Theorem yon11 18261

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	⊢ (𝜑 → 𝐶 ∈ Cat)
yon11.p	⊢ (𝜑 → 𝑋 ∈ 𝐵)
yon11.h	⊢ 𝐻 = (Hom ‘𝐶)
yon11.z	⊢ (𝜑 → 𝑍 ∈ 𝐵)

Assertion

Ref	Expression
yon11	⊢ (𝜑 → ((1st ‘((1st ‘𝑌)‘𝑋))‘𝑍) = (𝑍𝐻𝑋))

Proof of Theorem yon11

Step	Hyp	Ref	Expression
1		yon11.y	. . . . . . 7 ⊢ 𝑌 = (Yon‘𝐶)
2		yon11.c	. . . . . . 7 ⊢ (𝜑 → 𝐶 ∈ Cat)
3		eqid 2734	. . . . . . 7 ⊢ (oppCat‘𝐶) = (oppCat‘𝐶)
4		eqid 2734	. . . . . . 7 ⊢ (HomF‘(oppCat‘𝐶)) = (HomF‘(oppCat‘𝐶))
5	1, 2, 3, 4	yonval 18258	. . . . . 6 ⊢ (𝜑 → 𝑌 = (⟨𝐶, (oppCat‘𝐶)⟩ curryF (HomF‘(oppCat‘𝐶))))
6	5	fveq2d 6876	. . . . 5 ⊢ (𝜑 → (1st ‘𝑌) = (1st ‘(⟨𝐶, (oppCat‘𝐶)⟩ curryF (HomF‘(oppCat‘𝐶)))))
7	6	fveq1d 6874	. . . 4 ⊢ (𝜑 → ((1st ‘𝑌)‘𝑋) = ((1st ‘(⟨𝐶, (oppCat‘𝐶)⟩ curryF (HomF‘(oppCat‘𝐶))))‘𝑋))
8	7	fveq2d 6876	. . 3 ⊢ (𝜑 → (1st ‘((1st ‘𝑌)‘𝑋)) = (1st ‘((1st ‘(⟨𝐶, (oppCat‘𝐶)⟩ curryF (HomF‘(oppCat‘𝐶))))‘𝑋)))
9	8	fveq1d 6874	. 2 ⊢ (𝜑 → ((1st ‘((1st ‘𝑌)‘𝑋))‘𝑍) = ((1st ‘((1st ‘(⟨𝐶, (oppCat‘𝐶)⟩ curryF (HomF‘(oppCat‘𝐶))))‘𝑋))‘𝑍))
10		eqid 2734	. . 3 ⊢ (⟨𝐶, (oppCat‘𝐶)⟩ curryF (HomF‘(oppCat‘𝐶))) = (⟨𝐶, (oppCat‘𝐶)⟩ curryF (HomF‘(oppCat‘𝐶)))
11		yon11.b	. . 3 ⊢ 𝐵 = (Base‘𝐶)
12	3	oppccat 17719	. . . 4 ⊢ (𝐶 ∈ Cat → (oppCat‘𝐶) ∈ Cat)
13	2, 12	syl 17	. . 3 ⊢ (𝜑 → (oppCat‘𝐶) ∈ Cat)
14		eqid 2734	. . . 4 ⊢ (SetCat‘ran (Homf ‘𝐶)) = (SetCat‘ran (Homf ‘𝐶))
15		fvex 6885	. . . . . 6 ⊢ (Homf ‘𝐶) ∈ V
16	15	rnex 7900	. . . . 5 ⊢ ran (Homf ‘𝐶) ∈ V
17	16	a1i 11	. . . 4 ⊢ (𝜑 → ran (Homf ‘𝐶) ∈ V)
18		ssidd 3980	. . . 4 ⊢ (𝜑 → ran (Homf ‘𝐶) ⊆ ran (Homf ‘𝐶))
19	3, 4, 14, 2, 17, 18	oppchofcl 18257	. . 3 ⊢ (𝜑 → (HomF‘(oppCat‘𝐶)) ∈ ((𝐶 ×c (oppCat‘𝐶)) Func (SetCat‘ran (Homf ‘𝐶))))
20	3, 11	oppcbas 17715	. . 3 ⊢ 𝐵 = (Base‘(oppCat‘𝐶))
21		yon11.p	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
22		eqid 2734	. . 3 ⊢ ((1st ‘(⟨𝐶, (oppCat‘𝐶)⟩ curryF (HomF‘(oppCat‘𝐶))))‘𝑋) = ((1st ‘(⟨𝐶, (oppCat‘𝐶)⟩ curryF (HomF‘(oppCat‘𝐶))))‘𝑋)
23		yon11.z	. . 3 ⊢ (𝜑 → 𝑍 ∈ 𝐵)
24	10, 11, 2, 13, 19, 20, 21, 22, 23	curf11 18223	. 2 ⊢ (𝜑 → ((1st ‘((1st ‘(⟨𝐶, (oppCat‘𝐶)⟩ curryF (HomF‘(oppCat‘𝐶))))‘𝑋))‘𝑍) = (𝑋(1st ‘(HomF‘(oppCat‘𝐶)))𝑍))
25		eqid 2734	. . . 4 ⊢ (Hom ‘(oppCat‘𝐶)) = (Hom ‘(oppCat‘𝐶))
26	4, 13, 20, 25, 21, 23	hof1 18251	. . 3 ⊢ (𝜑 → (𝑋(1st ‘(HomF‘(oppCat‘𝐶)))𝑍) = (𝑋(Hom ‘(oppCat‘𝐶))𝑍))
27		yon11.h	. . . 4 ⊢ 𝐻 = (Hom ‘𝐶)
28	27, 3	oppchom 17712	. . 3 ⊢ (𝑋(Hom ‘(oppCat‘𝐶))𝑍) = (𝑍𝐻𝑋)
29	26, 28	eqtrdi 2785	. 2 ⊢ (𝜑 → (𝑋(1st ‘(HomF‘(oppCat‘𝐶)))𝑍) = (𝑍𝐻𝑋))
30	9, 24, 29	3eqtrd 2773	1 ⊢ (𝜑 → ((1st ‘((1st ‘𝑌)‘𝑋))‘𝑍) = (𝑍𝐻𝑋))

