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Theorem yonedalem21 16829
Description: Lemma for yoneda 16839. (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 6153 . . . . 5 (1st𝑍) = (1st ‘(𝐻func ((⟨(1st𝑌), tpos (2nd𝑌)⟩ ∘func (𝑄 2ndF 𝑂)) ⟨,⟩F (𝑄 1stF 𝑂))))
32oveqi 6618 . . . 4 (𝐹(1st𝑍)𝑋) = (𝐹(1st ‘(𝐻func ((⟨(1st𝑌), tpos (2nd𝑌)⟩ ∘func (𝑄 2ndF 𝑂)) ⟨,⟩F (𝑄 1stF 𝑂))))𝑋)
4 df-ov 6608 . . . 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 2648 . . 3 (𝐹(1st𝑍)𝑋) = ((1st ‘(𝐻func ((⟨(1st𝑌), tpos (2nd𝑌)⟩ ∘func (𝑄 2ndF 𝑂)) ⟨,⟩F (𝑄 1stF 𝑂))))‘⟨𝐹, 𝑋⟩)
6 eqid 2626 . . . . 5 (𝑄 ×c 𝑂) = (𝑄 ×c 𝑂)
7 yoneda.q . . . . . 6 𝑄 = (𝑂 FuncCat 𝑆)
87fucbas 16536 . . . . 5 (𝑂 Func 𝑆) = (Base‘𝑄)
9 yoneda.o . . . . . 6 𝑂 = (oppCat‘𝐶)
10 yoneda.b . . . . . 6 𝐵 = (Base‘𝐶)
119, 10oppcbas 16294 . . . . 5 𝐵 = (Base‘𝑂)
126, 8, 11xpcbas 16734 . . . 4 ((𝑂 Func 𝑆) × 𝐵) = (Base‘(𝑄 ×c 𝑂))
13 eqid 2626 . . . . 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 2626 . . . . 5 ((oppCat‘𝑄) ×c 𝑄) = ((oppCat‘𝑄) ×c 𝑄)
15 yoneda.c . . . . . . . . 9 (𝜑𝐶 ∈ Cat)
169oppccat 16298 . . . . . . . . 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 3774 . . . . . . . . . 10 (𝜑𝑈𝑉)
2118, 20ssexd 4770 . . . . . . . . 9 (𝜑𝑈 ∈ V)
22 yoneda.s . . . . . . . . . 10 𝑆 = (SetCat‘𝑈)
2322setccat 16651 . . . . . . . . 9 (𝑈 ∈ V → 𝑆 ∈ Cat)
2421, 23syl 17 . . . . . . . 8 (𝜑𝑆 ∈ Cat)
257, 17, 24fuccat 16546 . . . . . . 7 (𝜑𝑄 ∈ Cat)
26 eqid 2626 . . . . . . 7 (𝑄 2ndF 𝑂) = (𝑄 2ndF 𝑂)
276, 25, 17, 262ndfcl 16754 . . . . . 6 (𝜑 → (𝑄 2ndF 𝑂) ∈ ((𝑄 ×c 𝑂) Func 𝑂))
28 eqid 2626 . . . . . . . 8 (oppCat‘𝑄) = (oppCat‘𝑄)
29 relfunc 16438 . . . . . . . . 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 16818 . . . . . . . . 9 (𝜑𝑌 ∈ (𝐶 Func 𝑄))
33 1st2ndbr 7165 . . . . . . . . 9 ((Rel (𝐶 Func 𝑄) ∧ 𝑌 ∈ (𝐶 Func 𝑄)) → (1st𝑌)(𝐶 Func 𝑄)(2nd𝑌))
3429, 32, 33sylancr 694 . . . . . . . 8 (𝜑 → (1st𝑌)(𝐶 Func 𝑄)(2nd𝑌))
359, 28, 34funcoppc 16451 . . . . . . 7 (𝜑 → (1st𝑌)(𝑂 Func (oppCat‘𝑄))tpos (2nd𝑌))
