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Theorem yonedalem21 17134
Description: Lemma for yoneda 17144. (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 = (HomF𝑄)
yoneda.r 𝑅 = ((𝑄 ×c 𝑂) FuncCat 𝑇)
yoneda.e 𝐸 = (𝑂 evalF 𝑆)
yoneda.z 𝑍 = (𝐻func ((⟨(1st𝑌), tpos (2nd𝑌)⟩ ∘func (𝑄 2ndF 𝑂)) ⟨,⟩F (𝑄 1stF 𝑂)))
yoneda.c (𝜑𝐶 ∈ Cat)
yoneda.w (𝜑𝑉𝑊)
yoneda.u (𝜑 → ran (Homf𝐶) ⊆ 𝑈)
yoneda.v (𝜑 → (ran (Homf𝑄) ∪ 𝑈) ⊆ 𝑉)
yonedalem21.f (𝜑𝐹 ∈ (𝑂 Func 𝑆))
yonedalem21.x (𝜑𝑋𝐵)
Assertion
Ref Expression
yonedalem21 (𝜑 → (𝐹(1st𝑍)𝑋) = (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹))

Proof of Theorem yonedalem21
StepHypRef Expression
1 yoneda.z . . . . . 6 𝑍 = (𝐻func ((⟨(1st𝑌), tpos (2nd𝑌)⟩ ∘func (𝑄 2ndF 𝑂)) ⟨,⟩F (𝑄 1stF 𝑂)))
21fveq2i 6356 . . . . 5 (1st𝑍) = (1st ‘(𝐻func ((⟨(1st𝑌), tpos (2nd𝑌)⟩ ∘func (𝑄 2ndF 𝑂)) ⟨,⟩F (𝑄 1stF 𝑂))))
32oveqi 6827 . . . 4 (𝐹(1st𝑍)𝑋) = (𝐹(1st ‘(𝐻func ((⟨(1st𝑌), tpos (2nd𝑌)⟩ ∘func (𝑄 2ndF 𝑂)) ⟨,⟩F (𝑄 1stF 𝑂))))𝑋)
4 df-ov 6817 . . . 4 (𝐹(1st ‘(𝐻func ((⟨(1st𝑌), tpos (2nd𝑌)⟩ ∘func (𝑄 2ndF 𝑂)) ⟨,⟩F (𝑄 1stF 𝑂))))𝑋) = ((1st ‘(𝐻func ((⟨(1st𝑌), tpos (2nd𝑌)⟩ ∘func (𝑄 2ndF 𝑂)) ⟨,⟩F (𝑄 1stF 𝑂))))‘⟨𝐹, 𝑋⟩)
53, 4eqtri 2782 . . 3 (𝐹(1st𝑍)𝑋) = ((1st ‘(𝐻func ((⟨(1st𝑌), tpos (2nd𝑌)⟩ ∘func (𝑄 2ndF 𝑂)) ⟨,⟩F (𝑄 1stF 𝑂))))‘⟨𝐹, 𝑋⟩)
6 eqid 2760 . . . . 5 (𝑄 ×c 𝑂) = (𝑄 ×c 𝑂)
7 yoneda.q . . . . . 6 𝑄 = (𝑂 FuncCat 𝑆)
87fucbas 16841 . . . . 5 (𝑂 Func 𝑆) = (Base‘𝑄)
9 yoneda.o . . . . . 6 𝑂 = (oppCat‘𝐶)
10 yoneda.b . . . . . 6 𝐵 = (Base‘𝐶)
119, 10oppcbas 16599 . . . . 5 𝐵 = (Base‘𝑂)
126, 8, 11xpcbas 17039 . . . 4 ((𝑂 Func 𝑆) × 𝐵) = (Base‘(𝑄 ×c 𝑂))
13 eqid 2760 . . . . 5 ((⟨(1st𝑌), tpos (2nd𝑌)⟩ ∘func (𝑄 2ndF 𝑂)) ⟨,⟩F (𝑄 1stF 𝑂)) = ((⟨(1st𝑌), tpos (2nd𝑌)⟩ ∘func (𝑄 2ndF 𝑂)) ⟨,⟩F (𝑄 1stF 𝑂))
14 eqid 2760 . . . . 5 ((oppCat‘𝑄) ×c 𝑄) = ((oppCat‘𝑄) ×c 𝑄)
15 yoneda.c . . . . . . . . 9 (𝜑𝐶 ∈ Cat)
