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Mirrors > Home > MPE Home > Th. List > oyoncl | Structured version Visualization version GIF version |
Description: The opposite Yoneda embedding is a functor from oppCat‘𝐶 to the functor category 𝐶 → SetCat. (Contributed by Mario Carneiro, 26-Jan-2017.) |
Ref | Expression |
---|---|
oyoncl.o | ⊢ 𝑂 = (oppCat‘𝐶) |
oyoncl.y | ⊢ 𝑌 = (Yon‘𝑂) |
oyoncl.c | ⊢ (𝜑 → 𝐶 ∈ Cat) |
oyoncl.s | ⊢ 𝑆 = (SetCat‘𝑈) |
oyoncl.u | ⊢ (𝜑 → 𝑈 ∈ 𝑉) |
oyoncl.h | ⊢ (𝜑 → ran (Homf ‘𝐶) ⊆ 𝑈) |
oyoncl.q | ⊢ 𝑄 = (𝐶 FuncCat 𝑆) |
Ref | Expression |
---|---|
oyoncl | ⊢ (𝜑 → 𝑌 ∈ (𝑂 Func 𝑄)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oyoncl.y | . . 3 ⊢ 𝑌 = (Yon‘𝑂) | |
2 | oyoncl.c | . . . 4 ⊢ (𝜑 → 𝐶 ∈ Cat) | |
3 | oyoncl.o | . . . . 5 ⊢ 𝑂 = (oppCat‘𝐶) | |
4 | 3 | oppccat 16734 | . . . 4 ⊢ (𝐶 ∈ Cat → 𝑂 ∈ Cat) |
5 | 2, 4 | syl 17 | . . 3 ⊢ (𝜑 → 𝑂 ∈ Cat) |
6 | eqid 2825 | . . 3 ⊢ (oppCat‘𝑂) = (oppCat‘𝑂) | |
7 | oyoncl.s | . . 3 ⊢ 𝑆 = (SetCat‘𝑈) | |
8 | eqid 2825 | . . 3 ⊢ ((oppCat‘𝑂) FuncCat 𝑆) = ((oppCat‘𝑂) FuncCat 𝑆) | |
9 | oyoncl.u | . . 3 ⊢ (𝜑 → 𝑈 ∈ 𝑉) | |
10 | eqid 2825 | . . . . . . 7 ⊢ (Homf ‘𝐶) = (Homf ‘𝐶) | |
11 | 3, 10 | oppchomf 16732 | . . . . . 6 ⊢ tpos (Homf ‘𝐶) = (Homf ‘𝑂) |
12 | 11 | rneqi 5584 | . . . . 5 ⊢ ran tpos (Homf ‘𝐶) = ran (Homf ‘𝑂) |
13 | relxp 5360 | . . . . . . 7 ⊢ Rel ((Base‘𝐶) × (Base‘𝐶)) | |
14 | eqid 2825 | . . . . . . . . . 10 ⊢ (Base‘𝐶) = (Base‘𝐶) | |
15 | 10, 14 | homffn 16705 | . . . . . . . . 9 ⊢ (Homf ‘𝐶) Fn ((Base‘𝐶) × (Base‘𝐶)) |
16 | fndm 6223 | . . . . . . . . 9 ⊢ ((Homf ‘𝐶) Fn ((Base‘𝐶) × (Base‘𝐶)) → dom (Homf ‘𝐶) = ((Base‘𝐶) × (Base‘𝐶))) | |
17 | 15, 16 | ax-mp 5 | . . . . . . . 8 ⊢ dom (Homf ‘𝐶) = ((Base‘𝐶) × (Base‘𝐶)) |
18 | 17 | releqi 5437 | . . . . . . 7 ⊢ (Rel dom (Homf ‘𝐶) ↔ Rel ((Base‘𝐶) × (Base‘𝐶))) |
19 | 13, 18 | mpbir 223 | . . . . . 6 ⊢ Rel dom (Homf ‘𝐶) |
20 | rntpos 7630 | . . . . . 6 ⊢ (Rel dom (Homf ‘𝐶) → ran tpos (Homf ‘𝐶) = ran (Homf ‘𝐶)) | |
21 | 19, 20 | ax-mp 5 | . . . . 5 ⊢ ran tpos (Homf ‘𝐶) = ran (Homf ‘𝐶) |
22 | 12, 21 | eqtr3i 2851 | . . . 4 ⊢ ran (Homf ‘𝑂) = ran (Homf ‘𝐶) |
23 | oyoncl.h | . . . 4 ⊢ (𝜑 → ran (Homf ‘𝐶) ⊆ 𝑈) | |
24 | 22, 23 | syl5eqss 3874 | . . 3 ⊢ (𝜑 → ran (Homf ‘𝑂) ⊆ 𝑈) |
25 | 1, 5, 6, 7, 8, 9, 24 | yoncl 17255 | . 2 ⊢ (𝜑 → 𝑌 ∈ (𝑂 Func ((oppCat‘𝑂) FuncCat 𝑆))) |
26 | oyoncl.q | . . . 4 ⊢ 𝑄 = (𝐶 FuncCat 𝑆) | |
27 | 3 | 2oppchomf 16736 | . . . . . 6 ⊢ (Homf ‘𝐶) = (Homf ‘(oppCat‘𝑂)) |
28 | 27 | a1i 11 | . . . . 5 ⊢ (𝜑 → (Homf ‘𝐶) = (Homf ‘(oppCat‘𝑂))) |
29 | 3 | 2oppccomf 16737 | . . . . . 6 ⊢ (compf‘𝐶) = (compf‘(oppCat‘𝑂)) |
30 | 29 | a1i 11 | . . . . 5 ⊢ (𝜑 → (compf‘𝐶) = (compf‘(oppCat‘𝑂))) |
31 | eqidd 2826 | . . . . 5 ⊢ (𝜑 → (Homf ‘𝑆) = (Homf ‘𝑆)) | |
32 | eqidd 2826 | . . . . 5 ⊢ (𝜑 → (compf‘𝑆) = (compf‘𝑆)) | |
33 | 6 | oppccat 16734 | . . . . . 6 ⊢ (𝑂 ∈ Cat → (oppCat‘𝑂) ∈ Cat) |
34 | 5, 33 | syl 17 | . . . . 5 ⊢ (𝜑 → (oppCat‘𝑂) ∈ Cat) |