36 df-br 4619 . . . . . . 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 16464 . . . . 5 (𝜑 → (⟨(1st𝑌), tpos (2nd𝑌)⟩ ∘func (𝑄 2ndF 𝑂)) ∈ ((𝑄 ×c 𝑂) Func (oppCat‘𝑄)))
39 eqid 2626 . . . . . 6 (𝑄 1stF 𝑂) = (𝑄 1stF 𝑂)
406, 25, 17, 391stfcl 16753 . . . . 5 (𝜑 → (𝑄 1stF 𝑂) ∈ ((𝑄 ×c 𝑂) Func 𝑄))
4113, 14, 38, 40prfcl 16759 . . . 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 3773 . . . . 5 (𝜑 → ran (Homf𝑄) ⊆ 𝑉)
4542, 28, 43, 25, 18, 44hofcl 16815 . . . 4 (𝜑𝐻 ∈ (((oppCat‘𝑄) ×c 𝑄) Func 𝑇))
46 yonedalem21.f . . . . 5 (𝜑𝐹 ∈ (𝑂 Func 𝑆))
47 yonedalem21.x . . . . 5 (𝜑𝑋𝐵)
48 opelxpi 5113 . . . . 5 ((𝐹 ∈ (𝑂 Func 𝑆) ∧ 𝑋𝐵) → ⟨𝐹, 𝑋⟩ ∈ ((𝑂 Func 𝑆) × 𝐵))
4946, 47, 48syl2anc 692 . . . 4 (𝜑 → ⟨𝐹, 𝑋⟩ ∈ ((𝑂 Func 𝑆) × 𝐵))
5012, 41, 45, 49cofu1 16460 . . 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 2672 . 2 (𝜑 → (𝐹(1st𝑍)𝑋) = ((1st𝐻)‘((1st ‘((⟨(1st𝑌), tpos (2nd𝑌)⟩ ∘func (𝑄 2ndF 𝑂)) ⟨,⟩F (𝑄 1stF 𝑂)))‘⟨𝐹, 𝑋⟩)))
52 eqid 2626 . . . . . 6 (Hom ‘(𝑄 ×c 𝑂)) = (Hom ‘(𝑄 ×c 𝑂))
5313, 12, 52, 38, 40, 49prf1 16756 . . . . 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 16460 . . . . . . 7 (𝜑 → ((1st ‘(⟨(1st𝑌), tpos (2nd𝑌)⟩ ∘func (𝑄 2ndF 𝑂)))‘⟨𝐹, 𝑋⟩) = ((1st ‘⟨(1st𝑌), tpos (2nd𝑌)⟩)‘((1st ‘(𝑄 2ndF 𝑂))‘⟨𝐹, 𝑋⟩)))
55 fvex 6160 . . . . . . . . . 10 (1st𝑌) ∈ V
56 fvex 6160 . . . . . . . . . . 11 (2nd𝑌) ∈ V
5756tposex 7332 . . . . . . . . . 10 tpos (2nd𝑌) ∈ V
5855, 57op1st 7124 . . . . . . . . 9 (1st ‘⟨(1st𝑌), tpos (2nd𝑌)⟩) = (1st𝑌)
5958a1i 11 . . . . . . . 8 (𝜑 → (1st ‘⟨(1st𝑌), tpos (2nd𝑌)⟩) = (1st𝑌))
606, 12, 52, 25, 17, 26, 492ndf1 16751 . . . . . . . . 9 (𝜑 → ((1st ‘(𝑄 2ndF 𝑂))‘⟨𝐹, 𝑋⟩) = (2nd ‘⟨𝐹, 𝑋⟩))
61 op2ndg 7129 . . . . . . . . . 10 ((𝐹 ∈ (𝑂 Func 𝑆) ∧ 𝑋𝐵) → (2nd ‘⟨𝐹, 𝑋⟩) = 𝑋)
6246, 47, 61syl2anc 692 . . . . . . . . 9 (𝜑 → (2nd ‘⟨𝐹, 𝑋⟩) = 𝑋)
6360, 62eqtrd 2660 . . . . . . . 8 (𝜑 → ((1st ‘(𝑄 2ndF 𝑂))‘⟨𝐹, 𝑋⟩) = 𝑋)
6459, 63fveq12d 6156 . . . . . . 7 (𝜑 → ((1st ‘⟨(1st𝑌), tpos (2nd𝑌)⟩)‘((1st ‘(𝑄 2ndF 𝑂))‘⟨𝐹, 𝑋⟩)) = ((1st𝑌)‘𝑋))
