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Mirrors > Home > MPE Home > Th. List > yonedalem1 | Structured version Visualization version GIF version |
Description: Lemma for yoneda 17536. (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 2824 | . . . . 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 2824 | . . . . 5 ⊢ ((oppCat‘𝑄) ×c 𝑄) = ((oppCat‘𝑄) ×c 𝑄) | |
4 | eqid 2824 | . . . . . . 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 16995 | . . . . . . . . 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 4167 | . . . . . . . . . 10 ⊢ (𝜑 → 𝑈 ⊆ 𝑉) |
13 | 10, 12 | ssexd 5231 | . . . . . . . . 9 ⊢ (𝜑 → 𝑈 ∈ V) |
14 | yoneda.s | . . . . . . . . . 10 ⊢ 𝑆 = (SetCat‘𝑈) | |
15 | 14 | setccat 17348 | . . . . . . . . 9 ⊢ (𝑈 ∈ V → 𝑆 ∈ Cat) |
16 | 13, 15 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → 𝑆 ∈ Cat) |
17 | 5, 9, 16 | fuccat 17243 | . . . . . . 7 ⊢ (𝜑 → 𝑄 ∈ Cat) |
18 | eqid 2824 | . . . . . . 7 ⊢ (𝑄 2ndF 𝑂) = (𝑄 2ndF 𝑂) | |
19 | 4, 17, 9, 18 | 2ndfcl 17451 | . . . . . 6 ⊢ (𝜑 → (𝑄 2ndF 𝑂) ∈ ((𝑄 ×c 𝑂) Func 𝑂)) |
20 | eqid 2824 | . . . . . . . 8 ⊢ (oppCat‘𝑄) = (oppCat‘𝑄) | |
21 | relfunc 17135 | . . . . . . . . 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 17515 | . . . . . . . . 9 ⊢ (𝜑 → 𝑌 ∈ (𝐶 Func 𝑄)) |
25 | 1st2ndbr 7744 | . . . . . . . . 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 17148 | . . . . . . 7 ⊢ (𝜑 → (1st ‘𝑌)(𝑂 Func (oppCat‘𝑄))tpos (2nd ‘𝑌)) |
28 | df-br 5070 | . . . . . . 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 17161 | . . . . 5 ⊢ (𝜑 → (〈(1st ‘𝑌), tpos (2nd ‘𝑌)〉 ∘func (𝑄 2ndF 𝑂)) ∈ ((𝑄 ×c 𝑂) Func (oppCat‘𝑄))) |
31 | eqid 2824 | . . . . . 6 ⊢ (𝑄 1stF 𝑂) = (𝑄 1stF 𝑂) | |
32 | 4, 17, 9, 31 | 1stfcl 17450 | . . . . 5 ⊢ (𝜑 → (𝑄 1stF 𝑂) ∈ ((𝑄 ×c 𝑂) Func 𝑄)) |
33 | 2, 3, 30, 32 | prfcl 17456 | . . . 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 4166 | . . . . 5 ⊢ (𝜑 → ran (Homf ‘𝑄) ⊆ 𝑉) |
37 | 34, 20, 35, 17, 10, 36 | hofcl 17512 | . . . 4 ⊢ (𝜑 → 𝐻 ∈ (((oppCat‘𝑄) ×c 𝑄) Func 𝑇)) |
38 | 33, 37 | cofucl 17161 | . . 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 2920 | . 2 ⊢ (𝜑 → 𝑍 ∈ ((𝑄 ×c 𝑂) Func 𝑇)) |
40 | 35, 14, 10, 12 | funcsetcres2 17356 | . . 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 17475 | . . 3 ⊢ (𝜑 → 𝐸 ∈ ((𝑄 ×c 𝑂) Func 𝑆)) |
43 | 40, 42 | sseldd 3971 | . 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 1536 ∈ wcel 2113 Vcvv 3497 ∪ cun 3937 ⊆ wss 3939 〈cop 4576 class class class wbr 5069 ran crn 5559 Rel wrel 5563 ‘cfv 6358 (class class class)co 7159 1st c1st 7690 2nd c2nd 7691 tpos ctpos 7894 Basecbs 16486 Catccat 16938 Idccid 16939 Homf chomf 16940 oppCatcoppc 16984 Func cfunc 17127 ∘func ccofu 17129 FuncCat cfuc 17215 SetCatcsetc 17338 ×c cxpc 17421 1stF c1stf 17422 2ndF c2ndf 17423 〈,〉F cprf 17424 evalF cevlf 17462 HomFchof 17501 Yoncyon 17502 |
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 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-rep 5193 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 ax-cnex 10596 ax-resscn 10597 ax-1cn 10598 ax-icn 10599 ax-addcl 10600 ax-addrcl 10601 ax-mulcl 10602 ax-mulrcl 10603 ax-mulcom 10604 ax-addass 10605 ax-mulass 10606 ax-distr 10607 ax-i2m1 10608 ax-1ne0 10609 ax-1rid 10610 ax-rnegex 10611 ax-rrecex 10612 ax-cnre 10613 ax-pre-lttri 10614 ax-pre-lttrn 10615 ax-pre-ltadd 10616 ax-pre-mulgt0 10617 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1539 df-fal 1549 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-nel 3127 df-ral 3146 df-rex 3147 df-reu 3148 df-rmo 3149 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-pss 3957 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-tp 4575 df-op 4577 df-uni 4842 df-int 4880 df-iun 4924 df-br 5070 df-opab 5132 df-mpt 5150 df-tr 5176 df-id 5463 df-eprel 5468 df-po 5477 df-so 5478 df-fr 5517 df-we 5519 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-pred 6151 df-ord 6197 df-on 6198 df-lim 6199 df-suc 6200 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7117 df-ov 7162 df-oprab 7163 df-mpo 7164 df-om 7584 df-1st 7692 df-2nd 7693 df-tpos 7895 df-wrecs 7950 df-recs 8011 df-rdg 8049 df-1o 8105 df-oadd 8109 df-er 8292 df-map 8411 df-pm 8412 df-ixp 8465 df-en 8513 df-dom 8514 df-sdom 8515 df-fin 8516 df-pnf 10680 df-mnf 10681 df-xr 10682 df-ltxr 10683 df-le 10684 df-sub 10875 df-neg 10876 df-nn 11642 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 16488 df-ndx 16489 df-slot 16490 df-base 16492 df-sets 16493 df-ress 16494 df-hom 16592 df-cco 16593 df-cat 16942 df-cid 16943 df-homf 16944 df-comf 16945 df-oppc 16985 df-ssc 17083 df-resc 17084 df-subc 17085 df-func 17131 df-cofu 17133 df-nat 17216 df-fuc 17217 df-setc 17339 df-xpc 17425 df-1stf 17426 df-2ndf 17427 df-prf 17428 df-evlf 17466 df-curf 17467 df-hof 17503 df-yon 17504 |
This theorem is referenced by: yonedalem3b 17532 yonedalem3 17533 yonedainv 17534 yonffthlem 17535 yoneda 17536 |
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