Theorem yonedalem1 18317

Description: Lemma for yoneda 18328. (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 2737	. . . . 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 2737	. . . . 5 ⊢ ((oppCat‘𝑄) ×c 𝑄) = ((oppCat‘𝑄) ×c 𝑄)
4		eqid 2737	. . . . . . 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 17765	. . . . . . . . 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 4194	. . . . . . . . . 10 ⊢ (𝜑 → 𝑈 ⊆ 𝑉)
13	10, 12	ssexd 5324	. . . . . . . . 9 ⊢ (𝜑 → 𝑈 ∈ V)
14		yoneda.s	. . . . . . . . . 10 ⊢ 𝑆 = (SetCat‘𝑈)
15	14	setccat 18130	. . . . . . . . 9 ⊢ (𝑈 ∈ V → 𝑆 ∈ Cat)
16	13, 15	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝑆 ∈ Cat)
17	5, 9, 16	fuccat 18018	. . . . . . 7 ⊢ (𝜑 → 𝑄 ∈ Cat)
18		eqid 2737	. . . . . . 7 ⊢ (𝑄 2ndF 𝑂) = (𝑄 2ndF 𝑂)
19	4, 17, 9, 18	2ndfcl 18243	. . . . . 6 ⊢ (𝜑 → (𝑄 2ndF 𝑂) ∈ ((𝑄 ×c 𝑂) Func 𝑂))
20		eqid 2737	. . . . . . . 8 ⊢ (oppCat‘𝑄) = (oppCat‘𝑄)
21		relfunc 17907	. . . . . . . . 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 18307	. . . . . . . . 9 ⊢ (𝜑 → 𝑌 ∈ (𝐶 Func 𝑄))
25		1st2ndbr 8067	. . . . . . . . 9 ⊢ ((Rel (𝐶 Func 𝑄) ∧ 𝑌 ∈ (𝐶 Func 𝑄)) → (1st ‘𝑌)(𝐶 Func 𝑄)(2nd ‘𝑌))
26	21, 24, 25	sylancr 587	. . . . . . . 8 ⊢ (𝜑 → (1st ‘𝑌)(𝐶 Func 𝑄)(2nd ‘𝑌))
27	7, 20, 26	funcoppc 17920	. . . . . . 7 ⊢ (𝜑 → (1st ‘𝑌)(𝑂 Func (oppCat‘𝑄))tpos (2nd ‘𝑌))
28		df-br 5144	. . . . . . 7 ⊢ ((1st ‘𝑌)(𝑂 Func (oppCat‘𝑄))tpos (2nd ‘𝑌) ↔ ⟨(1st ‘𝑌), tpos (2nd ‘𝑌)⟩ ∈ (𝑂 Func (oppCat‘𝑄)))
29	27, 28	sylib 218	. . . . . 6 ⊢ (𝜑 → ⟨(1st ‘𝑌), tpos (2nd ‘𝑌)⟩ ∈ (𝑂 Func (oppCat‘𝑄)))
30	19, 29	cofucl 17933	. . . . 5 ⊢ (𝜑 → (⟨(1st ‘𝑌), tpos (2nd ‘𝑌)⟩ ∘func (𝑄 2ndF 𝑂)) ∈ ((𝑄 ×c 𝑂) Func (oppCat‘𝑄)))
31		eqid 2737	. . . . . 6 ⊢ (𝑄 1stF 𝑂) = (𝑄 1stF 𝑂)
32	4, 17, 9, 31	1stfcl 18242	. . . . 5 ⊢ (𝜑 → (𝑄 1stF 𝑂) ∈ ((𝑄 ×c 𝑂) Func 𝑄))
33	2, 3, 30, 32	prfcl 18248	. . . 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 4193	. . . . 5 ⊢ (𝜑 → ran (Homf ‘𝑄) ⊆ 𝑉)
37	34, 20, 35, 17, 10, 36	hofcl 18304	. . . 4 ⊢ (𝜑 → 𝐻 ∈ (((oppCat‘𝑄) ×c 𝑄) Func 𝑇))
38	33, 37	cofucl 17933	. . 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 2845	. 2 ⊢ (𝜑 → 𝑍 ∈ ((𝑄 ×c 𝑂) Func 𝑇))
40	35, 14, 10, 12	funcsetcres2 18138	. . 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 18267	. . 3 ⊢ (𝜑 → 𝐸 ∈ ((𝑄 ×c 𝑂) Func 𝑆))
43	40, 42	sseldd 3984	. 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 1540   ∈ wcel 2108  Vcvv 3480   ∪ cun 3949   ⊆ wss 3951  ⟨cop 4632   class class class wbr 5143  ran crn 5686  Rel wrel 5690  ‘cfv 6561  (class class class)co 7431  1^st c1st 8012  2^nd c2nd 8013  tpos ctpos 8250  Basecbs 17247  Catccat 17707  Idccid 17708  Hom_f chomf 17709  oppCatcoppc 17754   Func cfunc 17899   ∘_func ccofu 17901   FuncCat cfuc 17990  SetCatcsetc 18120   ×_c cxpc 18213   1^st_F c1stf 18214   2^nd_F c2ndf 18215   ⟨,⟩_F cprf 18216   eval_F cevlf 18254  Hom_Fchof 18293  Yoncyon 18294

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 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-tp 4631  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-tpos 8251  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-er 8745  df-map 8868  df-pm 8869  df-ixp 8938  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-nn 12267  df-2 12329  df-3 12330  df-4 12331  df-5 12332  df-6 12333  df-7 12334  df-8 12335  df-9 12336  df-n0 12527  df-z 12614  df-dec 12734  df-uz 12879  df-fz 13548  df-struct 17184  df-sets 17201  df-slot 17219  df-ndx 17231  df-base 17248  df-ress 17275  df-hom 17321  df-cco 17322  df-cat 17711  df-cid 17712  df-homf 17713  df-comf 17714  df-oppc 17755  df-ssc 17854  df-resc 17855  df-subc 17856  df-func 17903  df-cofu 17905  df-nat 17991  df-fuc 17992  df-setc 18121  df-xpc 18217  df-1stf 18218  df-2ndf 18219  df-prf 18220  df-evlf 18258  df-curf 18259  df-hof 18295  df-yon 18296

This theorem is referenced by:  yonedalem3b  18324  yonedalem3  18325  yonedainv  18326  yonffthlem  18327  yoneda  18328