6554, 64eqtrd 2660 . . . . . 6 (𝜑 → ((1st ‘(⟨(1st𝑌), tpos (2nd𝑌)⟩ ∘func (𝑄 2ndF 𝑂)))‘⟨𝐹, 𝑋⟩) = ((1st𝑌)‘𝑋))
666, 12, 52, 25, 17, 39, 491stf1 16748 . . . . . . 7 (𝜑 → ((1st ‘(𝑄 1stF 𝑂))‘⟨𝐹, 𝑋⟩) = (1st ‘⟨𝐹, 𝑋⟩))
67 op1stg 7128 . . . . . . . 8 ((𝐹 ∈ (𝑂 Func 𝑆) ∧ 𝑋𝐵) → (1st ‘⟨𝐹, 𝑋⟩) = 𝐹)
6846, 47, 67syl2anc 692 . . . . . . 7 (𝜑 → (1st ‘⟨𝐹, 𝑋⟩) = 𝐹)
6966, 68eqtrd 2660 . . . . . 6 (𝜑 → ((1st ‘(𝑄 1stF 𝑂))‘⟨𝐹, 𝑋⟩) = 𝐹)
7065, 69opeq12d 4383 . . . . 5 (𝜑 → ⟨((1st ‘(⟨(1st𝑌), tpos (2nd𝑌)⟩ ∘func (𝑄 2ndF 𝑂)))‘⟨𝐹, 𝑋⟩), ((1st ‘(𝑄 1stF 𝑂))‘⟨𝐹, 𝑋⟩)⟩ = ⟨((1st𝑌)‘𝑋), 𝐹⟩)
7153, 70eqtrd 2660 . . . 4 (𝜑 → ((1st ‘((⟨(1st𝑌), tpos (2nd𝑌)⟩ ∘func (𝑄 2ndF 𝑂)) ⟨,⟩F (𝑄 1stF 𝑂)))‘⟨𝐹, 𝑋⟩) = ⟨((1st𝑌)‘𝑋), 𝐹⟩)
7271fveq2d 6154 . . 3 (𝜑 → ((1st𝐻)‘((1st ‘((⟨(1st𝑌), tpos (2nd𝑌)⟩ ∘func (𝑄 2ndF 𝑂)) ⟨,⟩F (𝑄 1stF 𝑂)))‘⟨𝐹, 𝑋⟩)) = ((1st𝐻)‘⟨((1st𝑌)‘𝑋), 𝐹⟩))
73 df-ov 6608 . . 3 (((1st𝑌)‘𝑋)(1st𝐻)𝐹) = ((1st𝐻)‘⟨((1st𝑌)‘𝑋), 𝐹⟩)
7472, 73syl6eqr 2678 . 2 (𝜑 → ((1st𝐻)‘((1st ‘((⟨(1st𝑌), tpos (2nd𝑌)⟩ ∘func (𝑄 2ndF 𝑂)) ⟨,⟩F (𝑄 1stF 𝑂)))‘⟨𝐹, 𝑋⟩)) = (((1st𝑌)‘𝑋)(1st𝐻)𝐹))
75 eqid 2626 . . . 4 (𝑂 Nat 𝑆) = (𝑂 Nat 𝑆)
767, 75fuchom 16537 . . 3 (𝑂 Nat 𝑆) = (Hom ‘𝑄)
7730, 10, 15, 47, 9, 22, 21, 31yon1cl 16819 . . 3 (𝜑 → ((1st𝑌)‘𝑋) ∈ (𝑂 Func 𝑆))
7842, 25, 8, 76, 77, 46hof1 16810 . 2 (𝜑 → (((1st𝑌)‘𝑋)(1st𝐻)𝐹) = (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹))
7951, 74, 783eqtrd 2664 1 (𝜑 → (𝐹(1st𝑍)𝑋) = (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1480  wcel 1992  Vcvv 3191  cun 3558  wss 3560  cop 4159   class class class wbr 4618   × cxp 5077  ran crn 5080  Rel wrel 5084  cfv 5850  (class class class)co 6605  1st c1st 7114  2nd c2nd 7115  tpos ctpos 7297  Basecbs 15776  Hom chom 15868  Catccat 16241  Idccid 16242  Homf chomf 16243  oppCatcoppc 16287   Func cfunc 16430  func ccofu 16432   Nat cnat 16517   FuncCat cfuc 16518  SetCatcsetc 16641   ×c cxpc 16724   1stF c1stf 16725   2ndF c2ndf 16726   ⟨,⟩F cprf 16727   evalF cevlf 16765  HomFchof 16804  Yoncyon 16805
