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Theorem yonedalem4a 18038
Description: Lemma for yoneda 18046. (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 769 . . . . . 6 ((πœ‘ ∧ (𝑓 = 𝐹 ∧ π‘₯ = 𝑋)) β†’ 𝑓 = 𝐹)
43fveq2d 6808 . . . . 5 ((πœ‘ ∧ (𝑓 = 𝐹 ∧ π‘₯ = 𝑋)) β†’ (1st β€˜π‘“) = (1st β€˜πΉ))
5 simprr 771 . . . . 5 ((πœ‘ ∧ (𝑓 = 𝐹 ∧ π‘₯ = 𝑋)) β†’ π‘₯ = 𝑋)
64, 5fveq12d 6811 . . . 4 ((πœ‘ ∧ (𝑓 = 𝐹 ∧ π‘₯ = 𝑋)) β†’ ((1st β€˜π‘“)β€˜π‘₯) = ((1st β€˜πΉ)β€˜π‘‹))
7 simplrr 776 . . . . . . 7 (((πœ‘ ∧ (𝑓 = 𝐹 ∧ π‘₯ = 𝑋)) ∧ 𝑦 ∈ 𝐡) β†’ π‘₯ = 𝑋)
87oveq2d 7323 . . . . . 6 (((πœ‘ ∧ (𝑓 = 𝐹 ∧ π‘₯ = 𝑋)) ∧ 𝑦 ∈ 𝐡) β†’ (𝑦(Hom β€˜πΆ)π‘₯) = (𝑦(Hom β€˜πΆ)𝑋))
9 simplrl 775 . . . . . . . . . 10 (((πœ‘ ∧ (𝑓 = 𝐹 ∧ π‘₯ = 𝑋)) ∧ 𝑦 ∈ 𝐡) β†’ 𝑓 = 𝐹)
109fveq2d 6808 . . . . . . . . 9 (((πœ‘ ∧ (𝑓 = 𝐹 ∧ π‘₯ = 𝑋)) ∧ 𝑦 ∈ 𝐡) β†’ (2nd β€˜π‘“) = (2nd β€˜πΉ))
11 eqidd 2737 . . . . . . . . 9 (((πœ‘ ∧ (𝑓 = 𝐹 ∧ π‘₯ = 𝑋)) ∧ 𝑦 ∈ 𝐡) β†’ 𝑦 = 𝑦)
1210, 7, 11oveq123d 7328 . . . . . . . 8 (((πœ‘ ∧ (𝑓 = 𝐹 ∧ π‘₯ = 𝑋)) ∧ 𝑦 ∈ 𝐡) β†’ (π‘₯(2nd β€˜π‘“)𝑦) = (𝑋(2nd β€˜πΉ)𝑦))
1312fveq1d 6806 . . . . . . 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 6806 . . . . . 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 5172 . . . . 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 5181 . . . 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 5172 . . 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 6817 . . . . 5 ((1st β€˜πΉ)β€˜π‘‹) ∈ V
2120mptex 7131 . . . 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 7457 . 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 486 . . . . 5 ((πœ‘ ∧ 𝑒 = 𝐴) β†’ 𝑒 = 𝐴)
2524fveq2d 6808 . . . 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 5183 . . 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 5183 . 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 6818 . . . 4 𝐡 ∈ V
3130mptex 7131 . . 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 6914 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 397   = wceq 1539   ∈ wcel 2104  Vcvv 3437   βˆͺ cun 3890   βŠ† wss 3892  βŸ¨cop 4571   ↦ cmpt 5164  ran crn 5601  β€˜cfv 6458  (class class class)co 7307   ∈ cmpo 7309  1st c1st 7861  2nd c2nd 7862  tpos ctpos 8072  Basecbs 16957  Hom chom 17018  Catccat 17418  Idccid 17419  Homf chomf 17420  oppCatcoppc 17465   Func cfunc 17614   ∘func ccofu 17616   FuncCat cfuc 17703  SetCatcsetc 17835   Γ—c cxpc 17930   1stF c1stf 17931   2ndF c2ndf 17932   ⟨,⟩F cprf 17933   evalF cevlf 17972  HomFchof 18011  Yoncyon 18012
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2707  ax-rep 5218  ax-sep 5232  ax-nul 5239  ax-pr 5361
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 846  df-3an 1089  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3286  df-rab 3287  df-v 3439  df-sbc 3722  df-csb 3838  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-nul 4263  df-if 4466  df-sn 4566  df-pr 4568  df-op 4572  df-uni 4845  df-iun 4933  df-br 5082  df-opab 5144  df-mpt 5165  df-id 5500  df-xp 5606  df-rel 5607  df-cnv 5608  df-co 5609  df-dm 5610  df-rn 5611  df-res 5612  df-ima 5613  df-iota 6410  df-fun 6460  df-fn 6461  df-f 6462  df-f1 6463  df-fo 6464  df-f1o 6465  df-fv 6466  df-ov 7310  df-oprab 7311  df-mpo 7312
This theorem is referenced by:  yonedalem4b  18039  yonedalem4c  18040  yonffthlem  18045
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