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Mirrors > Home > MPE Home > Th. List > symgsubmefmnd | Structured version Visualization version GIF version |
Description: The symmetric group on a set π΄ is a submonoid of the monoid of endofunctions on π΄. (Contributed by AV, 18-Feb-2024.) |
Ref | Expression |
---|---|
symgsubmefmnd.m | β’ π = (EndoFMndβπ΄) |
symgsubmefmnd.g | β’ πΊ = (SymGrpβπ΄) |
symgsubmefmnd.b | β’ π΅ = (BaseβπΊ) |
Ref | Expression |
---|---|
symgsubmefmnd | β’ (π΄ β π β π΅ β (SubMndβπ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | symgsubmefmnd.g | . . 3 β’ πΊ = (SymGrpβπ΄) | |
2 | symgsubmefmnd.b | . . 3 β’ π΅ = (BaseβπΊ) | |
3 | 1, 2 | symgbas 19329 | . 2 β’ π΅ = {π β£ π:π΄β1-1-ontoβπ΄} |
4 | inab 4294 | . . . 4 β’ ({π β£ π:π΄β1-1βπ΄} β© {π β£ π:π΄βontoβπ΄}) = {π β£ (π:π΄β1-1βπ΄ β§ π:π΄βontoβπ΄)} | |
5 | df-f1o 6550 | . . . . . 6 β’ (π:π΄β1-1-ontoβπ΄ β (π:π΄β1-1βπ΄ β§ π:π΄βontoβπ΄)) | |
6 | 5 | bicomi 223 | . . . . 5 β’ ((π:π΄β1-1βπ΄ β§ π:π΄βontoβπ΄) β π:π΄β1-1-ontoβπ΄) |
7 | 6 | abbii 2795 | . . . 4 β’ {π β£ (π:π΄β1-1βπ΄ β§ π:π΄βontoβπ΄)} = {π β£ π:π΄β1-1-ontoβπ΄} |
8 | 4, 7 | eqtr2i 2754 | . . 3 β’ {π β£ π:π΄β1-1-ontoβπ΄} = ({π β£ π:π΄β1-1βπ΄} β© {π β£ π:π΄βontoβπ΄}) |
9 | symgsubmefmnd.m | . . . . 5 β’ π = (EndoFMndβπ΄) | |
10 | 9 | injsubmefmnd 18853 | . . . 4 β’ (π΄ β π β {π β£ π:π΄β1-1βπ΄} β (SubMndβπ)) |
11 | 9 | sursubmefmnd 18852 | . . . 4 β’ (π΄ β π β {π β£ π:π΄βontoβπ΄} β (SubMndβπ)) |
12 | insubm 18774 | . . . 4 β’ (({π β£ π:π΄β1-1βπ΄} β (SubMndβπ) β§ {π β£ π:π΄βontoβπ΄} β (SubMndβπ)) β ({π β£ π:π΄β1-1βπ΄} β© {π β£ π:π΄βontoβπ΄}) β (SubMndβπ)) | |
13 | 10, 11, 12 | syl2anc 582 | . . 3 β’ (π΄ β π β ({π β£ π:π΄β1-1βπ΄} β© {π β£ π:π΄βontoβπ΄}) β (SubMndβπ)) |
14 | 8, 13 | eqeltrid 2829 | . 2 β’ (π΄ β π β {π β£ π:π΄β1-1-ontoβπ΄} β (SubMndβπ)) |
15 | 3, 14 | eqeltrid 2829 | 1 β’ (π΄ β π β π΅ β (SubMndβπ)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 394 = wceq 1533 β wcel 2098 {cab 2702 β© cin 3938 β1-1βwf1 6540 βontoβwfo 6541 β1-1-ontoβwf1o 6542 βcfv 6543 Basecbs 17179 SubMndcsubmnd 18738 EndoFMndcefmnd 18824 SymGrpcsymg 19325 |
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 2696 ax-rep 5280 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7738 ax-cnex 11194 ax-resscn 11195 ax-1cn 11196 ax-icn 11197 ax-addcl 11198 ax-addrcl 11199 ax-mulcl 11200 ax-mulrcl 11201 ax-mulcom 11202 ax-addass 11203 ax-mulass 11204 ax-distr 11205 ax-i2m1 11206 ax-1ne0 11207 ax-1rid 11208 ax-rnegex 11209 ax-rrecex 11210 ax-cnre 11211 ax-pre-lttri 11212 ax-pre-lttrn 11213 ax-pre-ltadd 11214 ax-pre-mulgt0 11215 |
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 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3769 df-csb 3885 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-pss 3959 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-uni 4904 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7372 df-ov 7419 df-oprab 7420 df-mpo 7421 df-om 7869 df-1st 7991 df-2nd 7992 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-1o 8485 df-er 8723 df-map 8845 df-en 8963 df-dom 8964 df-sdom 8965 df-fin 8966 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 df-le 11284 df-sub 11476 df-neg 11477 df-nn 12243 df-2 12305 df-3 12306 df-4 12307 df-5 12308 df-6 12309 df-7 12310 df-8 12311 df-9 12312 df-n0 12503 df-z 12589 df-uz 12853 df-fz 13517 df-struct 17115 df-sets 17132 df-slot 17150 df-ndx 17162 df-base 17180 df-ress 17209 df-plusg 17245 df-tset 17251 df-0g 17422 df-mgm 18599 df-sgrp 18678 df-mnd 18694 df-submnd 18740 df-efmnd 18825 df-symg 19326 |
This theorem is referenced by: symgid 19360 symgtgp 24028 |
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