Theorem yonedalem1 18292

Description: Lemma for yoneda 18303. (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 2726	. . . . 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 2726	. . . . 5 ⊢ ((oppCat‘𝑄) ×c 𝑄) = ((oppCat‘𝑄) ×c 𝑄)
4		eqid 2726	. . . . . . 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 17732	. . . . . . . . 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 4186	. . . . . . . . . 10 ⊢ (𝜑 → 𝑈 ⊆ 𝑉)
13	10, 12	ssexd 5321	. . . . . . . . 9 ⊢ (𝜑 → 𝑈 ∈ V)
14		yoneda.s	. . . . . . . . . 10 ⊢ 𝑆 = (SetCat‘𝑈)
15	14	setccat 18102	. . . . . . . . 9 ⊢ (𝑈 ∈ V → 𝑆 ∈ Cat)
16	13, 15	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝑆 ∈ Cat)
17	5, 9, 16	fuccat 17990	. . . . . . 7 ⊢ (𝜑 → 𝑄 ∈ Cat)
18		eqid 2726	. . . . . . 7 ⊢ (𝑄 2ndF 𝑂) = (𝑄 2ndF 𝑂)
19	4, 17, 9, 18	2ndfcl 18217	. . . . . 6 ⊢ (𝜑 → (𝑄 2ndF 𝑂) ∈ ((𝑄 ×c 𝑂) Func 𝑂))
20		eqid 2726	. . . . . . . 8 ⊢ (oppCat‘𝑄) = (oppCat‘𝑄)
21		relfunc 17876	. . . . . . . . 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 18282	. . . . . . . . 9 ⊢ (𝜑 → 𝑌 ∈ (𝐶 Func 𝑄))
25		1st2ndbr 8048	. . . . . . . . 9 ⊢ ((Rel (𝐶 Func 𝑄) ∧ 𝑌 ∈ (𝐶 Func 𝑄)) → (1st ‘𝑌)(𝐶 Func 𝑄)(2nd ‘𝑌))
26	21, 24, 25	sylancr 585	. . . . . . . 8 ⊢ (𝜑 → (1st ‘𝑌)(𝐶 Func 𝑄)(2nd ‘𝑌))
27	7, 20, 26	funcoppc 17889	. . . . . . 7 ⊢ (𝜑 → (1st ‘𝑌)(𝑂 Func (oppCat‘𝑄))tpos (2nd ‘𝑌))
28		df-br 5146	. . . . . . 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 17902	. . . . 5 ⊢ (𝜑 → (⟨(1st ‘𝑌), tpos (2nd ‘𝑌)⟩ ∘func (𝑄 2ndF 𝑂)) ∈ ((𝑄 ×c 𝑂) Func (oppCat‘𝑄)))
31		eqid 2726	. . . . . 6 ⊢ (𝑄 1stF 𝑂) = (𝑄 1stF 𝑂)
32	4, 17, 9, 31	1stfcl 18216	. . . . 5 ⊢ (𝜑 → (𝑄 1stF 𝑂) ∈ ((𝑄 ×c 𝑂) Func 𝑄))
33	2, 3, 30, 32	prfcl 18222	. . . 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 4185	. . . . 5 ⊢ (𝜑 → ran (Homf ‘𝑄) ⊆ 𝑉)
37	34, 20, 35, 17, 10, 36	hofcl 18279	. . . 4 ⊢ (𝜑 → 𝐻 ∈ (((oppCat‘𝑄) ×c 𝑄) Func 𝑇))
38	33, 37	cofucl 17902	. . 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 2830	. 2 ⊢ (𝜑 → 𝑍 ∈ ((𝑄 ×c 𝑂) Func 𝑇))
40	35, 14, 10, 12	funcsetcres2 18110	. . 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 18242	. . 3 ⊢ (𝜑 → 𝐸 ∈ ((𝑄 ×c 𝑂) Func 𝑆))
43	40, 42	sseldd 3979	. 2 ⊢ (𝜑 → 𝐸 ∈ ((𝑄 ×c 𝑂) Func 𝑇))
44	39, 43	jca 510	1 ⊢ (𝜑 → (𝑍 ∈ ((𝑄 ×c 𝑂) Func 𝑇) ∧ 𝐸 ∈ ((𝑄 ×c 𝑂) Func 𝑇)))

