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Theorem yonedalem21 17310
Description: Lemma for yoneda 17320. (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 6451 . . . . 5 (1st𝑍) = (1st ‘(𝐻func ((⟨(1st𝑌), tpos (2nd𝑌)⟩ ∘func (𝑄 2ndF 𝑂)) ⟨,⟩F (𝑄 1stF 𝑂))))
32oveqi 6937 . . . 4 (𝐹(1st𝑍)𝑋) = (𝐹(1st ‘(𝐻func ((⟨(1st𝑌), tpos (2nd𝑌)⟩ ∘func (𝑄 2ndF 𝑂)) ⟨,⟩F (𝑄 1stF 𝑂))))𝑋)
4 df-ov 6927 . . . 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 2802 . . 3 (𝐹(1st𝑍)𝑋) = ((1st ‘(𝐻func ((⟨(1st𝑌), tpos (2nd𝑌)⟩ ∘func (𝑄 2ndF 𝑂)) ⟨,⟩F (𝑄 1stF 𝑂))))‘⟨𝐹, 𝑋⟩)
6 eqid 2778 . . . . 5 (𝑄 ×c 𝑂) = (𝑄 ×c 𝑂)
7 yoneda.q . . . . . 6 𝑄 = (𝑂 FuncCat 𝑆)
87fucbas 17016 . . . . 5 (𝑂 Func 𝑆) = (Base‘𝑄)
9 yoneda.o . . . . . 6 𝑂 = (oppCat‘𝐶)
10 yoneda.b . . . . . 6 𝐵 = (Base‘𝐶)
119, 10oppcbas 16774 . . . . 5 𝐵 = (Base‘𝑂)
126, 8, 11xpcbas 17215 . . . 4 ((𝑂 Func 𝑆) × 𝐵) = (Base‘(𝑄 ×c 𝑂))
13 eqid 2778 . . . . 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 2778 . . . . 5 ((oppCat‘𝑄) ×c 𝑄) = ((oppCat‘𝑄) ×c 𝑄)
15 yoneda.c . . . . . . . . 9 (𝜑𝐶 ∈ Cat)
169oppccat 16778 . . . . . . . . 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 4014 . . . . . . . . . 10 (𝜑𝑈𝑉)
2118, 20ssexd 5044 . . . . . . . . 9 (𝜑𝑈 ∈ V)
22 yoneda.s . . . . . . . . . 10 𝑆 = (SetCat‘𝑈)
2322setccat 17131 . . . . . . . . 9 (𝑈 ∈ V → 𝑆 ∈ Cat)
2421, 23syl 17 . . . . . . . 8 (𝜑𝑆 ∈ Cat)
257, 17, 24fuccat 17026 . . . . . . 7 (𝜑𝑄 ∈ Cat)
26 eqid 2778 . . . . . . 7 (𝑄 2ndF 𝑂) = (𝑄 2ndF 𝑂)
276, 25, 17, 262ndfcl 17235 . . . . . 6 (𝜑 → (𝑄 2ndF 𝑂) ∈ ((𝑄 ×c 𝑂) Func 𝑂))
28 eqid 2778 . . . . . . . 8 (oppCat‘𝑄) = (oppCat‘𝑄)
29 relfunc 16918 . . . . . . . . 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 17299 . . . . . . . . 9 (𝜑𝑌 ∈ (𝐶 Func 𝑄))
33 1st2ndbr 7498 . . . . . . . . 9 ((Rel (𝐶 Func 𝑄) ∧ 𝑌 ∈ (𝐶 Func 𝑄)) → (1st𝑌)(𝐶 Func 𝑄)(2nd𝑌))
3429, 32, 33sylancr 581 . . . . . . . 8 (𝜑 → (1st𝑌)(𝐶 Func 𝑄)(2nd𝑌))
359, 28, 34funcoppc 16931 . . . . . . 7 (𝜑 → (1st𝑌)(𝑂 Func (oppCat‘𝑄))tpos (2nd𝑌))
36 df-br 4889 . . . . . . 7 ((1st𝑌)(𝑂 Func (oppCat‘𝑄))tpos (2nd𝑌) ↔ ⟨(1st𝑌), tpos (2nd𝑌)⟩ ∈ (𝑂 Func (oppCat‘𝑄)))