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6903  ax-cnex 9937  ax-resscn 9938  ax-1cn 9939  ax-icn 9940  ax-addcl 9941  ax-addrcl 9942  ax-mulcl 9943  ax-mulrcl 9944  ax-mulcom 9945  ax-addass 9946  ax-mulass 9947  ax-distr 9948  ax-i2m1 9949  ax-1ne0 9950  ax-1rid 9951  ax-rnegex 9952  ax-rrecex 9953  ax-cnre 9954  ax-pre-lttri 9955  ax-pre-lttrn 9956  ax-pre-ltadd 9957  ax-pre-mulgt0 9958
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-fal 1486  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-nel 2900  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3193  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-int 4446  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5642  df-ord 5688  df-on 5689  df-lim 5690  df-suc 5691  df-iota 5813  df-fun 5852  df-fn 5853  df-f 5854  df-f1 5855  df-fo 5856  df-f1o 5857  df-fv 5858  df-riota 6566  df-ov 6608  df-oprab 6609  df-mpt2 6610  df-om 7014  df-1st 7116  df-2nd 7117  df-tpos 7298  df-wrecs 7353  df-recs 7414  df-rdg 7452  df-1o 7506  df-oadd 7510  df-er 7688  df-map 7805  df-ixp 7854  df-en 7901  df-dom 7902  df-sdom 7903  df-fin 7904  df-pnf 10021  df-mnf 10022  df-xr 10023  df-ltxr 10024  df-le 10025  df-sub 10213  df-neg 10214  df-nn 10966  df-2 11024  df-3 11025  df-4 11026  df-5 11027  df-6 11028  df-7 11029  df-8 11030  df-9 11031  df-n0 11238  df-z 11323  df-dec 11438  df-uz 11632  df-fz 12266  df-struct 15778  df-ndx 15779  df-slot 15780  df-base 15781  df-sets 15782  df-hom 15882  df-cco 15883  df-cat 16245  df-cid 16246  df-homf 16247  df-comf 16248  df-oppc 16288  df-func 16434  df-cofu 16436  df-nat 16519  df-fuc 16520  df-setc 16642  df-xpc 16728  df-1stf 16729  df-2ndf 16730  df-prf 16731  df-curf 16770  df-hof 16806  df-yon 16807
This theorem is referenced by:  yonedalem3a  16830  yonedalem3b  16835  yonedainv  16837  yonffthlem  16838
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