169oppccat 16603 . . . . . . . . 9 (𝐶 ∈ Cat → 𝑂 ∈ Cat)
1715, 16syl 17 . . . . . . . 8 (𝜑𝑂 ∈ Cat)
18 yoneda.w . . . . . . . . . 10 (𝜑𝑉𝑊)
19 yoneda.v . . . . . . . . . . 11 (𝜑 → (ran (Homf𝑄) ∪ 𝑈) ⊆ 𝑉)
2019unssbd 3934 . . . . . . . . . 10 (𝜑𝑈𝑉)
2118, 20ssexd 4957 . . . . . . . . 9 (𝜑𝑈 ∈ V)
22 yoneda.s . . . . . . . . . 10 𝑆 = (SetCat‘𝑈)
2322setccat 16956 . . . . . . . . 9 (𝑈 ∈ V → 𝑆 ∈ Cat)
2421, 23syl 17 . . . . . . . 8 (𝜑𝑆 ∈ Cat)
257, 17, 24fuccat 16851 . . . . . . 7 (𝜑𝑄 ∈ Cat)
26 eqid 2760 . . . . . . 7 (𝑄 2ndF 𝑂) = (𝑄 2ndF 𝑂)
276, 25, 17, 262ndfcl 17059 . . . . . 6 (𝜑 → (𝑄 2ndF 𝑂) ∈ ((𝑄 ×c 𝑂) Func 𝑂))
28 eqid 2760 . . . . . . . 8 (oppCat‘𝑄) = (oppCat‘𝑄)
29 relfunc 16743 . . . . . . . . 9 Rel (𝐶 Func 𝑄)
30 yoneda.y . . . . . . . . . 10 𝑌 = (Yon‘𝐶)
31 yoneda.u . . . . . . . . . 10 (𝜑 → ran (Homf𝐶) ⊆ 𝑈)
3230, 15, 9, 22, 7, 21, 31yoncl 17123 . . . . . . . . 9 (𝜑𝑌 ∈ (𝐶 Func 𝑄))
33 1st2ndbr 7385 . . . . . . . . 9 ((Rel (𝐶 Func 𝑄) ∧ 𝑌 ∈ (𝐶 Func 𝑄)) → (1st𝑌)(𝐶 Func 𝑄)(2nd𝑌))
3429, 32, 33sylancr 698 . . . . . . . 8 (𝜑 → (1st𝑌)(𝐶 Func 𝑄)(2nd𝑌))
359, 28, 34funcoppc 16756 . . . . . . 7 (𝜑 → (1st𝑌)(𝑂 Func (oppCat‘𝑄))tpos (2nd𝑌))
36 df-br 4805 . . . . . . 7 ((1st𝑌)(𝑂 Func (oppCat‘𝑄))tpos (2nd𝑌) ↔ ⟨(1st𝑌), tpos (2nd𝑌)⟩ ∈ (𝑂 Func (oppCat‘𝑄)))
3735, 36sylib 208 . . . . . 6 (𝜑 → ⟨(1st𝑌), tpos (2nd𝑌)⟩ ∈ (𝑂 Func (oppCat‘𝑄)))
3827, 37cofucl 16769 . . . . 5 (𝜑 → (⟨(1st𝑌), tpos (2nd𝑌)⟩ ∘func (𝑄 2ndF 𝑂)) ∈ ((𝑄 ×c 𝑂) Func (oppCat‘𝑄)))
39 eqid 2760 . . . . . 6 (𝑄 1stF 𝑂) = (𝑄 1stF 𝑂)
406, 25, 17, 391stfcl 17058 . . . . 5 (𝜑 → (𝑄 1stF 𝑂) ∈ ((𝑄 ×c 𝑂) Func 𝑄))
4113, 14, 38, 40prfcl 17064 . . . 4 (𝜑 → ((⟨(1st𝑌), tpos (2nd𝑌)⟩ ∘func (𝑄 2ndF 𝑂)) ⟨,⟩F (𝑄 1stF 𝑂)) ∈ ((𝑄 ×c 𝑂) Func ((oppCat‘𝑄) ×c 𝑄)))
42 yoneda.h . . . . 5 𝐻 = (HomF𝑄)
43 yoneda.t . . . . 5 𝑇 = (SetCat‘𝑉)
4419unssad 3933 . . . . 5 (𝜑 → ran (Homf𝑄) ⊆ 𝑉)