3735, 36sylib 210 . . . . . 6 (𝜑 → ⟨(1st𝑌), tpos (2nd𝑌)⟩ ∈ (𝑂 Func (oppCat‘𝑄)))
3827, 37cofucl 16944 . . . . 5 (𝜑 → (⟨(1st𝑌), tpos (2nd𝑌)⟩ ∘func (𝑄 2ndF 𝑂)) ∈ ((𝑄 ×c 𝑂) Func (oppCat‘𝑄)))
39 eqid 2778 . . . . . 6 (𝑄 1stF 𝑂) = (𝑄 1stF 𝑂)
406, 25, 17, 391stfcl 17234 . . . . 5 (𝜑 → (𝑄 1stF 𝑂) ∈ ((𝑄 ×c 𝑂) Func 𝑄))
4113, 14, 38, 40prfcl 17240 . . . 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 4013 . . . . 5 (𝜑 → ran (Homf𝑄) ⊆ 𝑉)
4542, 28, 43, 25, 18, 44hofcl 17296 . . . 4 (𝜑𝐻 ∈ (((oppCat‘𝑄) ×c 𝑄) Func 𝑇))
46 yonedalem21.f . . . . 5 (𝜑𝐹 ∈ (𝑂 Func 𝑆))
47 yonedalem21.x . . . . 5 (𝜑𝑋𝐵)
4846, 47opelxpd 5395 . . . 4 (𝜑 → ⟨𝐹, 𝑋⟩ ∈ ((𝑂 Func 𝑆) × 𝐵))
4912, 41, 45, 48cofu1 16940 . . 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⟩)))
505, 49syl5eq 2826 . 2 (𝜑 → (𝐹(1st𝑍)𝑋) = ((1st𝐻)‘((1st ‘((⟨(1st𝑌), tpos (2nd𝑌)⟩ ∘func (𝑄 2ndF 𝑂)) ⟨,⟩F (𝑄 1stF 𝑂)))‘⟨𝐹, 𝑋⟩)))
51 eqid 2778 . . . . . 6 (Hom ‘(𝑄 ×c 𝑂)) = (Hom ‘(𝑄 ×c 𝑂))
5213, 12, 51, 38, 40, 48prf1 17237 . . . . 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⟩)⟩)
5312, 27, 37, 48cofu1 16940 . . . . . . 7 (𝜑 → ((1st ‘(⟨(1st𝑌), tpos (2nd𝑌)⟩ ∘func (𝑄 2ndF 𝑂)))‘⟨𝐹, 𝑋⟩) = ((1st ‘⟨(1st𝑌), tpos (2nd𝑌)⟩)‘((1st ‘(𝑄 2ndF 𝑂))‘⟨𝐹, 𝑋⟩)))
54 fvex 6461 . . . . . . . . . 10 (1st𝑌) ∈ V
55 fvex 6461 . . . . . . . . . . 11 (2nd𝑌) ∈ V
5655tposex 7670 . . . . . . . . . 10 tpos (2nd𝑌) ∈ V
5754, 56op1st 7455 . . . . . . . . 9 (1st ‘⟨(1st𝑌), tpos (2nd𝑌)⟩) = (1st𝑌)
5857a1i 11 . . . . . . . 8 (𝜑 → (1st ‘⟨(1st𝑌), tpos (2nd𝑌)⟩) = (1st𝑌))
596, 12, 51, 25, 17, 26, 482ndf1 17232 . . . . . . . . 9 (𝜑 → ((1st ‘(𝑄 2ndF 𝑂))‘⟨𝐹, 𝑋⟩) = (2nd ‘⟨𝐹, 𝑋⟩))
60 op2ndg 7460 . . . . . . . . . 10 ((𝐹 ∈ (𝑂 Func 𝑆) ∧ 𝑋𝐵) → (2nd ‘⟨𝐹, 𝑋⟩) = 𝑋)
6146, 47, 60syl2anc 579 . . . . . . . . 9 (𝜑 → (2nd ‘⟨𝐹, 𝑋⟩) = 𝑋)
6259, 61eqtrd 2814 . . . . . . . 8 (𝜑 → ((1st ‘(𝑄 2ndF 𝑂))‘⟨𝐹, 𝑋⟩) = 𝑋)
6358, 62fveq12d 6455 . . . . . . 7 (𝜑 → ((1st ‘⟨(1st𝑌), tpos (2nd𝑌)⟩)‘((1st ‘(𝑄 2ndF 𝑂))‘⟨𝐹, 𝑋⟩)) = ((1st𝑌)‘𝑋))
6453, 63eqtrd 2814 . . . . . 6 (𝜑 → ((1st ‘(⟨(1st𝑌), tpos (2nd𝑌)⟩ ∘func (𝑄 2ndF 𝑂)))‘⟨𝐹, 𝑋⟩) = ((1st𝑌)‘𝑋))
