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Theorem yonedalem4a 18233
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‘‹))
Assertion
Ref Expression
yonedalem4a (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΄))))
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,𝑂,𝑔,𝑒,π‘₯,𝑦   𝑆,𝑓,𝑔,𝑒,π‘₯,𝑦   𝑄,𝑓,𝑔,𝑒,π‘₯   𝑇,𝑓,𝑔,𝑒,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)   𝑄(𝑦)   𝑅(π‘₯,𝑦,𝑓,𝑔)   𝑇(π‘₯)   π‘ˆ(π‘₯,𝑦,𝑒,𝑓,𝑔)   1 (𝑒)   𝐸(π‘₯)   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 yonedalem4a
StepHypRef Expression
1 yonedalem4.n . . . 4 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‘’)))))
21a1i 11 . . 3 (πœ‘ β†’ 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‘’))))))
3 simprl 768 . . . . . 6 ((πœ‘ ∧ (𝑓 = 𝐹 ∧ π‘₯ = 𝑋)) β†’ 𝑓 = 𝐹)
43fveq2d 6896 . . . . 5 ((πœ‘ ∧ (𝑓 = 𝐹 ∧ π‘₯ = 𝑋)) β†’ (1st β€˜π‘“) = (1st β€˜πΉ))
5 simprr 770 . . . . 5 ((πœ‘ ∧ (𝑓 = 𝐹 ∧ π‘₯ = 𝑋)) β†’ π‘₯ = 𝑋)
64, 5fveq12d 6899 . . . 4 ((πœ‘ ∧ (𝑓 = 𝐹 ∧ π‘₯ = 𝑋)) β†’ ((1st β€˜π‘“)β€˜π‘₯) = ((1st β€˜πΉ)β€˜π‘‹))
7 simplrr 775 . . . . . . 7 (((πœ‘ ∧ (𝑓 = 𝐹 ∧ π‘₯ = 𝑋)) ∧ 𝑦 ∈ 𝐡) β†’ π‘₯ = 𝑋)
87oveq2d 7428 . . . . . 6 (((πœ‘ ∧ (𝑓 = 𝐹 ∧ π‘₯ = 𝑋)) ∧ 𝑦 ∈ 𝐡) β†’ (𝑦(Hom β€˜πΆ)π‘₯) = (𝑦(Hom β€˜πΆ)𝑋))
9 simplrl 774 . . . . . . . . . 10 (((πœ‘ ∧ (𝑓 = 𝐹 ∧ π‘₯ = 𝑋)) ∧ 𝑦 ∈ 𝐡) β†’ 𝑓 = 𝐹)
109fveq2d 6896 . . . . . . . . 9 (((πœ‘ ∧ (𝑓 = 𝐹 ∧ π‘₯ = 𝑋)) ∧ 𝑦 ∈ 𝐡) β†’ (2nd β€˜π‘“) = (2nd β€˜πΉ))
11 eqidd 2732 . . . . . . . . 9 (((πœ‘ ∧ (𝑓 = 𝐹 ∧ π‘₯ = 𝑋)) ∧ 𝑦 ∈ 𝐡) β†’ 𝑦 = 𝑦)
1210, 7, 11oveq123d 7433 . . . . . . . 8 (((πœ‘ ∧ (𝑓 = 𝐹 ∧ π‘₯ = 𝑋)) ∧ 𝑦 ∈ 𝐡) β†’ (π‘₯(2nd β€˜π‘“)𝑦) = (𝑋(2nd β€˜πΉ)𝑦))
1312fveq1d 6894 . . . . . . 7 (((πœ‘ ∧ (𝑓 = 𝐹 ∧ π‘₯ = 𝑋)) ∧ 𝑦 ∈ 𝐡) β†’ ((π‘₯(2nd β€˜π‘“)𝑦)β€˜π‘”) = ((𝑋(2nd β€˜πΉ)𝑦)β€˜π‘”))
1413fveq1d 6894 . . . . . 6 (((πœ‘ ∧ (𝑓 = 𝐹 ∧ π‘₯ = 𝑋)) ∧ 𝑦 ∈ 𝐡) β†’ (((π‘₯(2nd β€˜π‘“)𝑦)β€˜π‘”)β€˜π‘’) = (((𝑋(2nd β€˜πΉ)𝑦)β€˜π‘”)β€˜π‘’))
158, 14mpteq12dv 5240 . . . . 5 (((πœ‘ ∧ (𝑓 = 𝐹 ∧ π‘₯ = 𝑋)) ∧ 𝑦 ∈ 𝐡) β†’ (𝑔 ∈ (𝑦(Hom β€˜πΆ)π‘₯) ↦ (((π‘₯(2nd β€˜π‘“)𝑦)β€˜π‘”)β€˜π‘’)) = (𝑔 ∈ (𝑦(Hom β€˜πΆ)𝑋) ↦ (((𝑋(2nd β€˜πΉ)𝑦)β€˜π‘”)β€˜π‘’)))
1615mpteq2dva 5249 . . . 4 ((πœ‘ ∧ (𝑓 = 𝐹 ∧ π‘₯ = 𝑋)) β†’ (𝑦 ∈ 𝐡 ↦ (𝑔 ∈ (𝑦(Hom β€˜πΆ)π‘₯) ↦ (((π‘₯(2nd β€˜π‘“)𝑦)β€˜π‘”)β€˜π‘’))) = (𝑦 ∈ 𝐡 ↦ (𝑔 ∈ (𝑦(Hom β€˜πΆ)𝑋) ↦ (((𝑋(2nd β€˜πΉ)𝑦)β€˜π‘”)β€˜π‘’))))
