Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  yon11 Structured version   Visualization version   GIF version

Theorem yon11 17526
 Description: Value of the Yoneda embedding at an object. The partially evaluated Yoneda embedding is also the contravariant Hom functor. (Contributed by Mario Carneiro, 17-Jan-2017.)
Hypotheses
Ref Expression
yon11.y 𝑌 = (Yon‘𝐶)
yon11.b 𝐵 = (Base‘𝐶)
yon11.c (𝜑𝐶 ∈ Cat)
yon11.p (𝜑𝑋𝐵)
yon11.h 𝐻 = (Hom ‘𝐶)
yon11.z (𝜑𝑍𝐵)
Assertion
Ref Expression
yon11 (𝜑 → ((1st ‘((1st𝑌)‘𝑋))‘𝑍) = (𝑍𝐻𝑋))

Proof of Theorem yon11
StepHypRef Expression
1 yon11.y . . . . . . 7 𝑌 = (Yon‘𝐶)
2 yon11.c . . . . . . 7 (𝜑𝐶 ∈ Cat)
3 eqid 2798 . . . . . . 7 (oppCat‘𝐶) = (oppCat‘𝐶)
4 eqid 2798 . . . . . . 7 (HomF‘(oppCat‘𝐶)) = (HomF‘(oppCat‘𝐶))
51, 2, 3, 4yonval 17523 . . . . . 6 (𝜑𝑌 = (⟨𝐶, (oppCat‘𝐶)⟩ curryF (HomF‘(oppCat‘𝐶))))
65fveq2d 6659 . . . . 5 (𝜑 → (1st𝑌) = (1st ‘(⟨𝐶, (oppCat‘𝐶)⟩ curryF (HomF‘(oppCat‘𝐶)))))
76fveq1d 6657 . . . 4 (𝜑 → ((1st𝑌)‘𝑋) = ((1st ‘(⟨𝐶, (oppCat‘𝐶)⟩ curryF (HomF‘(oppCat‘𝐶))))‘𝑋))
87fveq2d 6659 . . 3 (𝜑 → (1st ‘((1st𝑌)‘𝑋)) = (1st ‘((1st ‘(⟨𝐶, (oppCat‘𝐶)⟩ curryF (HomF‘(oppCat‘𝐶))))‘𝑋)))
98fveq1d 6657 . 2 (𝜑 → ((1st ‘((1st𝑌)‘𝑋))‘𝑍) = ((1st ‘((1st ‘(⟨𝐶, (oppCat‘𝐶)⟩ curryF (HomF‘(oppCat‘𝐶))))‘𝑋))‘𝑍))
10 eqid 2798 . . 3 (⟨𝐶, (oppCat‘𝐶)⟩ curryF (HomF‘(oppCat‘𝐶))) = (⟨𝐶, (oppCat‘𝐶)⟩ curryF (HomF‘(oppCat‘𝐶)))
11 yon11.b . . 3 𝐵 = (Base‘𝐶)
123oppccat 17004 . . . 4 (𝐶 ∈ Cat → (oppCat‘𝐶) ∈ Cat)
132, 12syl 17 . . 3 (𝜑 → (oppCat‘𝐶) ∈ Cat)
14 eqid 2798 . . . 4 (SetCat‘ran (Homf𝐶)) = (SetCat‘ran (Homf𝐶))
15 fvex 6668 . . . . . 6 (Homf𝐶) ∈ V
1615rnex 7612 . . . . 5 ran (Homf𝐶) ∈ V
1716a1i 11 . . . 4 (𝜑 → ran (Homf𝐶) ∈ V)
18 ssidd 3940 . . . 4 (𝜑 → ran (Homf𝐶) ⊆ ran (Homf𝐶))
193, 4, 14, 2, 17, 18oppchofcl 17522 . . 3 (𝜑 → (HomF‘(oppCat‘𝐶)) ∈ ((𝐶 ×c (oppCat‘𝐶)) Func (SetCat‘ran (Homf𝐶))))
203, 11oppcbas 17000 . . 3 𝐵 = (Base‘(oppCat‘𝐶))
21 yon11.p . . 3 (𝜑𝑋𝐵)
22 eqid 2798 . . 3 ((1st ‘(⟨𝐶, (oppCat‘𝐶)⟩ curryF (HomF‘(oppCat‘𝐶))))‘𝑋) = ((1st ‘(⟨𝐶, (oppCat‘𝐶)⟩ curryF (HomF‘(oppCat‘𝐶))))‘𝑋)
23 yon11.z . . 3 (𝜑𝑍𝐵)
2410, 11, 2, 13, 19, 20, 21, 22, 23curf11 17488 . 2 (𝜑 → ((1st ‘((1st ‘(⟨𝐶, (oppCat‘𝐶)⟩ curryF (HomF‘(oppCat‘𝐶))))‘𝑋))‘𝑍) = (𝑋(1st ‘(HomF‘(oppCat‘𝐶)))𝑍))
25 eqid 2798 . . . 4 (Hom ‘(oppCat‘𝐶)) = (Hom ‘(oppCat‘𝐶))
264, 13, 20, 25, 21, 23hof1 17516 . . 3 (𝜑 → (𝑋(1st ‘(HomF‘(oppCat‘𝐶)))𝑍) = (𝑋(Hom ‘(oppCat‘𝐶))𝑍))
27 yon11.h . . . 4 𝐻 = (Hom ‘𝐶)
2827, 3oppchom 16997 . . 3 (𝑋(Hom ‘(oppCat‘𝐶))𝑍) = (𝑍𝐻𝑋)
2926, 28eqtrdi 2849 . 2 (𝜑 → (𝑋(1st ‘(HomF‘(oppCat‘𝐶)))𝑍) = (𝑍𝐻𝑋))
