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Theorem yon12 18257
Description: Value of the Yoneda embedding at a morphism. The partially evaluated Yoneda embedding is also the contravariant Hom functor. (Contributed by Mario Carneiro, 17-Jan-2017.)
Hypotheses
Ref Expression
yon11.y π‘Œ = (Yonβ€˜πΆ)
yon11.b 𝐡 = (Baseβ€˜πΆ)
yon11.c (πœ‘ β†’ 𝐢 ∈ Cat)
yon11.p (πœ‘ β†’ 𝑋 ∈ 𝐡)
yon11.h 𝐻 = (Hom β€˜πΆ)
yon11.z (πœ‘ β†’ 𝑍 ∈ 𝐡)
yon12.x Β· = (compβ€˜πΆ)
yon12.w (πœ‘ β†’ π‘Š ∈ 𝐡)
yon12.f (πœ‘ β†’ 𝐹 ∈ (π‘Šπ»π‘))
yon12.g (πœ‘ β†’ 𝐺 ∈ (𝑍𝐻𝑋))
Assertion
Ref Expression
yon12 (πœ‘ β†’ (((𝑍(2nd β€˜((1st β€˜π‘Œ)β€˜π‘‹))π‘Š)β€˜πΉ)β€˜πΊ) = (𝐺(βŸ¨π‘Š, π‘βŸ© Β· 𝑋)𝐹))

Proof of Theorem yon12
StepHypRef Expression
1 yon11.y . . . . . . . . . 10 π‘Œ = (Yonβ€˜πΆ)
2 yon11.c . . . . . . . . . 10 (πœ‘ β†’ 𝐢 ∈ Cat)
3 eqid 2728 . . . . . . . . . 10 (oppCatβ€˜πΆ) = (oppCatβ€˜πΆ)
4 eqid 2728 . . . . . . . . . 10 (HomFβ€˜(oppCatβ€˜πΆ)) = (HomFβ€˜(oppCatβ€˜πΆ))
51, 2, 3, 4yonval 18253 . . . . . . . . 9 (πœ‘ β†’ π‘Œ = (⟨𝐢, (oppCatβ€˜πΆ)⟩ curryF (HomFβ€˜(oppCatβ€˜πΆ))))
65fveq2d 6901 . . . . . . . 8 (πœ‘ β†’ (1st β€˜π‘Œ) = (1st β€˜(⟨𝐢, (oppCatβ€˜πΆ)⟩ curryF (HomFβ€˜(oppCatβ€˜πΆ)))))
76fveq1d 6899 . . . . . . 7 (πœ‘ β†’ ((1st β€˜π‘Œ)β€˜π‘‹) = ((1st β€˜(⟨𝐢, (oppCatβ€˜πΆ)⟩ curryF (HomFβ€˜(oppCatβ€˜πΆ))))β€˜π‘‹))
87fveq2d 6901 . . . . . 6 (πœ‘ β†’ (2nd β€˜((1st β€˜π‘Œ)β€˜π‘‹)) = (2nd β€˜((1st β€˜(⟨𝐢, (oppCatβ€˜πΆ)⟩ curryF (HomFβ€˜(oppCatβ€˜πΆ))))β€˜π‘‹)))
98oveqd 7437 . . . . 5 (πœ‘ β†’ (𝑍(2nd β€˜((1st β€˜π‘Œ)β€˜π‘‹))π‘Š) = (𝑍(2nd β€˜((1st β€˜(⟨𝐢, (oppCatβ€˜πΆ)⟩ curryF (HomFβ€˜(oppCatβ€˜πΆ))))β€˜π‘‹))π‘Š))
109fveq1d 6899 . . . 4 (πœ‘ β†’ ((𝑍(2nd β€˜((1st β€˜π‘Œ)β€˜π‘‹))π‘Š)β€˜πΉ) = ((𝑍(2nd β€˜((1st β€˜(⟨𝐢, (oppCatβ€˜πΆ)⟩ curryF (HomFβ€˜(oppCatβ€˜πΆ))))β€˜π‘‹))π‘Š)β€˜πΉ))
11 eqid 2728 . . . . 5 (⟨𝐢, (oppCatβ€˜πΆ)⟩ curryF (HomFβ€˜(oppCatβ€˜πΆ))) = (⟨𝐢, (oppCatβ€˜πΆ)⟩ curryF (HomFβ€˜(oppCatβ€˜πΆ)))
12 yon11.b . . . . 5 𝐡 = (Baseβ€˜πΆ)
133oppccat 17704 . . . . . 6 (𝐢 ∈ Cat β†’ (oppCatβ€˜πΆ) ∈ Cat)