4542, 28, 43, 25, 18, 44hofcl 17120 . . . 4 (𝜑𝐻 ∈ (((oppCat‘𝑄) ×c 𝑄) Func 𝑇))
46 yonedalem21.f . . . . 5 (𝜑𝐹 ∈ (𝑂 Func 𝑆))
47 yonedalem21.x . . . . 5 (𝜑𝑋𝐵)
48 opelxpi 5305 . . . . 5 ((𝐹 ∈ (𝑂 Func 𝑆) ∧ 𝑋𝐵) → ⟨𝐹, 𝑋⟩ ∈ ((𝑂 Func 𝑆) × 𝐵))
4946, 47, 48syl2anc 696 . . . 4 (𝜑 → ⟨𝐹, 𝑋⟩ ∈ ((𝑂 Func 𝑆) × 𝐵))
5012, 41, 45, 49cofu1 16765 . . 3 (𝜑 → ((1st ‘(𝐻func ((⟨(1st𝑌), tpos (2nd𝑌)⟩ ∘func (𝑄 2ndF 𝑂)) ⟨,⟩F (𝑄 1stF 𝑂))))‘⟨𝐹, 𝑋⟩) = ((1st𝐻)‘((1st ‘((⟨(1st𝑌), tpos (2nd𝑌)⟩ ∘func (𝑄 2ndF 𝑂)) ⟨,⟩F (𝑄 1stF 𝑂)))‘⟨𝐹, 𝑋⟩)))
515, 50syl5eq 2806 . 2 (𝜑 → (𝐹(1st𝑍)𝑋) = ((1st𝐻)‘((1st ‘((⟨(1st𝑌), tpos (2nd𝑌)⟩ ∘func (𝑄 2ndF 𝑂)) ⟨,⟩F (𝑄 1stF 𝑂)))‘⟨𝐹, 𝑋⟩)))
52 eqid 2760 . . . . . 6 (Hom ‘(𝑄 ×c 𝑂)) = (Hom ‘(𝑄 ×c 𝑂))
5313, 12, 52, 38, 40, 49prf1 17061 . . . . 5 (𝜑 → ((1st ‘((⟨(1st𝑌), tpos (2nd𝑌)⟩ ∘func (𝑄 2ndF 𝑂)) ⟨,⟩F (𝑄 1stF 𝑂)))‘⟨𝐹, 𝑋⟩) = ⟨((1st ‘(⟨(1st𝑌), tpos (2nd𝑌)⟩ ∘func (𝑄 2ndF 𝑂)))‘⟨𝐹, 𝑋⟩), ((1st ‘(𝑄 1stF 𝑂))‘⟨𝐹, 𝑋⟩)⟩)
5412, 27, 37, 49cofu1 16765 . . . . . . 7 (𝜑 → ((1st ‘(⟨(1st𝑌), tpos (2nd𝑌)⟩ ∘func (𝑄 2ndF 𝑂)))‘⟨𝐹, 𝑋⟩) = ((1st ‘⟨(1st𝑌), tpos (2nd𝑌)⟩)‘((1st ‘(𝑄 2ndF 𝑂))‘⟨𝐹, 𝑋⟩)))
55 fvex 6363 . . . . . . . . . 10 (1st𝑌) ∈ V
56 fvex 6363 . . . . . . . . . . 11 (2nd𝑌) ∈ V
5756tposex 7556 . . . . . . . . . 10 tpos (2nd𝑌) ∈ V
5855, 57op1st 7342 . . . . . . . . 9 (1st ‘⟨(1st𝑌), tpos (2nd𝑌)⟩) = (1st𝑌)
5958a1i 11 . . . . . . . 8 (𝜑 → (1st ‘⟨(1st𝑌), tpos (2nd𝑌)⟩) = (1st𝑌))
606, 12, 52, 25, 17, 26, 492ndf1 17056 . . . . . . . . 9 (𝜑 → ((1st ‘(𝑄 2ndF 𝑂))‘⟨𝐹, 𝑋⟩) = (2nd ‘⟨𝐹, 𝑋⟩))
61 op2ndg 7347 . . . . . . . . . 10 ((𝐹 ∈ (𝑂 Func 𝑆) ∧ 𝑋𝐵) → (2nd ‘⟨𝐹, 𝑋⟩) = 𝑋)
6246, 47, 61syl2anc 696 . . . . . . . . 9 (𝜑 → (2nd ‘⟨𝐹, 𝑋⟩) = 𝑋)
6360, 62eqtrd 2794 . . . . . . . 8 (𝜑 → ((1st ‘(𝑄 2ndF 𝑂))‘⟨𝐹, 𝑋⟩) = 𝑋)
6459, 63fveq12d 6359 . . . . . . 7 (𝜑 → ((1st ‘⟨(1st𝑌), tpos (2nd𝑌)⟩)‘((1st ‘(𝑄 2ndF 𝑂))‘⟨𝐹, 𝑋⟩)) = ((1st𝑌)‘𝑋))
6554, 64eqtrd 2794 . . . . . 6 (𝜑 → ((1st ‘(⟨(1st𝑌), tpos (2nd𝑌)⟩ ∘func (𝑄 2ndF 𝑂)))‘⟨𝐹, 𝑋⟩) = ((1st𝑌)‘𝑋))
