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Theorem yonedalem21 18330
Description: Lemma for yoneda 18340. (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 6910 . . . . 5 (1st𝑍) = (1st ‘(𝐻func ((⟨(1st𝑌), tpos (2nd𝑌)⟩ ∘func (𝑄 2ndF 𝑂)) ⟨,⟩F (𝑄 1stF 𝑂))))
32oveqi 7444 . . . 4 (𝐹(1st𝑍)𝑋) = (𝐹(1st ‘(𝐻func ((⟨(1st𝑌), tpos (2nd𝑌)⟩ ∘func (𝑄 2ndF 𝑂)) ⟨,⟩F (𝑄 1stF 𝑂))))𝑋)
4 df-ov 7434 . . . 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 2763 . . 3 (𝐹(1st𝑍)𝑋) = ((1st ‘(𝐻func ((⟨(1st𝑌), tpos (2nd𝑌)⟩ ∘func (𝑄 2ndF 𝑂)) ⟨,⟩F (𝑄 1stF 𝑂))))‘⟨𝐹, 𝑋⟩)
6 eqid 2735 . . . . 5 (𝑄 ×c 𝑂) = (𝑄 ×c 𝑂)
7 yoneda.q . . . . . 6 𝑄 = (𝑂 FuncCat 𝑆)
87fucbas 18016 . . . . 5 (𝑂 Func 𝑆) = (Base‘𝑄)
9 yoneda.o . . . . . 6 𝑂 = (oppCat‘𝐶)
10 yoneda.b . . . . . 6 𝐵 = (Base‘𝐶)
119, 10oppcbas 17764 . . . . 5 𝐵 = (Base‘𝑂)
126, 8, 11xpcbas 18234 . . . 4 ((𝑂 Func 𝑆) × 𝐵) = (Base‘(𝑄 ×c 𝑂))
13 eqid 2735 . . . . 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 2735 . . . . 5 ((oppCat‘𝑄) ×c 𝑄) = ((oppCat‘𝑄) ×c 𝑄)
15 yoneda.c . . . . . . . . 9 (𝜑𝐶 ∈ Cat)
169oppccat 17769 . . . . . . . . 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 4204 . . . . . . . . . 10 (𝜑𝑈𝑉)
2118, 20ssexd 5330 . . . . . . . . 9 (𝜑𝑈 ∈ V)
22 yoneda.s . . . . . . . . . 10 𝑆 = (SetCat‘𝑈)
2322setccat 18139 . . . . . . . . 9 (𝑈 ∈ V → 𝑆 ∈ Cat)
2421, 23syl 17 . . . . . . . 8 (𝜑𝑆 ∈ Cat)
257, 17, 24fuccat 18027 . . . . . . 7 (𝜑𝑄 ∈ Cat)
26 eqid 2735 . . . . . . 7 (𝑄 2ndF 𝑂) = (𝑄 2ndF 𝑂)
276, 25, 17, 262ndfcl 18254 . . . . . 6 (𝜑 → (𝑄 2ndF 𝑂) ∈ ((𝑄 ×c 𝑂) Func 𝑂))
28 eqid 2735 . . . . . . . 8 (oppCat‘𝑄) = (oppCat‘𝑄)
29 relfunc 17913 . . . . . . . . 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 18319 . . . . . . . . 9 (𝜑𝑌 ∈ (𝐶 Func 𝑄))
33 1st2ndbr 8066 . . . . . . . . 9 ((Rel (𝐶 Func 𝑄) ∧ 𝑌 ∈ (𝐶 Func 𝑄)) → (1st𝑌)(𝐶 Func 𝑄)(2nd𝑌))
3429, 32, 33sylancr 587 . . . . . . . 8 (𝜑 → (1st𝑌)(𝐶 Func 𝑄)(2nd𝑌))
359, 28, 34funcoppc 17926 . . . . . . 7 (𝜑 → (1st𝑌)(𝑂 Func (oppCat‘𝑄))tpos (2nd𝑌))
36 df-br 5149 . . . . . . 7 ((1st𝑌)(𝑂 Func (oppCat‘𝑄))tpos (2nd𝑌) ↔ ⟨(1st𝑌), tpos (2nd𝑌)⟩ ∈ (𝑂 Func (oppCat‘𝑄)))
3735, 36sylib 218 . . . . . 6 (𝜑 → ⟨(1st𝑌), tpos (2nd𝑌)⟩ ∈ (𝑂 Func (oppCat‘𝑄)))
3827, 37cofucl 17939 . . . . 5 (𝜑 → (⟨(1st𝑌), tpos (2nd𝑌)⟩ ∘func (𝑄 2ndF 𝑂)) ∈ ((𝑄 ×c 𝑂) Func (oppCat‘𝑄)))
39 eqid 2735 . . . . . 6 (𝑄 1stF 𝑂) = (𝑄 1stF 𝑂)
