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Mirrors > Home > MPE Home > Th. List > funcsetcestrc | Structured version Visualization version GIF version |
Description: The "embedding functor" from the category of sets into the category of extensible structures which sends each set to an extensible structure consisting of the base set slot only, preserving the morphisms as mappings between the corresponding base sets. (Contributed by AV, 28-Mar-2020.) |
Ref | Expression |
---|---|
funcsetcestrc.s | β’ π = (SetCatβπ) |
funcsetcestrc.c | β’ πΆ = (Baseβπ) |
funcsetcestrc.f | β’ (π β πΉ = (π₯ β πΆ β¦ {β¨(Baseβndx), π₯β©})) |
funcsetcestrc.u | β’ (π β π β WUni) |
funcsetcestrc.o | β’ (π β Ο β π) |
funcsetcestrc.g | β’ (π β πΊ = (π₯ β πΆ, π¦ β πΆ β¦ ( I βΎ (π¦ βm π₯)))) |
funcsetcestrc.e | β’ πΈ = (ExtStrCatβπ) |
Ref | Expression |
---|---|
funcsetcestrc | β’ (π β πΉ(π Func πΈ)πΊ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | funcsetcestrc.c | . 2 β’ πΆ = (Baseβπ) | |
2 | eqid 2730 | . 2 β’ (BaseβπΈ) = (BaseβπΈ) | |
3 | eqid 2730 | . 2 β’ (Hom βπ) = (Hom βπ) | |
4 | eqid 2730 | . 2 β’ (Hom βπΈ) = (Hom βπΈ) | |
5 | eqid 2730 | . 2 β’ (Idβπ) = (Idβπ) | |
6 | eqid 2730 | . 2 β’ (IdβπΈ) = (IdβπΈ) | |
7 | eqid 2730 | . 2 β’ (compβπ) = (compβπ) | |
8 | eqid 2730 | . 2 β’ (compβπΈ) = (compβπΈ) | |
9 | funcsetcestrc.u | . . 3 β’ (π β π β WUni) | |
10 | funcsetcestrc.s | . . . 4 β’ π = (SetCatβπ) | |
11 | 10 | setccat 18039 | . . 3 β’ (π β WUni β π β Cat) |
12 | 9, 11 | syl 17 | . 2 β’ (π β π β Cat) |
13 | funcsetcestrc.e | . . . 4 β’ πΈ = (ExtStrCatβπ) | |
14 | 13 | estrccat 18088 | . . 3 β’ (π β WUni β πΈ β Cat) |
15 | 9, 14 | syl 17 | . 2 β’ (π β πΈ β Cat) |
16 | funcsetcestrc.f | . . 3 β’ (π β πΉ = (π₯ β πΆ β¦ {β¨(Baseβndx), π₯β©})) | |
17 | funcsetcestrc.o | . . 3 β’ (π β Ο β π) | |
18 | 10, 1, 16, 9, 17, 13, 2 | funcsetcestrclem3 18112 | . 2 β’ (π β πΉ:πΆβΆ(BaseβπΈ)) |
19 | funcsetcestrc.g | . . 3 β’ (π β πΊ = (π₯ β πΆ, π¦ β πΆ β¦ ( I βΎ (π¦ βm π₯)))) | |
20 | 10, 1, 16, 9, 17, 19 | funcsetcestrclem4 18114 | . 2 β’ (π β πΊ Fn (πΆ Γ πΆ)) |
21 | 10, 1, 16, 9, 17, 19, 13 | funcsetcestrclem8 18118 | . 2 β’ ((π β§ (π β πΆ β§ π β πΆ)) β (ππΊπ):(π(Hom βπ)π)βΆ((πΉβπ)(Hom βπΈ)(πΉβπ))) |
22 | 10, 1, 16, 9, 17, 19, 13 | funcsetcestrclem7 18117 | . 2 β’ ((π β§ π β πΆ) β ((ππΊπ)β((Idβπ)βπ)) = ((IdβπΈ)β(πΉβπ))) |
23 | 10, 1, 16, 9, 17, 19, 13 | funcsetcestrclem9 18119 | . 2 β’ ((π β§ (π β πΆ β§ π β πΆ β§ π β πΆ) β§ (β β (π(Hom βπ)π) β§ π β (π(Hom βπ)π))) β ((ππΊπ)β(π(β¨π, πβ©(compβπ)π)β)) = (((ππΊπ)βπ)(β¨(πΉβπ), (πΉβπ)β©(compβπΈ)(πΉβπ))((ππΊπ)ββ))) |
24 | 1, 2, 3, 4, 5, 6, 7, 8, 12, 15, 18, 20, 21, 22, 23 | isfuncd 17819 | 1 β’ (π β πΉ(π Func πΈ)πΊ) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1539 β wcel 2104 {csn 4627 β¨cop 4633 class class class wbr 5147 β¦ cmpt 5230 I cid 5572 βΎ cres 5677 βcfv 6542 (class class class)co 7411 β cmpo 7413 Οcom 7857 βm cmap 8822 WUnicwun 10697 ndxcnx 17130 Basecbs 17148 Hom chom 17212 compcco 17213 Catccat 17612 Idccid 17613 Func cfunc 17808 SetCatcsetc 18029 ExtStrCatcestrc 18077 |
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-inf2 9638 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-int 4950 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-oadd 8472 df-omul 8473 df-er 8705 df-ec 8707 df-qs 8711 df-map 8824 df-pm 8825 df-ixp 8894 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-wun 10699 df-ni 10869 df-pli 10870 df-mi 10871 df-lti 10872 df-plpq 10905 df-mpq 10906 df-ltpq 10907 df-enq 10908 df-nq 10909 df-erq 10910 df-plq 10911 df-mq 10912 df-1nq 10913 df-rq 10914 df-ltnq 10915 df-np 10978 df-plp 10980 df-ltp 10982 df-enr 11052 df-nr 11053 df-c 11118 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-func 17812 df-setc 18030 df-estrc 18078 |
This theorem is referenced by: fthsetcestrc 18121 fullsetcestrc 18122 |
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