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Mirrors > Home > MPE Home > Th. List > yonedalem1 | Structured version Visualization version GIF version |
Description: Lemma for yoneda 17535. (Contributed by Mario Carneiro, 28-Jan-2017.) |
Ref | Expression |
---|---|
yoneda.y | ⊢ 𝑌 = (Yon‘𝐶) |
yoneda.b | ⊢ 𝐵 = (Base‘𝐶) |
yoneda.1 | ⊢ 1 = (Id‘𝐶) |
yoneda.o | ⊢ 𝑂 = (oppCat‘𝐶) |
yoneda.s | ⊢ 𝑆 = (SetCat‘𝑈) |
yoneda.t | ⊢ 𝑇 = (SetCat‘𝑉) |
yoneda.q | ⊢ 𝑄 = (𝑂 FuncCat 𝑆) |
yoneda.h | ⊢ 𝐻 = (HomF‘𝑄) |
yoneda.r | ⊢ 𝑅 = ((𝑄 ×c 𝑂) FuncCat 𝑇) |
yoneda.e | ⊢ 𝐸 = (𝑂 evalF 𝑆) |
yoneda.z | ⊢ 𝑍 = (𝐻 ∘func ((〈(1st ‘𝑌), tpos (2nd ‘𝑌)〉 ∘func (𝑄 2ndF 𝑂)) 〈,〉F (𝑄 1stF 𝑂))) |
yoneda.c | ⊢ (𝜑 → 𝐶 ∈ Cat) |
yoneda.w | ⊢ (𝜑 → 𝑉 ∈ 𝑊) |
yoneda.u | ⊢ (𝜑 → ran (Homf ‘𝐶) ⊆ 𝑈) |
yoneda.v | ⊢ (𝜑 → (ran (Homf ‘𝑄) ∪ 𝑈) ⊆ 𝑉) |
Ref | Expression |
---|---|
yonedalem1 | ⊢ (𝜑 → (𝑍 ∈ ((𝑄 ×c 𝑂) Func 𝑇) ∧ 𝐸 ∈ ((𝑄 ×c 𝑂) Func 𝑇))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | yoneda.z | . . 3 ⊢ 𝑍 = (𝐻 ∘func ((〈(1st ‘𝑌), tpos (2nd ‘𝑌)〉 ∘func (𝑄 2ndF 𝑂)) 〈,〉F (𝑄 1stF 𝑂))) | |
2 | eqid 2823 | . . . . 5 ⊢ ((〈(1st ‘𝑌), tpos (2nd ‘𝑌)〉 ∘func (𝑄 2ndF 𝑂)) 〈,〉F (𝑄 1stF 𝑂)) = ((〈(1st ‘𝑌), tpos (2nd ‘𝑌)〉 ∘func (𝑄 2ndF 𝑂)) 〈,〉F (𝑄 1stF 𝑂)) | |
3 | eqid 2823 | . . . . 5 ⊢ ((oppCat‘𝑄) ×c 𝑄) = ((oppCat‘𝑄) ×c 𝑄) | |
4 | eqid 2823 | . . . . . . 7 ⊢ (𝑄 ×c 𝑂) = (𝑄 ×c 𝑂) | |
5 | yoneda.q | . . . . . . . 8 ⊢ 𝑄 = (𝑂 FuncCat 𝑆) | |
6 | yoneda.c | . . . . . . . . 9 ⊢ (𝜑 → 𝐶 ∈ Cat) | |
7 | yoneda.o | . . . . . . . . . 10 ⊢ 𝑂 = (oppCat‘𝐶) | |
8 | 7 | oppccat 16994 | . . . . . . . . 9 ⊢ (𝐶 ∈ Cat → 𝑂 ∈ Cat) |
9 | 6, 8 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → 𝑂 ∈ Cat) |
10 | yoneda.w | . . . . . . . . . 10 ⊢ (𝜑 → 𝑉 ∈ 𝑊) | |
11 | yoneda.v | . . . . . . . . . . 11 ⊢ (𝜑 → (ran (Homf ‘𝑄) ∪ 𝑈) ⊆ 𝑉) | |
12 | 11 | unssbd 4166 | . . . . . . . . . 10 ⊢ (𝜑 → 𝑈 ⊆ 𝑉) |
13 | 10, 12 | ssexd 5230 | . . . . . . . . 9 ⊢ (𝜑 → 𝑈 ∈ V) |
14 | yoneda.s | . . . . . . . . . 10 ⊢ 𝑆 = (SetCat‘𝑈) | |