656, 12, 51, 25, 17, 39, 481stf1 17229 . . . . . . 7 (𝜑 → ((1st ‘(𝑄 1stF 𝑂))‘⟨𝐹, 𝑋⟩) = (1st ‘⟨𝐹, 𝑋⟩))
66 op1stg 7459 . . . . . . . 8 ((𝐹 ∈ (𝑂 Func 𝑆) ∧ 𝑋𝐵) → (1st ‘⟨𝐹, 𝑋⟩) = 𝐹)
6746, 47, 66syl2anc 579 . . . . . . 7 (𝜑 → (1st ‘⟨𝐹, 𝑋⟩) = 𝐹)
6865, 67eqtrd 2814 . . . . . 6 (𝜑 → ((1st ‘(𝑄 1stF 𝑂))‘⟨𝐹, 𝑋⟩) = 𝐹)
6964, 68opeq12d 4646 . . . . 5 (𝜑 → ⟨((1st ‘(⟨(1st𝑌), tpos (2nd𝑌)⟩ ∘func (𝑄 2ndF 𝑂)))‘⟨𝐹, 𝑋⟩), ((1st ‘(𝑄 1stF 𝑂))‘⟨𝐹, 𝑋⟩)⟩ = ⟨((1st𝑌)‘𝑋), 𝐹⟩)
7052, 69eqtrd 2814 . . . 4 (𝜑 → ((1st ‘((⟨(1st𝑌), tpos (2nd𝑌)⟩ ∘func (𝑄 2ndF 𝑂)) ⟨,⟩F (𝑄 1stF 𝑂)))‘⟨𝐹, 𝑋⟩) = ⟨((1st𝑌)‘𝑋), 𝐹⟩)
7170fveq2d 6452 . . 3 (𝜑 → ((1st𝐻)‘((1st ‘((⟨(1st𝑌), tpos (2nd𝑌)⟩ ∘func (𝑄 2ndF 𝑂)) ⟨,⟩F (𝑄 1stF 𝑂)))‘⟨𝐹, 𝑋⟩)) = ((1st𝐻)‘⟨((1st𝑌)‘𝑋), 𝐹⟩))
72 df-ov 6927 . . 3 (((1st𝑌)‘𝑋)(1st𝐻)𝐹) = ((1st𝐻)‘⟨((1st𝑌)‘𝑋), 𝐹⟩)
7371, 72syl6eqr 2832 . 2 (𝜑 → ((1st𝐻)‘((1st ‘((⟨(1st𝑌), tpos (2nd𝑌)⟩ ∘func (𝑄 2ndF 𝑂)) ⟨,⟩F (𝑄 1stF 𝑂)))‘⟨𝐹, 𝑋⟩)) = (((1st𝑌)‘𝑋)(1st𝐻)𝐹))
74 eqid 2778 . . . 4 (𝑂 Nat 𝑆) = (𝑂 Nat 𝑆)
757, 74fuchom 17017 . . 3 (𝑂 Nat 𝑆) = (Hom ‘𝑄)
7630, 10, 15, 47, 9, 22, 21, 31yon1cl 17300 . . 3 (𝜑 → ((1st𝑌)‘𝑋) ∈ (𝑂 Func 𝑆))
7742, 25, 8, 75, 76, 46hof1 17291 . 2 (𝜑 → (((1st𝑌)‘𝑋)(1st𝐻)𝐹) = (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹))
7850, 73, 773eqtrd 2818 1 (𝜑 → (𝐹(1st𝑍)𝑋) = (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1601  wcel 2107  Vcvv 3398  cun 3790  wss 3792  cop 4404   class class class wbr 4888   × cxp 5355  ran crn 5358  Rel wrel 5362  cfv 6137  (class class class)co 6924  1st c1st 7445  2nd c2nd 7446  tpos ctpos 7635  Basecbs 16266  Hom chom 16360  Catccat 16721  Idccid 16722  Homf chomf 16723  oppCatcoppc 16767   Func cfunc 16910  func ccofu 16912   Nat cnat 16997   FuncCat cfuc 16998  SetCatcsetc 17121   ×c cxpc 17205   1stF c1stf 17206   2ndF c2ndf 17207   ⟨,⟩F cprf 17208   evalF cevlf 17246  HomFchof 17285  Yoncyon 17286
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-rep 5008  ax-sep 5019  ax-nul 5027  ax-pow 5079  ax-pr 5140  ax-un 7228  ax-cnex 10330  ax-resscn 10331  ax-1cn 10332  ax-icn 10333  ax-addcl 10334  ax-addrcl 10335  ax-mulcl 10336  ax-mulrcl 10337  ax-mulcom 10338  ax-addass 10339  ax-mulass 10340  ax-distr 10341  ax-i2m1 10342  ax-1ne0 10343  ax-1rid 10344  ax-rnegex 10345  ax-rrecex 10346  ax-cnre 10347  ax-pre-lttri 10348  ax-pre-lttrn 10349  ax-pre-ltadd 10350  ax-pre-mulgt0 10351