666, 12, 52, 25, 17, 39, 491stf1 17053 . . . . . . 7 (𝜑 → ((1st ‘(𝑄 1stF 𝑂))‘⟨𝐹, 𝑋⟩) = (1st ‘⟨𝐹, 𝑋⟩))
67 op1stg 7346 . . . . . . . 8 ((𝐹 ∈ (𝑂 Func 𝑆) ∧ 𝑋𝐵) → (1st ‘⟨𝐹, 𝑋⟩) = 𝐹)
6846, 47, 67syl2anc 696 . . . . . . 7 (𝜑 → (1st ‘⟨𝐹, 𝑋⟩) = 𝐹)
6966, 68eqtrd 2794 . . . . . 6 (𝜑 → ((1st ‘(𝑄 1stF 𝑂))‘⟨𝐹, 𝑋⟩) = 𝐹)
7065, 69opeq12d 4561 . . . . 5 (𝜑 → ⟨((1st ‘(⟨(1st𝑌), tpos (2nd𝑌)⟩ ∘func (𝑄 2ndF 𝑂)))‘⟨𝐹, 𝑋⟩), ((1st ‘(𝑄 1stF 𝑂))‘⟨𝐹, 𝑋⟩)⟩ = ⟨((1st𝑌)‘𝑋), 𝐹⟩)
7153, 70eqtrd 2794 . . . 4 (𝜑 → ((1st ‘((⟨(1st𝑌), tpos (2nd𝑌)⟩ ∘func (𝑄 2ndF 𝑂)) ⟨,⟩F (𝑄 1stF 𝑂)))‘⟨𝐹, 𝑋⟩) = ⟨((1st𝑌)‘𝑋), 𝐹⟩)
7271fveq2d 6357 . . 3 (𝜑 → ((1st𝐻)‘((1st ‘((⟨(1st𝑌), tpos (2nd𝑌)⟩ ∘func (𝑄 2ndF 𝑂)) ⟨,⟩F (𝑄 1stF 𝑂)))‘⟨𝐹, 𝑋⟩)) = ((1st𝐻)‘⟨((1st𝑌)‘𝑋), 𝐹⟩))
73 df-ov 6817 . . 3 (((1st𝑌)‘𝑋)(1st𝐻)𝐹) = ((1st𝐻)‘⟨((1st𝑌)‘𝑋), 𝐹⟩)
7472, 73syl6eqr 2812 . 2 (𝜑 → ((1st𝐻)‘((1st ‘((⟨(1st𝑌), tpos (2nd𝑌)⟩ ∘func (𝑄 2ndF 𝑂)) ⟨,⟩F (𝑄 1stF 𝑂)))‘⟨𝐹, 𝑋⟩)) = (((1st𝑌)‘𝑋)(1st𝐻)𝐹))
75 eqid 2760 . . . 4 (𝑂 Nat 𝑆) = (𝑂 Nat 𝑆)
767, 75fuchom 16842 . . 3 (𝑂 Nat 𝑆) = (Hom ‘𝑄)
7730, 10, 15, 47, 9, 22, 21, 31yon1cl 17124 . . 3 (𝜑 → ((1st𝑌)‘𝑋) ∈ (𝑂 Func 𝑆))
7842, 25, 8, 76, 77, 46hof1 17115 . 2 (𝜑 → (((1st𝑌)‘𝑋)(1st𝐻)𝐹) = (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹))
7951, 74, 783eqtrd 2798 1 (𝜑 → (𝐹(1st𝑍)𝑋) = (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1632  wcel 2139  Vcvv 3340  cun 3713  wss 3715  cop 4327   class class class wbr 4804   × cxp 5264  ran crn 5267  Rel wrel 5271  cfv 6049  (class class class)co 6814  1st c1st 7332  2nd c2nd 7333  tpos ctpos 7521  Basecbs 16079  Hom chom 16174  Catccat 16546  Idccid 16547  Homf chomf 16548  oppCatcoppc 16592   Func cfunc 16735  func ccofu 16737   Nat cnat 16822   FuncCat cfuc 16823  SetCatcsetc 16946   ×c cxpc 17029   1stF c1stf 17030   2ndF c2ndf 17031   ⟨,⟩F cprf 17032   evalF cevlf 17070  HomFchof 17109  Yoncyon 17110
