Theorem yonedalem1 17906

Description: Lemma for yoneda 17917. (Contributed by Mario Carneiro, 28-Jan-2017.)

Hypotheses

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)

Assertion

Ref	Expression
yonedalem1	⊢ (𝜑 → (𝑍 ∈ ((𝑄 ×c 𝑂) Func 𝑇) ∧ 𝐸 ∈ ((𝑄 ×c 𝑂) Func 𝑇)))

Proof of Theorem yonedalem1

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 2738	. . . . 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 2738	. . . . 5 ⊢ ((oppCat‘𝑄) ×c 𝑄) = ((oppCat‘𝑄) ×c 𝑄)
4		eqid 2738	. . . . . . 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 17350	. . . . . . . . 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 4118	. . . . . . . . . 10 ⊢ (𝜑 → 𝑈 ⊆ 𝑉)
13	10, 12	ssexd 5243	. . . . . . . . 9 ⊢ (𝜑 → 𝑈 ∈ V)
14		yoneda.s	. . . . . . . . . 10 ⊢ 𝑆 = (SetCat‘𝑈)
15	14	setccat 17716	. . . . . . . . 9 ⊢ (𝑈 ∈ V → 𝑆 ∈ Cat)
16	13, 15	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝑆 ∈ Cat)
17	5, 9, 16	fuccat 17604	. . . . . . 7 ⊢ (𝜑 → 𝑄 ∈ Cat)
18		eqid 2738	. . . . . . 7 ⊢ (𝑄 2ndF 𝑂) = (𝑄 2ndF 𝑂)
19	4, 17, 9, 18	2ndfcl 17831	. . . . . 6 ⊢ (𝜑 → (𝑄 2ndF 𝑂) ∈ ((𝑄 ×c 𝑂) Func 𝑂))
20		eqid 2738	. . . . . . . 8 ⊢ (oppCat‘𝑄) = (oppCat‘𝑄)
21		relfunc 17493	. . . . . . . . 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 17896	. . . . . . . . 9 ⊢ (𝜑 → 𝑌 ∈ (𝐶 Func 𝑄))
25		1st2ndbr 7856	. . . . . . . . 9 ⊢ ((Rel (𝐶 Func 𝑄) ∧ 𝑌 ∈ (𝐶 Func 𝑄)) → (1st ‘𝑌)(𝐶 Func 𝑄)(2nd ‘𝑌))
26	21, 24, 25	sylancr 586	. . . . . . . 8 ⊢ (𝜑 → (1st ‘𝑌)(𝐶 Func 𝑄)(2nd ‘𝑌))
27	7, 20, 26	funcoppc 17506	. . . . . . 7 ⊢ (𝜑 → (1st ‘𝑌)(𝑂 Func (oppCat‘𝑄))tpos (2nd ‘𝑌))
28		df-br 5071	. . . . . . 7 ⊢ ((1st ‘𝑌)(𝑂 Func (oppCat‘𝑄))tpos (2nd ‘𝑌) ↔ ⟨(1st ‘𝑌), tpos (2nd ‘𝑌)⟩ ∈ (𝑂 Func (oppCat‘𝑄)))
29	27, 28	sylib 217	. . . . . 6 ⊢ (𝜑 → ⟨(1st ‘𝑌), tpos (2nd ‘𝑌)⟩ ∈ (𝑂 Func (oppCat‘𝑄)))
30	19, 29	cofucl 17519	. . . . 5 ⊢ (𝜑 → (⟨(1st ‘𝑌), tpos (2nd ‘𝑌)⟩ ∘func (𝑄 2ndF 𝑂)) ∈ ((𝑄 ×c 𝑂) Func (oppCat‘𝑄)))
31		eqid 2738	. . . . . 6 ⊢ (𝑄 1stF 𝑂) = (𝑄 1stF 𝑂)
32	4, 17, 9, 31	1stfcl 17830	. . . . 5 ⊢ (𝜑 → (𝑄 1stF 𝑂) ∈ ((𝑄 ×c 𝑂) Func 𝑄))
33	2, 3, 30, 32	prfcl 17836	. . . 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 4117	. . . . 5 ⊢ (𝜑 → ran (Homf ‘𝑄) ⊆ 𝑉)
37	34, 20, 35, 17, 10, 36	hofcl 17893	. . . 4 ⊢ (𝜑 → 𝐻 ∈ (((oppCat‘𝑄) ×c 𝑄) Func 𝑇))
38	33, 37	cofucl 17519	. . 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 2843	. 2 ⊢ (𝜑 → 𝑍 ∈ ((𝑄 ×c 𝑂) Func 𝑇))
40	35, 14, 10, 12	funcsetcres2 17724	. . 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 17856	. . 3 ⊢ (𝜑 → 𝐸 ∈ ((𝑄 ×c 𝑂) Func 𝑆))
43	40, 42	sseldd 3918	. 2 ⊢ (𝜑 → 𝐸 ∈ ((𝑄 ×c 𝑂) Func 𝑇))
44	39, 43	jca 511	1 ⊢ (𝜑 → (𝑍 ∈ ((𝑄 ×c 𝑂) Func 𝑇) ∧ 𝐸 ∈ ((𝑄 ×c 𝑂) Func 𝑇)))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1539   ∈ wcel 2108  Vcvv 3422   ∪ cun 3881   ⊆ wss 3883  ⟨cop 4564   class class class wbr 5070  ran crn 5581  Rel wrel 5585  ‘cfv 6418  (class class class)co 7255  1^st c1st 7802  2^nd c2nd 7803  tpos ctpos 8012  Basecbs 16840  Catccat 17290  Idccid 17291  Hom_f chomf 17292  oppCatcoppc 17337   Func cfunc 17485   ∘_func ccofu 17487   FuncCat cfuc 17574  SetCatcsetc 17706   ×_c cxpc 17801   1^st_F c1stf 17802   2^nd_F c2ndf 17803   ⟨,⟩_F cprf 17804   eval_F cevlf 17843  Hom_Fchof 17882  Yoncyon 17883