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3or 1072  df-3an 1073  df-tru 1605  df-fal 1615  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-nel 3076  df-ral 3095  df-rex 3096  df-reu 3097  df-rmo 3098  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-pss 3808  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-tp 4403  df-op 4405  df-uni 4674  df-int 4713  df-iun 4757  df-br 4889  df-opab 4951  df-mpt 4968  df-tr 4990  df-id 5263  df-eprel 5268  df-po 5276  df-so 5277  df-fr 5316  df-we 5318  df-xp 5363  df-rel 5364  df-cnv 5365  df-co 5366  df-dm 5367  df-rn 5368  df-res 5369  df-ima 5370  df-pred 5935  df-ord 5981  df-on 5982  df-lim 5983  df-suc 5984  df-iota 6101  df-fun 6139  df-fn 6140  df-f 6141  df-f1 6142  df-fo 6143  df-f1o 6144  df-fv 6145  df-riota 6885  df-ov 6927  df-oprab 6928  df-mpt2 6929  df-om 7346  df-1st 7447  df-2nd 7448  df-tpos 7636  df-wrecs 7691  df-recs 7753  df-rdg 7791  df-1o 7845  df-oadd 7849  df-er 8028  df-map 8144  df-ixp 8197  df-en 8244  df-dom 8245  df-sdom 8246  df-fin 8247  df-pnf 10415  df-mnf 10416  df-xr 10417  df-ltxr 10418  df-le 10419  df-sub 10610  df-neg 10611  df-nn 11380  df-2 11443  df-3 11444  df-4 11445  df-5 11446  df-6 11447  df-7 11448  df-8 11449  df-9 11450  df-n0 11648  df-z 11734  df-dec 11851  df-uz 11998  df-fz 12649  df-struct 16268  df-ndx 16269  df-slot 16270  df-base 16272  df-sets 16273  df-hom 16373  df-cco 16374  df-cat 16725  df-cid 16726  df-homf 16727  df-comf 16728  df-oppc 16768  df-func 16914  df-cofu 16916  df-nat 16999  df-fuc 17000  df-setc 17122  df-xpc 17209  df-1stf 17210  df-2ndf 17211  df-prf 17212  df-curf 17251  df-hof 17287  df-yon 17288
This theorem is referenced by:  yonedalem3a  17311  yonedalem3b  17316  yonedainv  17318  yonffthlem  17319
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