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Theorem yonedalem4b 18234
Description: Lemma for yoneda 18241. (Contributed by Mario Carneiro, 29-Jan-2017.)
Hypotheses
Ref Expression
yoneda.y π‘Œ = (Yonβ€˜πΆ)
yoneda.b 𝐡 = (Baseβ€˜πΆ)
yoneda.1 1 = (Idβ€˜πΆ)
yoneda.o 𝑂 = (oppCatβ€˜πΆ)
yoneda.s 𝑆 = (SetCatβ€˜π‘ˆ)
yoneda.t 𝑇 = (SetCatβ€˜π‘‰)
yoneda.q 𝑄 = (𝑂 FuncCat 𝑆)
yoneda.h 𝐻 = (HomFβ€˜π‘„)
yoneda.r 𝑅 = ((𝑄 Γ—c 𝑂) FuncCat 𝑇)
yoneda.e 𝐸 = (𝑂 evalF 𝑆)
yoneda.z 𝑍 = (𝐻 ∘func ((⟨(1st β€˜π‘Œ), tpos (2nd β€˜π‘Œ)⟩ ∘func (𝑄 2ndF 𝑂)) ⟨,⟩F (𝑄 1stF 𝑂)))
yoneda.c (πœ‘ β†’ 𝐢 ∈ Cat)
yoneda.w (πœ‘ β†’ 𝑉 ∈ π‘Š)
yoneda.u (πœ‘ β†’ ran (Homf β€˜πΆ) βŠ† π‘ˆ)
yoneda.v (πœ‘ β†’ (ran (Homf β€˜π‘„) βˆͺ π‘ˆ) βŠ† 𝑉)
yonedalem21.f (πœ‘ β†’ 𝐹 ∈ (𝑂 Func 𝑆))
yonedalem21.x (πœ‘ β†’ 𝑋 ∈ 𝐡)
yonedalem4.n 𝑁 = (𝑓 ∈ (𝑂 Func 𝑆), π‘₯ ∈ 𝐡 ↦ (𝑒 ∈ ((1st β€˜π‘“)β€˜π‘₯) ↦ (𝑦 ∈ 𝐡 ↦ (𝑔 ∈ (𝑦(Hom β€˜πΆ)π‘₯) ↦ (((π‘₯(2nd β€˜π‘“)𝑦)β€˜π‘”)β€˜π‘’)))))
yonedalem4.p (πœ‘ β†’ 𝐴 ∈ ((1st β€˜πΉ)β€˜π‘‹))
yonedalem4b.p (πœ‘ β†’ 𝑃 ∈ 𝐡)
yonedalem4b.g (πœ‘ β†’ 𝐺 ∈ (𝑃(Hom β€˜πΆ)𝑋))
Assertion
Ref Expression
yonedalem4b (πœ‘ β†’ ((((𝐹𝑁𝑋)β€˜π΄)β€˜π‘ƒ)β€˜πΊ) = (((𝑋(2nd β€˜πΉ)𝑃)β€˜πΊ)β€˜π΄))
Distinct variable groups:   𝑓,𝑔,π‘₯,𝑦, 1   𝑒,𝑔,𝐴,𝑦   𝑒,𝑓,𝐢,𝑔,π‘₯,𝑦   𝑓,𝐸,𝑔,𝑒,𝑦   𝑓,𝐹,𝑔,𝑒,π‘₯,𝑦   𝐡,𝑓,𝑔,𝑒,π‘₯,𝑦   𝑓,𝐺,𝑔,π‘₯,𝑦   𝑓,𝑂,𝑔,𝑒,π‘₯,𝑦   𝑆,𝑓,𝑔,𝑒,π‘₯,𝑦   𝑄,𝑓,𝑔,𝑒,π‘₯   𝑇,𝑓,𝑔,𝑒,𝑦   𝑃,𝑓,𝑔,π‘₯,𝑦   πœ‘,𝑓,𝑔,𝑒,π‘₯,𝑦   𝑒,𝑅   𝑓,π‘Œ,𝑔,𝑒,π‘₯,𝑦   𝑓,𝑍,𝑔,𝑒,π‘₯,𝑦   𝑓,𝑋,𝑔,𝑒,π‘₯,𝑦
Allowed substitution hints:   𝐴(π‘₯,𝑓)   𝑃(𝑒)   𝑄(𝑦)   𝑅(π‘₯,𝑦,𝑓,𝑔)   𝑇(π‘₯)   π‘ˆ(π‘₯,𝑦,𝑒,𝑓,𝑔)   1 (𝑒)   𝐸(π‘₯)   𝐺(𝑒)   𝐻(π‘₯,𝑦,𝑒,𝑓,𝑔)   𝑁(π‘₯,𝑦,𝑒,𝑓,𝑔)   𝑉(π‘₯,𝑦,𝑒,𝑓,𝑔)   π‘Š(π‘₯,𝑦,𝑒,𝑓,𝑔)