142, 13syl 17 . . . . 5 (πœ‘ β†’ (oppCatβ€˜πΆ) ∈ Cat)
15 eqid 2728 . . . . . 6 (SetCatβ€˜ran (Homf β€˜πΆ)) = (SetCatβ€˜ran (Homf β€˜πΆ))
16 fvex 6910 . . . . . . . 8 (Homf β€˜πΆ) ∈ V
1716rnex 7918 . . . . . . 7 ran (Homf β€˜πΆ) ∈ V
1817a1i 11 . . . . . 6 (πœ‘ β†’ ran (Homf β€˜πΆ) ∈ V)
19 ssidd 4003 . . . . . 6 (πœ‘ β†’ ran (Homf β€˜πΆ) βŠ† ran (Homf β€˜πΆ))
203, 4, 15, 2, 18, 19oppchofcl 18252 . . . . 5 (πœ‘ β†’ (HomFβ€˜(oppCatβ€˜πΆ)) ∈ ((𝐢 Γ—c (oppCatβ€˜πΆ)) Func (SetCatβ€˜ran (Homf β€˜πΆ))))
213, 12oppcbas 17699 . . . . 5 𝐡 = (Baseβ€˜(oppCatβ€˜πΆ))
22 yon11.p . . . . 5 (πœ‘ β†’ 𝑋 ∈ 𝐡)
23 eqid 2728 . . . . 5 ((1st β€˜(⟨𝐢, (oppCatβ€˜πΆ)⟩ curryF (HomFβ€˜(oppCatβ€˜πΆ))))β€˜π‘‹) = ((1st β€˜(⟨𝐢, (oppCatβ€˜πΆ)⟩ curryF (HomFβ€˜(oppCatβ€˜πΆ))))β€˜π‘‹)
24 yon11.z . . . . 5 (πœ‘ β†’ 𝑍 ∈ 𝐡)
25 eqid 2728 . . . . 5 (Hom β€˜(oppCatβ€˜πΆ)) = (Hom β€˜(oppCatβ€˜πΆ))
26 eqid 2728 . . . . 5 (Idβ€˜πΆ) = (Idβ€˜πΆ)
27 yon12.w . . . . 5 (πœ‘ β†’ π‘Š ∈ 𝐡)
28 yon12.f . . . . . 6 (πœ‘ β†’ 𝐹 ∈ (π‘Šπ»π‘))
29 yon11.h . . . . . . 7 𝐻 = (Hom β€˜πΆ)
3029, 3oppchom 17696 . . . . . 6 (𝑍(Hom β€˜(oppCatβ€˜πΆ))π‘Š) = (π‘Šπ»π‘)
3128, 30eleqtrrdi 2840 . . . . 5 (πœ‘ β†’ 𝐹 ∈ (𝑍(Hom β€˜(oppCatβ€˜πΆ))π‘Š))
3211, 12, 2, 14, 20, 21, 22, 23, 24, 25, 26, 27, 31curf12 18219 . . . 4 (πœ‘ β†’ ((𝑍(2nd β€˜((1st β€˜(⟨𝐢, (oppCatβ€˜πΆ)⟩ curryF (HomFβ€˜(oppCatβ€˜πΆ))))β€˜π‘‹))π‘Š)β€˜πΉ) = (((Idβ€˜πΆ)β€˜π‘‹)(βŸ¨π‘‹, π‘βŸ©(2nd β€˜(HomFβ€˜(oppCatβ€˜πΆ)))βŸ¨π‘‹, π‘ŠβŸ©)𝐹))
3310, 32eqtrd 2768 . . 3 (πœ‘ β†’ ((𝑍(2nd β€˜((1st β€˜π‘Œ)β€˜π‘‹))π‘Š)β€˜πΉ) = (((Idβ€˜πΆ)β€˜π‘‹)(βŸ¨π‘‹, π‘βŸ©(2nd β€˜(HomFβ€˜(oppCatβ€˜πΆ)))βŸ¨π‘‹, π‘ŠβŸ©)𝐹))
3433fveq1d 6899 . 2 (πœ‘ β†’ (((𝑍(2nd β€˜((1st β€˜π‘Œ)β€˜π‘‹))π‘Š)β€˜πΉ)β€˜πΊ) = ((((Idβ€˜πΆ)β€˜π‘‹)(βŸ¨π‘‹, π‘βŸ©(2nd β€˜(HomFβ€˜(oppCatβ€˜πΆ)))βŸ¨π‘‹, π‘ŠβŸ©)𝐹)β€˜πΊ))
35 eqid 2728 . . 3 (compβ€˜(oppCatβ€˜πΆ)) = (compβ€˜(oppCatβ€˜πΆ))
3612, 29, 26, 2, 22catidcl 17662 . . . 4 (πœ‘ β†’ ((Idβ€˜πΆ)β€˜π‘‹) ∈ (𝑋𝐻𝑋))