15 | 14 | setccat 17347 | . . . . . . . . 9 ⊢ (𝑈 ∈ V → 𝑆 ∈ Cat) |
16 | 13, 15 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → 𝑆 ∈ Cat) |
17 | 5, 9, 16 | fuccat 17242 | . . . . . . 7 ⊢ (𝜑 → 𝑄 ∈ Cat) |
18 | eqid 2823 | . . . . . . 7 ⊢ (𝑄 2ndF 𝑂) = (𝑄 2ndF 𝑂) | |
19 | 4, 17, 9, 18 | 2ndfcl 17450 | . . . . . 6 ⊢ (𝜑 → (𝑄 2ndF 𝑂) ∈ ((𝑄 ×c 𝑂) Func 𝑂)) |
20 | eqid 2823 | . . . . . . . 8 ⊢ (oppCat‘𝑄) = (oppCat‘𝑄) | |
21 | relfunc 17134 | . . . . . . . . 9 ⊢ Rel (𝐶 Func 𝑄) | |
22 | yoneda.y | . . . . . . . . . 10 ⊢ 𝑌 = (Yon‘𝐶) | |
23 | yoneda.u | . . . . . . . . . 10 ⊢ (𝜑 → ran (Homf ‘𝐶) ⊆ 𝑈) | |
24 | 22, 6, 7, 14, 5, 13, 23 | yoncl 17514 | . . . . . . . . 9 ⊢ (𝜑 → 𝑌 ∈ (𝐶 Func 𝑄)) |
25 | 1st2ndbr 7743 | . . . . . . . . 9 ⊢ ((Rel (𝐶 Func 𝑄) ∧ 𝑌 ∈ (𝐶 Func 𝑄)) → (1st ‘𝑌)(𝐶 Func 𝑄)(2nd ‘𝑌)) | |
26 | 21, 24, 25 | sylancr 589 | . . . . . . . 8 ⊢ (𝜑 → (1st ‘𝑌)(𝐶 Func 𝑄)(2nd ‘𝑌)) |
27 | 7, 20, 26 | funcoppc 17147 | . . . . . . 7 ⊢ (𝜑 → (1st ‘𝑌)(𝑂 Func (oppCat‘𝑄))tpos (2nd ‘𝑌)) |
28 | df-br 5069 | . . . . . . 7 ⊢ ((1st ‘𝑌)(𝑂 Func (oppCat‘𝑄))tpos (2nd ‘𝑌) ↔ 〈(1st ‘𝑌), tpos (2nd ‘𝑌)〉 ∈ (𝑂 Func (oppCat‘𝑄))) | |
29 | 27, 28 | sylib 220 | . . . . . 6 ⊢ (𝜑 → 〈(1st ‘𝑌), tpos (2nd ‘𝑌)〉 ∈ (𝑂 Func (oppCat‘𝑄))) |
30 | 19, 29 | cofucl 17160 | . . . . 5 ⊢ (𝜑 → (〈(1st ‘𝑌), tpos (2nd ‘𝑌)〉 ∘func (𝑄 2ndF 𝑂)) ∈ ((𝑄 ×c 𝑂) Func (oppCat‘𝑄))) |
31 | eqid 2823 | . . . . . 6 ⊢ (𝑄 1stF 𝑂) = (𝑄 1stF 𝑂) | |
32 | 4, 17, 9, 31 | 1stfcl 17449 | . . . . 5 ⊢ (𝜑 → (𝑄 1stF 𝑂) ∈ ((𝑄 ×c 𝑂) Func 𝑄)) |
33 | 2, 3, 30, 32 | prfcl 17455 | . . . 4 ⊢ (𝜑 → ((〈(1st ‘𝑌), tpos (2nd ‘𝑌)〉 ∘func (𝑄 2ndF 𝑂)) 〈,〉F (𝑄 1stF 𝑂)) ∈ ((𝑄 ×c 𝑂) Func ((oppCat‘𝑄) ×c 𝑄))) |
34 | yoneda.h | . . . . 5 ⊢ 𝐻 = (HomF‘𝑄) | |
35 | yoneda.t | . . . . 5 ⊢ 𝑇 = (SetCat‘𝑉) | |
36 | 11 | unssad 4165 | . . . . 5 ⊢ (𝜑 → ran (Homf ‘𝑄) ⊆ 𝑉) |
37 | 34, 20, 35, 17, 10, 36 | hofcl 17511 | . . . 4 ⊢ (𝜑 → 𝐻 ∈ (((oppCat‘𝑄) ×c 𝑄) Func 𝑇)) |