Proof of Theorem yonedalem4b
StepHypRef Expression
1 yoneda.y . . . . 5 π‘Œ = (Yonβ€˜πΆ)
2 yoneda.b . . . . 5 𝐡 = (Baseβ€˜πΆ)
3 yoneda.1 . . . . 5 1 = (Idβ€˜πΆ)
4 yoneda.o . . . . 5 𝑂 = (oppCatβ€˜πΆ)
5 yoneda.s . . . . 5 𝑆 = (SetCatβ€˜π‘ˆ)
6 yoneda.t . . . . 5 𝑇 = (SetCatβ€˜π‘‰)
7 yoneda.q . . . . 5 𝑄 = (𝑂 FuncCat 𝑆)
8 yoneda.h . . . . 5 𝐻 = (HomFβ€˜π‘„)
9 yoneda.r . . . . 5 𝑅 = ((𝑄 Γ—c 𝑂) FuncCat 𝑇)
10 yoneda.e . . . . 5 𝐸 = (𝑂 evalF 𝑆)
11 yoneda.z . . . . 5 𝑍 = (𝐻 ∘func ((⟨(1st β€˜π‘Œ), tpos (2nd β€˜π‘Œ)⟩ ∘func (𝑄 2ndF 𝑂)) ⟨,⟩F (𝑄 1stF 𝑂)))
12 yoneda.c . . . . 5 (πœ‘ β†’ 𝐢 ∈ Cat)
13 yoneda.w . . . . 5 (πœ‘ β†’ 𝑉 ∈ π‘Š)
14 yoneda.u . . . . 5 (πœ‘ β†’ ran (Homf β€˜πΆ) βŠ† π‘ˆ)
15 yoneda.v . . . . 5 (πœ‘ β†’ (ran (Homf β€˜π‘„) βˆͺ π‘ˆ) βŠ† 𝑉)
16 yonedalem21.f . . . . 5 (πœ‘ β†’ 𝐹 ∈ (𝑂 Func 𝑆))
17 yonedalem21.x . . . . 5 (πœ‘ β†’ 𝑋 ∈ 𝐡)
18 yonedalem4.n . . . . 5 𝑁 = (𝑓 ∈ (𝑂 Func 𝑆), π‘₯ ∈ 𝐡 ↦ (𝑒 ∈ ((1st β€˜π‘“)β€˜π‘₯) ↦ (𝑦 ∈ 𝐡 ↦ (𝑔 ∈ (𝑦(Hom β€˜πΆ)π‘₯) ↦ (((π‘₯(2nd β€˜π‘“)𝑦)β€˜π‘”)β€˜π‘’)))))
19 yonedalem4.p . . . . 5 (πœ‘ β†’ 𝐴 ∈ ((1st β€˜πΉ)β€˜π‘‹))
201, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19yonedalem4a 18233 . . . 4 (πœ‘ β†’ ((𝐹𝑁𝑋)β€˜π΄) = (𝑦 ∈ 𝐡 ↦ (𝑔 ∈ (𝑦(Hom β€˜πΆ)𝑋) ↦ (((𝑋(2nd β€˜πΉ)𝑦)β€˜π‘”)β€˜π΄))))
2120fveq1d 6893 . . 3 (πœ‘ β†’ (((𝐹𝑁𝑋)β€˜π΄)β€˜π‘ƒ) = ((𝑦 ∈ 𝐡 ↦ (𝑔 ∈ (𝑦(Hom β€˜πΆ)𝑋) ↦ (((𝑋(2nd β€˜πΉ)𝑦)β€˜π‘”)β€˜π΄)))β€˜π‘ƒ))
2221fveq1d 6893 . 2 (πœ‘ β†’ ((((𝐹𝑁𝑋)β€˜π΄)β€˜π‘ƒ)β€˜πΊ) = (((𝑦 ∈ 𝐡 ↦ (𝑔 ∈ (𝑦(Hom β€˜πΆ)𝑋) ↦ (((𝑋(2nd β€˜πΉ)𝑦)β€˜π‘”)β€˜π΄)))β€˜π‘ƒ)β€˜πΊ))
23 eqidd 2732 . . 3 (πœ‘ β†’ (𝑦 ∈ 𝐡 ↦ (𝑔 ∈ (𝑦(Hom β€˜πΆ)𝑋) ↦ (((𝑋(2nd β€˜πΉ)𝑦)β€˜π‘”)β€˜π΄))) = (𝑦 ∈ 𝐡 ↦ (𝑔 ∈ (𝑦(Hom β€˜πΆ)𝑋) ↦ (((𝑋(2nd β€˜πΉ)𝑦)β€˜π‘”)β€˜π΄))))