Syntax hints:   → wi 4   = wceq 1539   ∈ wcel 2107  Vcvv 3457  ⟨cop 4605  ran crn 5652  ‘cfv 6527  (class class class)co 7399  1^st c1st 7980  Basecbs 17213  Hom chom 17267  Catccat 17661  Hom_f chomf 17663  oppCatcoppc 17708  SetCatcsetc 18073   curry_F ccurf 18207  Hom_Fchof 18245  Yoncyon 18246

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2706  ax-rep 5246  ax-sep 5263  ax-nul 5273  ax-pow 5332  ax-pr 5399  ax-un 7723  ax-cnex 11177  ax-resscn 11178  ax-1cn 11179  ax-icn 11180  ax-addcl 11181  ax-addrcl 11182  ax-mulcl 11183  ax-mulrcl 11184  ax-mulcom 11185  ax-addass 11186  ax-mulass 11187  ax-distr 11188  ax-i2m1 11189  ax-1ne0 11190  ax-1rid 11191  ax-rnegex 11192  ax-rrecex 11193  ax-cnre 11194  ax-pre-lttri 11195  ax-pre-lttrn 11196  ax-pre-ltadd 11197  ax-pre-mulgt0 11198

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2808  df-nfc 2884  df-ne 2932  df-nel 3036  df-ral 3051  df-rex 3060  df-rmo 3357  df-reu 3358  df-rab 3414  df-v 3459  df-sbc 3764  df-csb 3873  df-dif 3927  df-un 3929  df-in 3931  df-ss 3941  df-pss 3944  df-nul 4307  df-if 4499  df-pw 4575  df-sn 4600  df-pr 4602  df-tp 4604  df-op 4606  df-uni 4881  df-iun 4966  df-br 5117  df-opab 5179  df-mpt 5199  df-tr 5227  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-we 5605  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6287  df-ord 6352  df-on 6353  df-lim 6354  df-suc 6355  df-iota 6480  df-fun 6529  df-fn 6530  df-f 6531  df-f1 6532  df-fo 6533  df-f1o 6534  df-fv 6535  df-riota 7356  df-ov 7402  df-oprab 7403  df-mpo 7404  df-om 7856  df-1st 7982  df-2nd 7983  df-tpos 8219  df-frecs 8274  df-wrecs 8305  df-recs 8379  df-rdg 8418  df-1o 8474  df-er 8713  df-map 8836  df-ixp 8906  df-en 8954  df-dom 8955  df-sdom 8956  df-fin 8957  df-pnf 11263  df-mnf 11264  df-xr 11265  df-ltxr 11266  df-le 11267  df-sub 11460  df-neg 11461  df-nn 12233  df-2 12295  df-3 12296  df-4 12297  df-5 12298  df-6 12299  df-7 12300  df-8 12301  df-9 12302  df-n0 12494  df-z 12581  df-dec 12701  df-uz 12845  df-fz 13514  df-struct 17151  df-sets 17168  df-slot 17186  df-ndx 17198  df-base 17214  df-hom 17280  df-cco 17281  df-cat 17665  df-cid 17666  df-homf 17667  df-comf 17668  df-oppc 17709  df-func 17856  df-setc 18074  df-xpc 18169  df-curf 18211  df-hof 18247  df-yon 18248

This theorem is referenced by:  yonedalem3a  18271  yonedalem4c  18274  yonedalem3b  18276  yonedainv  18278  yonffthlem  18279  yoniso  18282