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115  ax-cnex 10204  ax-resscn 10205  ax-1cn 10206  ax-icn 10207  ax-addcl 10208  ax-addrcl 10209  ax-mulcl 10210  ax-mulrcl 10211  ax-mulcom 10212  ax-addass 10213  ax-mulass 10214  ax-distr 10215  ax-i2m1 10216  ax-1ne0 10217  ax-1rid 10218  ax-rnegex 10219  ax-rrecex 10220  ax-cnre 10221  ax-pre-lttri 10222  ax-pre-lttrn 10223  ax-pre-ltadd 10224  ax-pre-mulgt0 10225
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-fal 1638  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-nel 3036  df-ral 3055  df-rex 3056  df-reu 3057  df-rmo 3058  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-int 4628  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-riota 6775  df-ov 6817  df-oprab 6818  df-mpt2 6819  df-om 7232  df-1st 7334  df-2nd 7335  df-tpos 7522  df-wrecs 7577  df-recs 7638  df-rdg 7676  df-1o 7730  df-oadd 7734  df-er 7913  df-map 8027  df-ixp 8077  df-en 8124  df-dom 8125  df-sdom 8126  df-fin 8127  df-pnf 10288  df-mnf 10289  df-xr 10290  df-ltxr 10291  df-le 10292  df-sub 10480  df-neg 10481  df-nn 11233  df-2 11291  df-3 11292  df-4 11293  df-5 11294  df-6 11295  df-7 11296  df-8 11297  df-9 11298  df-n0 11505  df-z 11590  df-dec 11706  df-uz 11900  df-fz 12540  df-struct 16081  df-ndx 16082  df-slot 16083  df-base 16085  df-sets 16086  df-hom 16188  df-cco 16189  df-cat 16550  df-cid 16551  df-homf 16552  df-comf 16553  df-oppc 16593  df-func 16739  df-cofu 16741  df-nat 16824  df-fuc 16825  df-setc 16947  df-xpc 17033  df-1stf 17034  df-2ndf 17035  df-prf 17036  df-curf 17075  df-hof 17111  df-yon 17112
This theorem is referenced by:  yonedalem3a  17135  yonedalem3b  17140  yonedainv  17142  yonffthlem  17143
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