35 | 7 | setccat 17087 | . . . . . 6 ⊢ (𝑈 ∈ 𝑉 → 𝑆 ∈ Cat) |
36 | 9, 35 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝑆 ∈ Cat) |
37 | 28, 30, 31, 32, 2, 34, 36, 36 | fucpropd 16989 | . . . 4 ⊢ (𝜑 → (𝐶 FuncCat 𝑆) = ((oppCat‘𝑂) FuncCat 𝑆)) |
38 | 26, 37 | syl5eq 2873 | . . 3 ⊢ (𝜑 → 𝑄 = ((oppCat‘𝑂) FuncCat 𝑆)) |
39 | 38 | oveq2d 6921 | . 2 ⊢ (𝜑 → (𝑂 Func 𝑄) = (𝑂 Func ((oppCat‘𝑂) FuncCat 𝑆))) |
40 | 25, 39 | eleqtrrd 2909 | 1 ⊢ (𝜑 → 𝑌 ∈ (𝑂 Func 𝑄)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1656 ∈ wcel 2164 ⊆ wss 3798 × cxp 5340 dom cdm 5342 ran crn 5343 Rel wrel 5347 Fn wfn 6118 ‘cfv 6123 (class class class)co 6905 tpos ctpos 7616 Basecbs 16222 Catccat 16677 Homf chomf 16679 compfccomf 16680 oppCatcoppc 16723 Func cfunc 16866 FuncCat cfuc 16954 SetCatcsetc 17077 Yoncyon 17242 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-rep 4994 ax-sep 5005 ax-nul 5013 ax-pow 5065 ax-pr 5127 ax-un 7209 ax-cnex 10308 ax-resscn 10309 ax-1cn 10310 ax-icn 10311 ax-addcl 10312 ax-addrcl 10313 ax-mulcl 10314 ax-mulrcl 10315 ax-mulcom 10316 ax-addass 10317 ax-mulass 10318 ax-distr 10319 ax-i2m1 10320 ax-1ne0 10321 ax-1rid 10322 ax-rnegex 10323 ax-rrecex 10324 ax-cnre 10325 ax-pre-lttri 10326 ax-pre-lttrn 10327 ax-pre-ltadd 10328 ax-pre-mulgt0 10329 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3or 1112 df-3an 1113 df-tru 1660 df-fal 1670 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-ral 3122 df-rex 3123 df-reu 3124 df-rmo 3125 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-pss 3814 df-nul 4145 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-tp 4402 df-op 4404 df-uni 4659 df-int 4698 df-iun 4742 df-br 4874 df-opab 4936 df-mpt 4953 df-tr 4976 df-id 5250 df-eprel 5255 df-po 5263 df-so 5264 df-fr 5301 df-we 5303 df-xp 5348 df-rel 5349 df-cnv 5350 df-co 5351 df-dm 5352 df-rn 5353 df-res 5354 df-ima 5355 df-pred 5920 df-ord 5966 df-on 5967 df-lim 5968 df-suc 5969 df-iota 6086 df-fun 6125 df-fn 6126 df-f 6127 df-f1 6128 df-fo 6129 df-f1o 6130 df-fv 6131 df-riota 6866 df-ov 6908 df-oprab 6909 df-mpt2 6910 df-om 7327 df-1st 7428 df-2nd 7429 df-tpos 7617 df-wrecs 7672 df-recs 7734 df-rdg 7772 df-1o 7826 df-oadd 7830 df-er 8009 df-map 8124 df-ixp 8176 df-en 8223 df-dom 8224 df-sdom 8225 df-fin 8226 df-pnf 10393 df-mnf 10394 df-xr 10395 df-ltxr 10396 df-le 10397 df-sub 10587 df-neg 10588 df-nn 11351 df-2 11414 df-3 11415 df-4 11416 df-5 11417 df-6 11418 df-7 11419 df-8 11420 df-9 11421 df-n0 11619 df-z 11705 df-dec 11822 df-uz 11969 df-fz 12620 df-struct 16224 df-ndx 16225 df-slot 16226 df-base 16228 df-sets 16229 df-hom 16329 df-cco 16330 df-cat 16681 df-cid 16682 df-homf 16683 df-comf 16684 df-oppc 16724 df-func 16870 df-nat 16955 df-fuc 16956 df-setc 17078 df-xpc 17165 df-curf 17207 df-hof 17243 df-yon 17244 |
This theorem is referenced by: oyon1cl 17264 |
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