Syntax hints:   → wi 4   ∧ wa 394   = wceq 1534   ∈ wcel 2099  Vcvv 3462   ∪ cun 3944   ⊆ wss 3946  ⟨cop 4629   class class class wbr 5145  ran crn 5675  Rel wrel 5679  ‘cfv 6546  (class class class)co 7416  1^st c1st 7993  2^nd c2nd 7994  tpos ctpos 8232  Basecbs 17208  Catccat 17672  Idccid 17673  Hom_f chomf 17674  oppCatcoppc 17719   Func cfunc 17868   ∘_func ccofu 17870   FuncCat cfuc 17960  SetCatcsetc 18092   ×_c cxpc 18187   1^st_F c1stf 18188   2^nd_F c2ndf 18189   ⟨,⟩_F cprf 18190   eval_F cevlf 18229  Hom_Fchof 18268  Yoncyon 18269

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-rep 5282  ax-sep 5296  ax-nul 5303  ax-pow 5361  ax-pr 5425  ax-un 7738  ax-cnex 11205  ax-resscn 11206  ax-1cn 11207  ax-icn 11208  ax-addcl 11209  ax-addrcl 11210  ax-mulcl 11211  ax-mulrcl 11212  ax-mulcom 11213  ax-addass 11214  ax-mulass 11215  ax-distr 11216  ax-i2m1 11217  ax-1ne0 11218  ax-1rid 11219  ax-rnegex 11220  ax-rrecex 11221  ax-cnre 11222  ax-pre-lttri 11223  ax-pre-lttrn 11224  ax-pre-ltadd 11225  ax-pre-mulgt0 11226

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3364  df-reu 3365  df-rab 3420  df-v 3464  df-sbc 3776  df-csb 3892  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-pss 3966  df-nul 4323  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-tp 4628  df-op 4630  df-uni 4906  df-iun 4995  df-br 5146  df-opab 5208  df-mpt 5229  df-tr 5263  df-id 5572  df-eprel 5578  df-po 5586  df-so 5587  df-fr 5629  df-we 5631  df-xp 5680  df-rel 5681  df-cnv 5682  df-co 5683  df-dm 5684  df-rn 5685  df-res 5686  df-ima 5687  df-pred 6304  df-ord 6371  df-on 6372  df-lim 6373  df-suc 6374  df-iota 6498  df-fun 6548  df-fn 6549  df-f 6550  df-f1 6551  df-fo 6552  df-f1o 6553  df-fv 6554  df-riota 7372  df-ov 7419  df-oprab 7420  df-mpo 7421  df-om 7869  df-1st 7995  df-2nd 7996  df-tpos 8233  df-frecs 8288  df-wrecs 8319  df-recs 8393  df-rdg 8432  df-1o 8488  df-er 8726  df-map 8849  df-pm 8850  df-ixp 8919  df-en 8967  df-dom 8968  df-sdom 8969  df-fin 8970  df-pnf 11291  df-mnf 11292  df-xr 11293  df-ltxr 11294  df-le 11295  df-sub 11487  df-neg 11488  df-nn 12259  df-2 12321  df-3 12322  df-4 12323  df-5 12324  df-6 12325  df-7 12326  df-8 12327  df-9 12328  df-n0 12519  df-z 12605  df-dec 12724  df-uz 12869  df-fz 13533  df-struct 17144  df-sets 17161  df-slot 17179  df-ndx 17191  df-base 17209  df-ress 17238  df-hom 17285  df-cco 17286  df-cat 17676  df-cid 17677  df-homf 17678  df-comf 17679  df-oppc 17720  df-ssc 17821  df-resc 17822  df-subc 17823  df-func 17872  df-cofu 17874  df-nat 17961  df-fuc 17962  df-setc 18093  df-xpc 18191  df-1stf 18192  df-2ndf 18193  df-prf 18194  df-evlf 18233  df-curf 18234  df-hof 18270  df-yon 18271

This theorem is referenced by:  yonedalem3b  18299  yonedalem3  18300  yonedainv  18301  yonffthlem  18302  yoneda  18303