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Mirrors > Home > MPE Home > Th. List > Mathboxes > setcthin | Structured version Visualization version GIF version |
Description: A category of sets all of whose objects contain at most one element is thin. (Contributed by Zhi Wang, 20-Sep-2024.) |
Ref | Expression |
---|---|
setcthin.c | β’ (π β πΆ = (SetCatβπ)) |
setcthin.u | β’ (π β π β π) |
setcthin.x | β’ (π β βπ₯ β π β*π π β π₯) |
Ref | Expression |
---|---|
setcthin | β’ (π β πΆ β ThinCat) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | setcthin.c | . 2 β’ (π β πΆ = (SetCatβπ)) | |
2 | eqid 2730 | . . . 4 β’ (SetCatβπ) = (SetCatβπ) | |
3 | setcthin.u | . . . 4 β’ (π β π β π) | |
4 | 2, 3 | setcbas 18032 | . . 3 β’ (π β π = (Baseβ(SetCatβπ))) |
5 | eqidd 2731 | . . 3 β’ (π β (Hom β(SetCatβπ)) = (Hom β(SetCatβπ))) | |
6 | elequ2 2119 | . . . . . . 7 β’ (π₯ = π§ β (π β π₯ β π β π§)) | |
7 | 6 | mobidv 2541 | . . . . . 6 β’ (π₯ = π§ β (β*π π β π₯ β β*π π β π§)) |
8 | setcthin.x | . . . . . . 7 β’ (π β βπ₯ β π β*π π β π₯) | |
9 | 8 | adantr 479 | . . . . . 6 β’ ((π β§ (π¦ β π β§ π§ β π)) β βπ₯ β π β*π π β π₯) |
10 | simprr 769 | . . . . . 6 β’ ((π β§ (π¦ β π β§ π§ β π)) β π§ β π) | |
11 | 7, 9, 10 | rspcdva 3612 | . . . . 5 β’ ((π β§ (π¦ β π β§ π§ β π)) β β*π π β π§) |
12 | mofmo 47600 | . . . . 5 β’ (β*π π β π§ β β*π π:π¦βΆπ§) | |
13 | 11, 12 | syl 17 | . . . 4 β’ ((π β§ (π¦ β π β§ π§ β π)) β β*π π:π¦βΆπ§) |
14 | 3 | adantr 479 | . . . . . 6 β’ ((π β§ (π¦ β π β§ π§ β π)) β π β π) |
15 | eqid 2730 | . . . . . 6 β’ (Hom β(SetCatβπ)) = (Hom β(SetCatβπ)) | |
16 | simprl 767 | . . . . . 6 β’ ((π β§ (π¦ β π β§ π§ β π)) β π¦ β π) | |
17 | 2, 14, 15, 16, 10 | elsetchom 18035 | . . . . 5 β’ ((π β§ (π¦ β π β§ π§ β π)) β (π β (π¦(Hom β(SetCatβπ))π§) β π:π¦βΆπ§)) |
18 | 17 | mobidv 2541 | . . . 4 β’ ((π β§ (π¦ β π β§ π§ β π)) β (β*π π β (π¦(Hom β(SetCatβπ))π§) β β*π π:π¦βΆπ§)) |
19 | 13, 18 | mpbird 256 | . . 3 β’ ((π β§ (π¦ β π β§ π§ β π)) β β*π π β (π¦(Hom β(SetCatβπ))π§)) |
20 | 2 | setccat 18039 | . . . 4 β’ (π β π β (SetCatβπ) β Cat) |
21 | 3, 20 | syl 17 | . . 3 β’ (π β (SetCatβπ) β Cat) |
22 | 4, 5, 19, 21 | isthincd 47744 | . 2 β’ (π β (SetCatβπ) β ThinCat) |
23 | 1, 22 | eqeltrd 2831 | 1 β’ (π β πΆ β ThinCat) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 394 = wceq 1539 β wcel 2104 β*wmo 2530 βwral 3059 βΆwf 6538 βcfv 6542 (class class class)co 7411 Hom chom 17212 Catccat 17612 SetCatcsetc 18029 ThinCatcthinc 47726 |
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 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7727 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 395 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3474 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-tp 4632 df-op 4634 df-uni 4908 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7367 df-ov 7414 df-oprab 7415 df-mpo 7416 df-om 7858 df-1st 7977 df-2nd 7978 df-frecs 8268 df-wrecs 8299 df-recs 8373 df-rdg 8412 df-1o 8468 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-dec 12682 df-uz 12827 df-fz 13489 df-struct 17084 df-slot 17119 df-ndx 17131 df-base 17149 df-hom 17225 df-cco 17226 df-cat 17616 df-cid 17617 df-setc 18030 df-thinc 47727 |
This theorem is referenced by: setc2othin 47763 |
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