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Theorem yonedalem4b 18213
Description: Lemma for yoneda 18220. (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 18212 . . . 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 6881 . . 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 6881 . 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 2733 . . 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 7427 . . . . . 6 (𝑦(Hom β€˜πΆ)𝑋) ∈ V
2625mptex 7210 . . . . 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 481 . . . . . 6 ((πœ‘ ∧ 𝑦 = 𝑃) β†’ 𝐺 ∈ (𝑃(Hom β€˜πΆ)𝑋))
30 simpr 485 . . . . . . 7 ((πœ‘ ∧ 𝑦 = 𝑃) β†’ 𝑦 = 𝑃)
3130oveq1d 7409 . . . . . 6 ((πœ‘ ∧ 𝑦 = 𝑃) β†’ (𝑦(Hom β€˜πΆ)𝑋) = (𝑃(Hom β€˜πΆ)𝑋))
3229, 31eleqtrrd 2836 . . . . 5 ((πœ‘ ∧ 𝑦 = 𝑃) β†’ 𝐺 ∈ (𝑦(Hom β€˜πΆ)𝑋))
33 fvexd 6894 . . . . 5 (((πœ‘ ∧ 𝑦 = 𝑃) ∧ 𝑔 = 𝐺) β†’ (((𝑋(2nd β€˜πΉ)𝑦)β€˜π‘”)β€˜π΄) ∈ V)
34 simplr 767 . . . . . . . 8 (((πœ‘ ∧ 𝑦 = 𝑃) ∧ 𝑔 = 𝐺) β†’ 𝑦 = 𝑃)
3534oveq2d 7410 . . . . . . 7 (((πœ‘ ∧ 𝑦 = 𝑃) ∧ 𝑔 = 𝐺) β†’ (𝑋(2nd β€˜πΉ)𝑦) = (𝑋(2nd β€˜πΉ)𝑃))
36 simpr 485 . . . . . . 7 (((πœ‘ ∧ 𝑦 = 𝑃) ∧ 𝑔 = 𝐺) β†’ 𝑔 = 𝐺)
3735, 36fveq12d 6886 . . . . . 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 6881 . . . . 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 7003 . . . 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 5250 . . . 4 Ⅎ𝑦(𝑦 ∈ 𝐡 ↦ (𝑔 ∈ (𝑦(Hom β€˜πΆ)𝑋) ↦ (((𝑋(2nd β€˜πΉ)𝑦)β€˜π‘”)β€˜π΄)))
41 nffvmpt1 6890 . . . . . 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 2903 . . . . . 6 Ⅎ𝑦𝐺
4341, 42nffv 6889 . . . . 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 2918 . . . 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 7001 . . 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 2772 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 396   = wceq 1541   ∈ wcel 2106  Vcvv 3474   βˆͺ cun 3943   βŠ† wss 3945  βŸ¨cop 4629   ↦ cmpt 5225  ran crn 5671  β€˜cfv 6533  (class class class)co 7394   ∈ cmpo 7396  1st c1st 7957  2nd c2nd 7958  tpos ctpos 8194  Basecbs 17128  Hom chom 17192  Catccat 17592  Idccid 17593  Homf chomf 17594  oppCatcoppc 17639   Func cfunc 17788   ∘func ccofu 17790   FuncCat cfuc 17877  SetCatcsetc 18009   Γ—c cxpc 18104   1stF c1stf 18105   2ndF c2ndf 18106   ⟨,⟩F cprf 18107   evalF cevlf 18146  HomFchof 18185  Yoncyon 18186
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pr 5421
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3775  df-csb 3891  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-nul 4320  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-id 5568  df-xp 5676  df-rel 5677  df-cnv 5678  df-co 5679  df-dm 5680  df-rn 5681  df-res 5682  df-ima 5683  df-iota 6485  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-ov 7397  df-oprab 7398  df-mpo 7399
This theorem is referenced by:  yonedalem4c  18214  yonedainv  18218
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