3729, 3oppchom 17696 . . . 4 (𝑋(Hom β€˜(oppCatβ€˜πΆ))𝑋) = (𝑋𝐻𝑋)
3836, 37eleqtrrdi 2840 . . 3 (πœ‘ β†’ ((Idβ€˜πΆ)β€˜π‘‹) ∈ (𝑋(Hom β€˜(oppCatβ€˜πΆ))𝑋))
39 yon12.g . . . 4 (πœ‘ β†’ 𝐺 ∈ (𝑍𝐻𝑋))
4029, 3oppchom 17696 . . . 4 (𝑋(Hom β€˜(oppCatβ€˜πΆ))𝑍) = (𝑍𝐻𝑋)
4139, 40eleqtrrdi 2840 . . 3 (πœ‘ β†’ 𝐺 ∈ (𝑋(Hom β€˜(oppCatβ€˜πΆ))𝑍))
424, 14, 21, 25, 22, 24, 22, 27, 35, 38, 31, 41hof2 18249 . 2 (πœ‘ β†’ ((((Idβ€˜πΆ)β€˜π‘‹)(βŸ¨π‘‹, π‘βŸ©(2nd β€˜(HomFβ€˜(oppCatβ€˜πΆ)))βŸ¨π‘‹, π‘ŠβŸ©)𝐹)β€˜πΊ) = ((𝐹(βŸ¨π‘‹, π‘βŸ©(compβ€˜(oppCatβ€˜πΆ))π‘Š)𝐺)(βŸ¨π‘‹, π‘‹βŸ©(compβ€˜(oppCatβ€˜πΆ))π‘Š)((Idβ€˜πΆ)β€˜π‘‹)))
43 yon12.x . . . . 5 Β· = (compβ€˜πΆ)
4412, 43, 3, 22, 24, 27oppcco 17698 . . . 4 (πœ‘ β†’ (𝐹(βŸ¨π‘‹, π‘βŸ©(compβ€˜(oppCatβ€˜πΆ))π‘Š)𝐺) = (𝐺(βŸ¨π‘Š, π‘βŸ© Β· 𝑋)𝐹))
4544oveq1d 7435 . . 3 (πœ‘ β†’ ((𝐹(βŸ¨π‘‹, π‘βŸ©(compβ€˜(oppCatβ€˜πΆ))π‘Š)𝐺)(βŸ¨π‘‹, π‘‹βŸ©(compβ€˜(oppCatβ€˜πΆ))π‘Š)((Idβ€˜πΆ)β€˜π‘‹)) = ((𝐺(βŸ¨π‘Š, π‘βŸ© Β· 𝑋)𝐹)(βŸ¨π‘‹, π‘‹βŸ©(compβ€˜(oppCatβ€˜πΆ))π‘Š)((Idβ€˜πΆ)β€˜π‘‹)))
4612, 43, 3, 22, 22, 27oppcco 17698 . . 3 (πœ‘ β†’ ((𝐺(βŸ¨π‘Š, π‘βŸ© Β· 𝑋)𝐹)(βŸ¨π‘‹, π‘‹βŸ©(compβ€˜(oppCatβ€˜πΆ))π‘Š)((Idβ€˜πΆ)β€˜π‘‹)) = (((Idβ€˜πΆ)β€˜π‘‹)(βŸ¨π‘Š, π‘‹βŸ© Β· 𝑋)(𝐺(βŸ¨π‘Š, π‘βŸ© Β· 𝑋)𝐹)))
4712, 29, 43, 2, 27, 24, 22, 28, 39catcocl 17665 . . . 4 (πœ‘ β†’ (𝐺(βŸ¨π‘Š, π‘βŸ© Β· 𝑋)𝐹) ∈ (π‘Šπ»π‘‹))
4812, 29, 26, 2, 27, 43, 22, 47catlid 17663 . . 3 (πœ‘ β†’ (((Idβ€˜πΆ)β€˜π‘‹)(βŸ¨π‘Š, π‘‹βŸ© Β· 𝑋)(𝐺(βŸ¨π‘Š, π‘βŸ© Β· 𝑋)𝐹)) = (𝐺(βŸ¨π‘Š, π‘βŸ© Β· 𝑋)𝐹))
4945, 46, 483eqtrd 2772 . 2 (πœ‘ β†’ ((𝐹(βŸ¨π‘‹, π‘βŸ©(compβ€˜(oppCatβ€˜πΆ))π‘Š)𝐺)(βŸ¨π‘‹, π‘‹βŸ©(compβ€˜(oppCatβ€˜πΆ))π‘Š)((Idβ€˜πΆ)β€˜π‘‹)) = (𝐺(βŸ¨π‘Š, π‘βŸ© Β· 𝑋)𝐹))
5034, 42, 493eqtrd 2772 1 (πœ‘ β†’ (((𝑍(2nd β€˜((1st β€˜π‘Œ)β€˜π‘‹))π‘Š)β€˜πΉ)β€˜πΊ) = (𝐺(βŸ¨π‘Š, π‘βŸ© Β· 𝑋)𝐹))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   = wceq 1534   ∈ wcel 2099  Vcvv 3471  βŸ¨cop 4635  ran crn 5679  β€˜cfv 6548  (class class class)co 7420  1st c1st 7991  2nd c2nd 7992  Basecbs 17180  Hom chom 17244  compcco 17245  Catccat 17644  Idccid 17645  Homf chomf 17646  oppCatcoppc 17691  SetCatcsetc 18064   curryF ccurf 18202  HomFchof 18240  Yoncyon 18241