38 | 33, 37 | cofucl 17160 | . . 3 ⊢ (𝜑 → (𝐻 ∘func ((〈(1st ‘𝑌), tpos (2nd ‘𝑌)〉 ∘func (𝑄 2ndF 𝑂)) 〈,〉F (𝑄 1stF 𝑂))) ∈ ((𝑄 ×c 𝑂) Func 𝑇)) |
39 | 1, 38 | eqeltrid 2919 | . 2 ⊢ (𝜑 → 𝑍 ∈ ((𝑄 ×c 𝑂) Func 𝑇)) |
40 | 35, 14, 10, 12 | funcsetcres2 17355 | . . 3 ⊢ (𝜑 → ((𝑄 ×c 𝑂) Func 𝑆) ⊆ ((𝑄 ×c 𝑂) Func 𝑇)) |
41 | yoneda.e | . . . 4 ⊢ 𝐸 = (𝑂 evalF 𝑆) | |
42 | 41, 5, 9, 16 | evlfcl 17474 | . . 3 ⊢ (𝜑 → 𝐸 ∈ ((𝑄 ×c 𝑂) Func 𝑆)) |
43 | 40, 42 | sseldd 3970 | . 2 ⊢ (𝜑 → 𝐸 ∈ ((𝑄 ×c 𝑂) Func 𝑇)) |
44 | 39, 43 | jca 514 | 1 ⊢ (𝜑 → (𝑍 ∈ ((𝑄 ×c 𝑂) Func 𝑇) ∧ 𝐸 ∈ ((𝑄 ×c 𝑂) Func 𝑇))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 Vcvv 3496 ∪ cun 3936 ⊆ wss 3938 〈cop 4575 class class class wbr 5068 ran crn 5558 Rel wrel 5562 ‘cfv 6357 (class class class)co 7158 1st c1st 7689 2nd c2nd 7690 tpos ctpos 7893 Basecbs 16485 Catccat 16937 Idccid 16938 Homf chomf 16939 oppCatcoppc 16983 Func cfunc 17126 ∘func ccofu 17128 FuncCat cfuc 17214 SetCatcsetc 17337 ×c cxpc 17420 1stF c1stf 17421 2ndF c2ndf 17422 〈,〉F cprf 17423 evalF cevlf 17461 HomFchof 17500 Yoncyon 17501 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-tpos 7894 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-oadd 8108 df-er 8291 df-map 8410 df-pm 8411 df-ixp 8464 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-nn 11641 df-2 11703 df-3 11704 df-4 11705 df-5 11706 df-6 11707 df-7 11708 df-8 11709 df-9 11710 df-n0 11901 df-z 11985 df-dec 12102 df-uz 12247 df-fz 12896 df-struct 16487 df-ndx 16488 df-slot 16489 df-base 16491 df-sets 16492 df-ress 16493 df-hom 16591 df-cco 16592 df-cat 16941 df-cid 16942 df-homf 16943 df-comf 16944 df-oppc 16984 df-ssc 17082 df-resc 17083 df-subc 17084 df-func 17130 df-cofu 17132 df-nat 17215 df-fuc 17216 df-setc 17338 df-xpc 17424 df-1stf 17425 df-2ndf 17426 df-prf 17427 df-evlf 17465 df-curf 17466 df-hof 17502 df-yon 17503 |
This theorem is referenced by: yonedalem3b 17531 yonedalem3 17532 yonedainv 17533 yonffthlem 17534 yoneda 17535 |
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