176, 16mpteq12dv 5240 . . 3 ((πœ‘ ∧ (𝑓 = 𝐹 ∧ π‘₯ = 𝑋)) β†’ (𝑒 ∈ ((1st β€˜π‘“)β€˜π‘₯) ↦ (𝑦 ∈ 𝐡 ↦ (𝑔 ∈ (𝑦(Hom β€˜πΆ)π‘₯) ↦ (((π‘₯(2nd β€˜π‘“)𝑦)β€˜π‘”)β€˜π‘’)))) = (𝑒 ∈ ((1st β€˜πΉ)β€˜π‘‹) ↦ (𝑦 ∈ 𝐡 ↦ (𝑔 ∈ (𝑦(Hom β€˜πΆ)𝑋) ↦ (((𝑋(2nd β€˜πΉ)𝑦)β€˜π‘”)β€˜π‘’)))))
18 yonedalem21.f . . 3 (πœ‘ β†’ 𝐹 ∈ (𝑂 Func 𝑆))
19 yonedalem21.x . . 3 (πœ‘ β†’ 𝑋 ∈ 𝐡)
20 fvex 6905 . . . . 5 ((1st β€˜πΉ)β€˜π‘‹) ∈ V
2120mptex 7228 . . . 4 (𝑒 ∈ ((1st β€˜πΉ)β€˜π‘‹) ↦ (𝑦 ∈ 𝐡 ↦ (𝑔 ∈ (𝑦(Hom β€˜πΆ)𝑋) ↦ (((𝑋(2nd β€˜πΉ)𝑦)β€˜π‘”)β€˜π‘’)))) ∈ V
2221a1i 11 . . 3 (πœ‘ β†’ (𝑒 ∈ ((1st β€˜πΉ)β€˜π‘‹) ↦ (𝑦 ∈ 𝐡 ↦ (𝑔 ∈ (𝑦(Hom β€˜πΆ)𝑋) ↦ (((𝑋(2nd β€˜πΉ)𝑦)β€˜π‘”)β€˜π‘’)))) ∈ V)
232, 17, 18, 19, 22ovmpod 7563 . 2 (πœ‘ β†’ (𝐹𝑁𝑋) = (𝑒 ∈ ((1st β€˜πΉ)β€˜π‘‹) ↦ (𝑦 ∈ 𝐡 ↦ (𝑔 ∈ (𝑦(Hom β€˜πΆ)𝑋) ↦ (((𝑋(2nd β€˜πΉ)𝑦)β€˜π‘”)β€˜π‘’)))))
24 simpr 484 . . . . 5 ((πœ‘ ∧ 𝑒 = 𝐴) β†’ 𝑒 = 𝐴)
2524fveq2d 6896 . . . 4 ((πœ‘ ∧ 𝑒 = 𝐴) β†’ (((𝑋(2nd β€˜πΉ)𝑦)β€˜π‘”)β€˜π‘’) = (((𝑋(2nd β€˜πΉ)𝑦)β€˜π‘”)β€˜π΄))
2625mpteq2dv 5251 . . 3 ((πœ‘ ∧ 𝑒 = 𝐴) β†’ (𝑔 ∈ (𝑦(Hom β€˜πΆ)𝑋) ↦ (((𝑋(2nd β€˜πΉ)𝑦)β€˜π‘”)β€˜π‘’)) = (𝑔 ∈ (𝑦(Hom β€˜πΆ)𝑋) ↦ (((𝑋(2nd β€˜πΉ)𝑦)β€˜π‘”)β€˜π΄)))
2726mpteq2dv 5251 . 2 ((πœ‘ ∧ 𝑒 = 𝐴) 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΄))))
28 yonedalem4.p . 2 (πœ‘ β†’ 𝐴 ∈ ((1st β€˜πΉ)β€˜π‘‹))
29 yoneda.b . . . . 5 𝐡 = (Baseβ€˜πΆ)
3029fvexi 6906 . . . 4 𝐡 ∈ V
3130mptex 7228 . . 3 (𝑦 ∈ 𝐡 ↦ (𝑔 ∈ (𝑦(Hom β€˜πΆ)𝑋) ↦ (((𝑋(2nd β€˜πΉ)𝑦)β€˜π‘”)β€˜π΄))) ∈ V
3231a1i 11 . 2 (πœ‘ β†’ (𝑦 ∈ 𝐡 ↦ (𝑔 ∈ (𝑦(Hom β€˜πΆ)𝑋) ↦ (((𝑋(2nd β€˜πΉ)𝑦)β€˜π‘”)β€˜π΄))) ∈ V)
3323, 27, 28, 32fvmptd 7006 1 (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΄))))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2105  Vcvv 3473   βˆͺ cun 3947   βŠ† wss 3949  βŸ¨cop 4635   ↦ cmpt 5232  ran crn 5678  β€˜cfv 6544  (class class class)co 7412   ∈ cmpo 7414  1st c1st 7976  2nd c2nd 7977  tpos ctpos 8213  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 5286  ax-sep 5300  ax-nul 5307  ax-pr 5428
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 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-ov 7415  df-oprab 7416  df-mpo 7417
This theorem is referenced by:  yonedalem4b  18234  yonedalem4c  18235  yonffthlem  18240
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