309, 24, 293eqtrd 2837 1 (𝜑 → ((1st ‘((1st𝑌)‘𝑋))‘𝑍) = (𝑍𝐻𝑋))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1538   ∈ wcel 2111  Vcvv 3442  ⟨cop 4534  ran crn 5524  ‘cfv 6332  (class class class)co 7145  1st c1st 7682  Basecbs 16495  Hom chom 16588  Catccat 16947  Homf chomf 16949  oppCatcoppc 16993  SetCatcsetc 17347   curryF ccurf 17472  HomFchof 17510  Yoncyon 17511 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5158  ax-sep 5171  ax-nul 5178  ax-pow 5235  ax-pr 5299  ax-un 7454  ax-cnex 10600  ax-resscn 10601  ax-1cn 10602  ax-icn 10603  ax-addcl 10604  ax-addrcl 10605  ax-mulcl 10606  ax-mulrcl 10607  ax-mulcom 10608  ax-addass 10609  ax-mulass 10610  ax-distr 10611  ax-i2m1 10612  ax-1ne0 10613  ax-1rid 10614  ax-rnegex 10615  ax-rrecex 10616  ax-cnre 10617  ax-pre-lttri 10618  ax-pre-lttrn 10619  ax-pre-ltadd 10620  ax-pre-mulgt0 10621 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-nel 3092  df-ral 3111  df-rex 3112  df-reu 3113  df-rmo 3114  df-rab 3115  df-v 3444  df-sbc 3723  df-csb 3831  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-tp 4533  df-op 4535  df-uni 4805  df-int 4843  df-iun 4887  df-br 5035  df-opab 5097  df-mpt 5115  df-tr 5141  df-id 5429  df-eprel 5434  df-po 5442  df-so 5443  df-fr 5482  df-we 5484  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-pred 6123  df-ord 6169  df-on 6170  df-lim 6171  df-suc 6172  df-iota 6291  df-fun 6334  df-fn 6335  df-f 6336  df-f1 6337  df-fo 6338  df-f1o 6339  df-fv 6340  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-om 7574  df-1st 7684  df-2nd 7685  df-tpos 7893  df-wrecs 7948  df-recs 8009  df-rdg 8047  df-1o 8103  df-oadd 8107  df-er 8290  df-map 8409  df-ixp 8463  df-en 8511  df-dom 8512  df-sdom 8513  df-fin 8514  df-pnf 10684  df-mnf 10685  df-xr 10686  df-ltxr 10687  df-le 10688  df-sub 10879  df-neg 10880  df-nn 11644  df-2 11706  df-3 11707  df-4 11708  df-5 11709  df-6 11710  df-7 11711  df-8 11712  df-9 11713  df-n0 11904  df-z 11990  df-dec 12107  df-uz 12252  df-fz 12906  df-struct 16497  df-ndx 16498  df-slot 16499  df-base 16501  df-sets 16502  df-hom 16601  df-cco 16602  df-cat 16951  df-cid 16952  df-homf 16953  df-comf 16954  df-oppc 16994  df-func 17140  df-setc 17348  df-xpc 17434  df-curf 17476  df-hof 17512  df-yon 17513 This theorem is referenced by:  yonedalem3a  17536  yonedalem4c  17539  yonedalem3b  17541  yonedainv  17543  yonffthlem  17544  yoniso  17547
 Copyright terms: Public domain W3C validator