406, 25, 17, 391stfcl 18253 . . . . 5 (𝜑 → (𝑄 1stF 𝑂) ∈ ((𝑄 ×c 𝑂) Func 𝑄))
4113, 14, 38, 40prfcl 18259 . . . 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 4203 . . . . 5 (𝜑 → ran (Homf𝑄) ⊆ 𝑉)
4542, 28, 43, 25, 18, 44hofcl 18316 . . . 4 (𝜑𝐻 ∈ (((oppCat‘𝑄) ×c 𝑄) Func 𝑇))
46 yonedalem21.f . . . . 5 (𝜑𝐹 ∈ (𝑂 Func 𝑆))
47 yonedalem21.x . . . . 5 (𝜑𝑋𝐵)
4846, 47opelxpd 5728 . . . 4 (𝜑 → ⟨𝐹, 𝑋⟩ ∈ ((𝑂 Func 𝑆) × 𝐵))
4912, 41, 45, 48cofu1 17935 . . 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, 49eqtrid 2787 . 2 (𝜑 → (𝐹(1st𝑍)𝑋) = ((1st𝐻)‘((1st ‘((⟨(1st𝑌), tpos (2nd𝑌)⟩ ∘func (𝑄 2ndF 𝑂)) ⟨,⟩F (𝑄 1stF 𝑂)))‘⟨𝐹, 𝑋⟩)))
51 eqid 2735 . . . . . 6 (Hom ‘(𝑄 ×c 𝑂)) = (Hom ‘(𝑄 ×c 𝑂))
5213, 12, 51, 38, 40, 48prf1 18256 . . . . 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 17935 . . . . . . 7 (𝜑 → ((1st ‘(⟨(1st𝑌), tpos (2nd𝑌)⟩ ∘func (𝑄 2ndF 𝑂)))‘⟨𝐹, 𝑋⟩) = ((1st ‘⟨(1st𝑌), tpos (2nd𝑌)⟩)‘((1st ‘(𝑄 2ndF 𝑂))‘⟨𝐹, 𝑋⟩)))
54 fvex 6920 . . . . . . . . . 10 (1st𝑌) ∈ V
55 fvex 6920 . . . . . . . . . . 11 (2nd𝑌) ∈ V
5655tposex 8284 . . . . . . . . . 10 tpos (2nd𝑌) ∈ V
5754, 56op1st 8021 . . . . . . . . 9 (1st ‘⟨(1st𝑌), tpos (2nd𝑌)⟩) = (1st𝑌)
5857a1i 11 . . . . . . . 8 (𝜑 → (1st ‘⟨(1st𝑌), tpos (2nd𝑌)⟩) = (1st𝑌))
596, 12, 51, 25, 17, 26, 482ndf1 18251 . . . . . . . . 9 (𝜑 → ((1st ‘(𝑄 2ndF 𝑂))‘⟨𝐹, 𝑋⟩) = (2nd ‘⟨𝐹, 𝑋⟩))
60 op2ndg 8026 . . . . . . . . . 10 ((𝐹 ∈ (𝑂 Func 𝑆) ∧ 𝑋𝐵) → (2nd ‘⟨𝐹, 𝑋⟩) = 𝑋)
6146, 47, 60syl2anc 584 . . . . . . . . 9 (𝜑 → (2nd ‘⟨𝐹, 𝑋⟩) = 𝑋)
6259, 61eqtrd 2775 . . . . . . . 8 (𝜑 → ((1st ‘(𝑄 2ndF 𝑂))‘⟨𝐹, 𝑋⟩) = 𝑋)
6358, 62fveq12d 6914 . . . . . . 7 (𝜑 → ((1st ‘⟨(1st𝑌), tpos (2nd𝑌)⟩)‘((1st ‘(𝑄 2ndF 𝑂))‘⟨𝐹, 𝑋⟩)) = ((1st𝑌)‘𝑋))
6453, 63eqtrd 2775 . . . . . 6 (𝜑 → ((1st ‘(⟨(1st𝑌), tpos (2nd𝑌)⟩ ∘func (𝑄 2ndF 𝑂)))‘⟨𝐹, 𝑋⟩) = ((1st𝑌)‘𝑋))
656, 12, 51, 25, 17, 39, 481stf1 18248 . . . . . . 7 (𝜑 → ((1st ‘(𝑄 1stF 𝑂))‘⟨𝐹, 𝑋⟩) = (1st ‘⟨𝐹, 𝑋⟩))
66 op1stg 8025 . . . . . . . 8 ((𝐹 ∈ (𝑂 Func 𝑆) ∧ 𝑋𝐵) → (1st ‘⟨𝐹, 𝑋⟩) = 𝐹)
6746, 47, 66syl2anc 584 . . . . . . 7 (𝜑 → (1st ‘⟨𝐹, 𝑋⟩) = 𝐹)
6865, 67eqtrd 2775 . . . . . 6 (𝜑 → ((1st ‘(𝑄 1stF 𝑂))‘⟨𝐹, 𝑋⟩) = 𝐹)
6964, 68opeq12d 4886 . . . . 5 (𝜑 → ⟨((1st ‘(⟨(1st𝑌), tpos (2nd𝑌)⟩ ∘func (𝑄 2ndF 𝑂)))‘⟨𝐹, 𝑋⟩), ((1st ‘(𝑄 1stF 𝑂))‘⟨𝐹, 𝑋⟩)⟩ = ⟨((1st𝑌)‘𝑋), 𝐹⟩)