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5365  ax-pr 5429  ax-un 7740  ax-cnex 11195  ax-resscn 11196  ax-1cn 11197  ax-icn 11198  ax-addcl 11199  ax-addrcl 11200  ax-mulcl 11201  ax-mulrcl 11202  ax-mulcom 11203  ax-addass 11204  ax-mulass 11205  ax-distr 11206  ax-i2m1 11207  ax-1ne0 11208  ax-1rid 11209  ax-rnegex 11210  ax-rrecex 11211  ax-cnre 11212  ax-pre-lttri 11213  ax-pre-lttrn 11214  ax-pre-ltadd 11215  ax-pre-mulgt0 11216
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-rmo 3373  df-reu 3374  df-rab 3430  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-tp 4634  df-op 4636  df-uni 4909  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-we 5635  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-pred 6305  df-ord 6372  df-on 6373  df-lim 6374  df-suc 6375  df-iota 6500  df-fun 6550  df-fn 6551  df-f 6552  df-f1 6553  df-fo 6554  df-f1o 6555  df-fv 6556  df-riota 7376  df-ov 7423  df-oprab 7424  df-mpo 7425  df-om 7871  df-1st 7993  df-2nd 7994  df-tpos 8232  df-frecs 8287  df-wrecs 8318  df-recs 8392  df-rdg 8431  df-1o 8487  df-er 8725  df-map 8847  df-ixp 8917  df-en 8965  df-dom 8966  df-sdom 8967  df-fin 8968  df-pnf 11281  df-mnf 11282  df-xr 11283  df-ltxr 11284  df-le 11285  df-sub 11477  df-neg 11478  df-nn 12244  df-2 12306  df-3 12307  df-4 12308  df-5 12309  df-6 12310  df-7 12311  df-8 12312  df-9 12313  df-n0 12504  df-z 12590  df-dec 12709  df-uz 12854  df-fz 13518  df-struct 17116  df-sets 17133  df-slot 17151  df-ndx 17163  df-base 17181  df-hom 17257  df-cco 17258  df-cat 17648  df-cid 17649  df-homf 17650  df-comf 17651  df-oppc 17692  df-func 17844  df-setc 18065  df-xpc 18163  df-curf 18206  df-hof 18242  df-yon 18243
This theorem is referenced by:  yonedalem4c  18269  yonedalem3b  18271  yonedainv  18273  yonffthlem  18274
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