24 yonedalem4b.p . . . 4 (πœ‘ β†’ 𝑃 ∈ 𝐡)
25 ovex 7445 . . . . . 6 (𝑦(Hom β€˜πΆ)𝑋) ∈ V
2625mptex 7227 . . . . 5 (𝑔 ∈ (𝑦(Hom β€˜πΆ)𝑋) ↦ (((𝑋(2nd β€˜πΉ)𝑦)β€˜π‘”)β€˜π΄)) ∈ V
2726a1i 11 . . . 4 ((πœ‘ ∧ 𝑦 = 𝑃) β†’ (𝑔 ∈ (𝑦(Hom β€˜πΆ)𝑋) ↦ (((𝑋(2nd β€˜πΉ)𝑦)β€˜π‘”)β€˜π΄)) ∈ V)
28 yonedalem4b.g . . . . . . 7 (πœ‘ β†’ 𝐺 ∈ (𝑃(Hom β€˜πΆ)𝑋))
2928adantr 480 . . . . . 6 ((πœ‘ ∧ 𝑦 = 𝑃) β†’ 𝐺 ∈ (𝑃(Hom β€˜πΆ)𝑋))
30 simpr 484 . . . . . . 7 ((πœ‘ ∧ 𝑦 = 𝑃) β†’ 𝑦 = 𝑃)
3130oveq1d 7427 . . . . . 6 ((πœ‘ ∧ 𝑦 = 𝑃) β†’ (𝑦(Hom β€˜πΆ)𝑋) = (𝑃(Hom β€˜πΆ)𝑋))
3229, 31eleqtrrd 2835 . . . . 5 ((πœ‘ ∧ 𝑦 = 𝑃) β†’ 𝐺 ∈ (𝑦(Hom β€˜πΆ)𝑋))
33 fvexd 6906 . . . . 5 (((πœ‘ ∧ 𝑦 = 𝑃) ∧ 𝑔 = 𝐺) β†’ (((𝑋(2nd β€˜πΉ)𝑦)β€˜π‘”)β€˜π΄) ∈ V)
34 simplr 766 . . . . . . . 8 (((πœ‘ ∧ 𝑦 = 𝑃) ∧ 𝑔 = 𝐺) β†’ 𝑦 = 𝑃)
3534oveq2d 7428 . . . . . . 7 (((πœ‘ ∧ 𝑦 = 𝑃) ∧ 𝑔 = 𝐺) β†’ (𝑋(2nd β€˜πΉ)𝑦) = (𝑋(2nd β€˜πΉ)𝑃))
36 simpr 484 . . . . . . 7 (((πœ‘ ∧ 𝑦 = 𝑃) ∧ 𝑔 = 𝐺) β†’ 𝑔 = 𝐺)
3735, 36fveq12d 6898 . . . . . 6 (((πœ‘ ∧ 𝑦 = 𝑃) ∧ 𝑔 = 𝐺) β†’ ((𝑋(2nd β€˜πΉ)𝑦)β€˜π‘”) = ((𝑋(2nd β€˜πΉ)𝑃)β€˜πΊ))
3837fveq1d 6893 . . . . 5 (((πœ‘ ∧ 𝑦 = 𝑃) ∧ 𝑔 = 𝐺) β†’ (((𝑋(2nd β€˜πΉ)𝑦)β€˜π‘”)β€˜π΄) = (((𝑋(2nd β€˜πΉ)𝑃)β€˜πΊ)β€˜π΄))
3932, 33, 38fvmptdv2 7016 . . . 4 ((πœ‘ ∧ 𝑦 = 𝑃) β†’ (((𝑦 ∈ 𝐡 ↦ (𝑔 ∈ (𝑦(Hom β€˜πΆ)𝑋) ↦ (((𝑋(2nd β€˜πΉ)𝑦)β€˜π‘”)β€˜π΄)))β€˜π‘ƒ) = (𝑔 ∈ (𝑦(Hom β€˜πΆ)𝑋) ↦ (((𝑋(2nd β€˜πΉ)𝑦)β€˜π‘”)β€˜π΄)) β†’ (((𝑦 ∈ 𝐡 ↦ (𝑔 ∈ (𝑦(Hom β€˜πΆ)𝑋) ↦ (((𝑋(2nd β€˜πΉ)𝑦)β€˜π‘”)β€˜π΄)))β€˜π‘ƒ)β€˜πΊ) = (((𝑋(2nd β€˜πΉ)𝑃)β€˜πΊ)β€˜π΄)))
40 nfmpt1 5256 . . . 4 Ⅎ𝑦(𝑦 ∈ 𝐡 ↦ (𝑔 ∈ (𝑦(Hom β€˜πΆ)𝑋) ↦ (((𝑋(2nd β€˜πΉ)𝑦)β€˜π‘”)β€˜π΄)))
