 | 3eqtr4a 2881 | . . 3 ⊢ (𝐴 = ∅ → (EndoFMnd‘𝐴) = (SymGrp‘𝐴)) |
27 | 22, 26 | syl6bi 255 | . 2 ⊢ (¬ 𝑋 ∈ V → (𝐴 = {𝑋} → (EndoFMnd‘𝐴) = (SymGrp‘𝐴))) |
28 | 19, 27 | pm2.61i 184 | 1 ⊢ (𝐴 = {𝑋} → (EndoFMnd‘𝐴) = (SymGrp‘𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1536 ∈ wcel 2113 Vcvv 3491 ⊆ wss 3929 ∅c0 4284 {csn 4560 〈cop 4566 ‘cfv 6348 Basecbs 16476 EndoFMndcefmnd 18026 SymGrpcsymg 18488 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5323 ax-un 7454 ax-cnex 10586 ax-resscn 10587 ax-1cn 10588 ax-icn 10589 ax-addcl 10590 ax-addrcl 10591 ax-mulcl 10592 ax-mulrcl 10593 ax-mulcom 10594 ax-addass 10595 ax-mulass 10596 ax-distr 10597 ax-i2m1 10598 ax-1ne0 10599 ax-1rid 10600 ax-rnegex 10601 ax-rrecex 10602 ax-cnre 10603 ax-pre-lttri 10604 ax-pre-lttrn 10605 ax-pre-ltadd 10606 ax-pre-mulgt0 10607 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ne 3016 df-nel 3123 df-ral 3142 df-rex 3143 df-reu 3144 df-rab 3146 df-v 3493 df-sbc 3769 df-csb 3877 df-dif 3932 df-un 3934 df-in 3936 df-ss 3945 df-pss 3947 df-nul 4285 df-if 4461 df-pw 4534 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-int 4870 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7107 df-ov 7152 df-oprab 7153 df-mpo 7154 df-om 7574 df-1st 7682 df-2nd 7683 df-wrecs 7940 df-recs 8001 df-rdg 8039 df-1o 8095 df-oadd 8099 df-er 8282 df-map 8401 df-en 8503 df-dom 8504 df-sdom 8505 df-fin 8506 df-pnf 10670 df-mnf 10671 df-xr 10672 df-ltxr 10673 df-le 10674 df-sub 10865 df-neg 10866 df-nn 11632 df-2 11694 df-3 11695 df-4 11696 df-5 11697 df-6 11698 df-7 11699 df-8 11700 df-9 11701 df-n0 11892 df-z 11976 df-uz 12238 df-fz 12890 df-struct 16478 df-ndx 16479 df-slot 16480 df-base 16482 df-sets 16483 df-ress 16484 df-plusg 16571 df-tset 16577 df-efmnd 18027 df-symg 18489 |
This theorem is referenced by: symgvalstruct 18518 |
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