7052, 69eqtrd 2775 . . . 4 (𝜑 → ((1st ‘((⟨(1st𝑌), tpos (2nd𝑌)⟩ ∘func (𝑄 2ndF 𝑂)) ⟨,⟩F (𝑄 1stF 𝑂)))‘⟨𝐹, 𝑋⟩) = ⟨((1st𝑌)‘𝑋), 𝐹⟩)
7170fveq2d 6911 . . 3 (𝜑 → ((1st𝐻)‘((1st ‘((⟨(1st𝑌), tpos (2nd𝑌)⟩ ∘func (𝑄 2ndF 𝑂)) ⟨,⟩F (𝑄 1stF 𝑂)))‘⟨𝐹, 𝑋⟩)) = ((1st𝐻)‘⟨((1st𝑌)‘𝑋), 𝐹⟩))
72 df-ov 7434 . . 3 (((1st𝑌)‘𝑋)(1st𝐻)𝐹) = ((1st𝐻)‘⟨((1st𝑌)‘𝑋), 𝐹⟩)
7371, 72eqtr4di 2793 . 2 (𝜑 → ((1st𝐻)‘((1st ‘((⟨(1st𝑌), tpos (2nd𝑌)⟩ ∘func (𝑄 2ndF 𝑂)) ⟨,⟩F (𝑄 1stF 𝑂)))‘⟨𝐹, 𝑋⟩)) = (((1st𝑌)‘𝑋)(1st𝐻)𝐹))
74 eqid 2735 . . . 4 (𝑂 Nat 𝑆) = (𝑂 Nat 𝑆)
757, 74fuchom 18017 . . 3 (𝑂 Nat 𝑆) = (Hom ‘𝑄)
7630, 10, 15, 47, 9, 22, 21, 31yon1cl 18320 . . 3 (𝜑 → ((1st𝑌)‘𝑋) ∈ (𝑂 Func 𝑆))
7742, 25, 8, 75, 76, 46hof1 18311 . 2 (𝜑 → (((1st𝑌)‘𝑋)(1st𝐻)𝐹) = (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹))
7850, 73, 773eqtrd 2779 1 (𝜑 → (𝐹(1st𝑍)𝑋) = (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1537  wcel 2106  Vcvv 3478  cun 3961  wss 3963  cop 4637   class class class wbr 5148   × cxp 5687  ran crn 5690  Rel wrel 5694  cfv 6563  (class class class)co 7431  1st c1st 8011  2nd c2nd 8012  tpos ctpos 8249  Basecbs 17245  Hom chom 17309  Catccat 17709  Idccid 17710  Homf chomf 17711  oppCatcoppc 17756   Func cfunc 17905  func ccofu 17907   Nat cnat 17996   FuncCat cfuc 17997  SetCatcsetc 18129   ×c cxpc 18224   1stF c1stf 18225   2ndF c2ndf 18226   ⟨,⟩F cprf 18227   evalF cevlf 18266  HomFchof 18305  Yoncyon 18306
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-tp 4636  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-tpos 8250  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-er 8744  df-map 8867  df-ixp 8937  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-nn 12265  df-2 12327  df-3 12328  df-4 12329  df-5 12330  df-6 12331  df-7 12332  df-8 12333  df-9 12334  df-n0 12525  df-z 12612  df-dec 12732  df-uz 12877  df-fz 13545  df-struct 17181  df-sets 17198  df-slot 17216  df-ndx 17228  df-base 17246  df-hom 17322  df-cco 17323  df-cat 17713  df-cid 17714  df-homf 17715  df-comf 17716  df-oppc 17757  df-func 17909  df-cofu 17911  df-nat 17998  df-fuc 17999  df-setc 18130  df-xpc 18228  df-1stf 18229  df-2ndf 18230  df-prf 18231  df-curf 18271  df-hof 18307  df-yon 18308
This theorem is referenced by:  yonedalem3a  18331  yonedalem3b  18336  yonedainv  18338  yonffthlem  18339
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