Theorem yonedalem1 18207

Description: Lemma for yoneda 18218. (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 17657	. . . . . . . . 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 4148	. . . . . . . . . 10 ⊢ (𝜑 → 𝑈 ⊆ 𝑉)
13	10, 12	ssexd 5271	. . . . . . . . 9 ⊢ (𝜑 → 𝑈 ∈ V)
14		yoneda.s	. . . . . . . . . 10 ⊢ 𝑆 = (SetCat‘𝑈)
15	14	setccat 18021	. . . . . . . . 9 ⊢ (𝑈 ∈ V → 𝑆 ∈ Cat)
16	13, 15	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝑆 ∈ Cat)
17	5, 9, 16	fuccat 17909	. . . . . . 7 ⊢ (𝜑 → 𝑄 ∈ Cat)
18		eqid 2737	. . . . . . 7 ⊢ (𝑄 2ndF 𝑂) = (𝑄 2ndF 𝑂)
19	4, 17, 9, 18	2ndfcl 18133	. . . . . 6 ⊢ (𝜑 → (𝑄 2ndF 𝑂) ∈ ((𝑄 ×c 𝑂) Func 𝑂))
20		eqid 2737	. . . . . . . 8 ⊢ (oppCat‘𝑄) = (oppCat‘𝑄)
21		relfunc 17798	. . . . . . . . 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 18197	. . . . . . . . 9 ⊢ (𝜑 → 𝑌 ∈ (𝐶 Func 𝑄))
25		1st2ndbr 7996	. . . . . . . . 9 ⊢ ((Rel (𝐶 Func 𝑄) ∧ 𝑌 ∈ (𝐶 Func 𝑄)) → (1st ‘𝑌)(𝐶 Func 𝑄)(2nd ‘𝑌))
26	21, 24, 25	sylancr 588	. . . . . . . 8 ⊢ (𝜑 → (1st ‘𝑌)(𝐶 Func 𝑄)(2nd ‘𝑌))
27	7, 20, 26	funcoppc 17811	. . . . . . 7 ⊢ (𝜑 → (1st ‘𝑌)(𝑂 Func (oppCat‘𝑄))tpos (2nd ‘𝑌))
28		df-br 5101	. . . . . . 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 17824	. . . . 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 18132	. . . . 5 ⊢ (𝜑 → (𝑄 1stF 𝑂) ∈ ((𝑄 ×c 𝑂) Func 𝑄))
33	2, 3, 30, 32	prfcl 18138	. . . 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 4147	. . . . 5 ⊢ (𝜑 → ran (Homf ‘𝑄) ⊆ 𝑉)
37	34, 20, 35, 17, 10, 36	hofcl 18194	. . . 4 ⊢ (𝜑 → 𝐻 ∈ (((oppCat‘𝑄) ×c 𝑄) Func 𝑇))
38	33, 37	cofucl 17824	. . 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 2841	. 2 ⊢ (𝜑 → 𝑍 ∈ ((𝑄 ×c 𝑂) Func 𝑇))
40	35, 14, 10, 12	funcsetcres2 18029	. . 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 18157	. . 3 ⊢ (𝜑 → 𝐸 ∈ ((𝑄 ×c 𝑂) Func 𝑆))
43	40, 42	sseldd 3936	. 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 1542   ∈ wcel 2114  Vcvv 3442   ∪ cun 3901   ⊆ wss 3903  ⟨cop 4588   class class class wbr 5100  ran crn 5633  Rel wrel 5637  ‘cfv 6500  (class class class)co 7368  1^st c1st 7941  2^nd c2nd 7942  tpos ctpos 8177  Basecbs 17148  Catccat 17599  Idccid 17600  Hom_f chomf 17601  oppCatcoppc 17646   Func cfunc 17790   ∘_func ccofu 17792   FuncCat cfuc 17881  SetCatcsetc 18011   ×_c cxpc 18103   1^st_F c1stf 18104   2^nd_F c2ndf 18105   ⟨,⟩_F cprf 18106   eval_F cevlf 18144  Hom_Fchof 18183  Yoncyon 18184

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5226  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690  ax-cnex 11094  ax-resscn 11095  ax-1cn 11096  ax-icn 11097  ax-addcl 11098  ax-addrcl 11099  ax-mulcl 11100  ax-mulrcl 11101  ax-mulcom 11102  ax-addass 11103  ax-mulass 11104  ax-distr 11105  ax-i2m1 11106  ax-1ne0 11107  ax-1rid 11108  ax-rnegex 11109  ax-rrecex 11110  ax-cnre 11111  ax-pre-lttri 11112  ax-pre-lttrn 11113  ax-pre-ltadd 11114  ax-pre-mulgt0 11115

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-rmo 3352  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-tp 4587  df-op 4589  df-uni 4866  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5527  df-eprel 5532  df-po 5540  df-so 5541  df-fr 5585  df-we 5587  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-pred 6267  df-ord 6328  df-on 6329  df-lim 6330  df-suc 6331  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-riota 7325  df-ov 7371  df-oprab 7372  df-mpo 7373  df-om 7819  df-1st 7943  df-2nd 7944  df-tpos 8178  df-frecs 8233  df-wrecs 8264  df-recs 8313  df-rdg 8351  df-1o 8407  df-er 8645  df-map 8777  df-pm 8778  df-ixp 8848  df-en 8896  df-dom 8897  df-sdom 8898  df-fin 8899  df-pnf 11180  df-mnf 11181  df-xr 11182  df-ltxr 11183  df-le 11184  df-sub 11378  df-neg 11379  df-nn 12158  df-2 12220  df-3 12221  df-4 12222  df-5 12223  df-6 12224  df-7 12225  df-8 12226  df-9 12227  df-n0 12414  df-z 12501  df-dec 12620  df-uz 12764  df-fz 13436  df-struct 17086  df-sets 17103  df-slot 17121  df-ndx 17133  df-base 17149  df-ress 17170  df-hom 17213  df-cco 17214  df-cat 17603  df-cid 17604  df-homf 17605  df-comf 17606  df-oppc 17647  df-ssc 17746  df-resc 17747  df-subc 17748  df-func 17794  df-cofu 17796  df-nat 17882  df-fuc 17883  df-setc 18012  df-xpc 18107  df-1stf 18108  df-2ndf 18109  df-prf 18110  df-evlf 18148  df-curf 18149  df-hof 18185  df-yon 18186

This theorem is referenced by:  yonedalem3b  18214  yonedalem3  18215  yonedainv  18216  yonffthlem  18217  yoneda  18218