41 nffvmpt1 6902 . . . . . 6 Ⅎ𝑦((𝑦 ∈ 𝐡 ↦ (𝑔 ∈ (𝑦(Hom β€˜πΆ)𝑋) ↦ (((𝑋(2nd β€˜πΉ)𝑦)β€˜π‘”)β€˜π΄)))β€˜π‘ƒ)
42 nfcv 2902 . . . . . 6 Ⅎ𝑦𝐺
4341, 42nffv 6901 . . . . 5 Ⅎ𝑦(((𝑦 ∈ 𝐡 ↦ (𝑔 ∈ (𝑦(Hom β€˜πΆ)𝑋) ↦ (((𝑋(2nd β€˜πΉ)𝑦)β€˜π‘”)β€˜π΄)))β€˜π‘ƒ)β€˜πΊ)
4443nfeq1 2917 . . . 4 Ⅎ𝑦(((𝑦 ∈ 𝐡 ↦ (𝑔 ∈ (𝑦(Hom β€˜πΆ)𝑋) ↦ (((𝑋(2nd β€˜πΉ)𝑦)β€˜π‘”)β€˜π΄)))β€˜π‘ƒ)β€˜πΊ) = (((𝑋(2nd β€˜πΉ)𝑃)β€˜πΊ)β€˜π΄)
4524, 27, 39, 40, 44fvmptd2f 7014 . . 3 (πœ‘ β†’ ((𝑦 ∈ 𝐡 ↦ (𝑔 ∈ (𝑦(Hom β€˜πΆ)𝑋) ↦ (((𝑋(2nd β€˜πΉ)𝑦)β€˜π‘”)β€˜π΄))) = (𝑦 ∈ 𝐡 ↦ (𝑔 ∈ (𝑦(Hom β€˜πΆ)𝑋) ↦ (((𝑋(2nd β€˜πΉ)𝑦)β€˜π‘”)β€˜π΄))) β†’ (((𝑦 ∈ 𝐡 ↦ (𝑔 ∈ (𝑦(Hom β€˜πΆ)𝑋) ↦ (((𝑋(2nd β€˜πΉ)𝑦)β€˜π‘”)β€˜π΄)))β€˜π‘ƒ)β€˜πΊ) = (((𝑋(2nd β€˜πΉ)𝑃)β€˜πΊ)β€˜π΄)))
4623, 45mpd 15 . 2 (πœ‘ β†’ (((𝑦 ∈ 𝐡 ↦ (𝑔 ∈ (𝑦(Hom β€˜πΆ)𝑋) ↦ (((𝑋(2nd β€˜πΉ)𝑦)β€˜π‘”)β€˜π΄)))β€˜π‘ƒ)β€˜πΊ) = (((𝑋(2nd β€˜πΉ)𝑃)β€˜πΊ)β€˜π΄))
4722, 46eqtrd 2771 1 (πœ‘ β†’ ((((𝐹𝑁𝑋)β€˜π΄)β€˜π‘ƒ)β€˜πΊ) = (((𝑋(2nd β€˜πΉ)𝑃)β€˜πΊ)β€˜π΄))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2105  Vcvv 3473   βˆͺ cun 3946   βŠ† wss 3948  βŸ¨cop 4634   ↦ cmpt 5231  ran crn 5677  β€˜cfv 6543  (class class class)co 7412   ∈ cmpo 7414  1st c1st 7977  2nd c2nd 7978  tpos ctpos 8214  Basecbs 17149  Hom chom 17213  Catccat 17613  Idccid 17614  Homf chomf 17615  oppCatcoppc 17660   Func cfunc 17809   ∘func ccofu 17811   FuncCat cfuc 17898  SetCatcsetc 18030   Γ—c cxpc 18125   1stF c1stf 18126   2ndF c2ndf 18127   ⟨,⟩F cprf 18128   evalF cevlf 18167  HomFchof 18206  Yoncyon 18207
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7415  df-oprab 7416  df-mpo 7417
This theorem is referenced by:  yonedalem4c  18235  yonedainv  18239
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