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566  ax-cnex 10858  ax-resscn 10859  ax-1cn 10860  ax-icn 10861  ax-addcl 10862  ax-addrcl 10863  ax-mulcl 10864  ax-mulrcl 10865  ax-mulcom 10866  ax-addass 10867  ax-mulass 10868  ax-distr 10869  ax-i2m1 10870  ax-1ne0 10871  ax-1rid 10872  ax-rnegex 10873  ax-rrecex 10874  ax-cnre 10875  ax-pre-lttri 10876  ax-pre-lttrn 10877  ax-pre-ltadd 10878  ax-pre-mulgt0 10879

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-nel 3049  df-ral 3068  df-rex 3069  df-reu 3070  df-rmo 3071  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-pred 6191  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-riota 7212  df-ov 7258  df-oprab 7259  df-mpo 7260  df-om 7688  df-1st 7804  df-2nd 7805  df-tpos 8013  df-frecs 8068  df-wrecs 8099  df-recs 8173  df-rdg 8212  df-1o 8267  df-er 8456  df-map 8575  df-pm 8576  df-ixp 8644  df-en 8692  df-dom 8693  df-sdom 8694  df-fin 8695  df-pnf 10942  df-mnf 10943  df-xr 10944  df-ltxr 10945  df-le 10946  df-sub 11137  df-neg 11138  df-nn 11904  df-2 11966  df-3 11967  df-4 11968  df-5 11969  df-6 11970  df-7 11971  df-8 11972  df-9 11973  df-n0 12164  df-z 12250  df-dec 12367  df-uz 12512  df-fz 13169  df-struct 16776  df-sets 16793  df-slot 16811  df-ndx 16823  df-base 16841  df-ress 16868  df-hom 16912  df-cco 16913  df-cat 17294  df-cid 17295  df-homf 17296  df-comf 17297  df-oppc 17338  df-ssc 17439  df-resc 17440  df-subc 17441  df-func 17489  df-cofu 17491  df-nat 17575  df-fuc 17576  df-setc 17707  df-xpc 17805  df-1stf 17806  df-2ndf 17807  df-prf 17808  df-evlf 17847  df-curf 17848  df-hof 17884  df-yon 17885

This theorem is referenced by:  yonedalem3b  17913  yonedalem3  17914  yonedainv  17915  yonffthlem  17916  yoneda  17917