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Theorem yonedalem4a 18232
Description: Lemma for yoneda 18240. (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 767 . . . . . 6 ((πœ‘ ∧ (𝑓 = 𝐹 ∧ π‘₯ = 𝑋)) β†’ 𝑓 = 𝐹)
43fveq2d 6894 . . . . 5 ((πœ‘ ∧ (𝑓 = 𝐹 ∧ π‘₯ = 𝑋)) β†’ (1st β€˜π‘“) = (1st β€˜πΉ))
5 simprr 769 . . . . 5 ((πœ‘ ∧ (𝑓 = 𝐹 ∧ π‘₯ = 𝑋)) β†’ π‘₯ = 𝑋)
64, 5fveq12d 6897 . . . 4 ((πœ‘ ∧ (𝑓 = 𝐹 ∧ π‘₯ = 𝑋)) β†’ ((1st β€˜π‘“)β€˜π‘₯) = ((1st β€˜πΉ)β€˜π‘‹))
7 simplrr 774 . . . . . . 7 (((πœ‘ ∧ (𝑓 = 𝐹 ∧ π‘₯ = 𝑋)) ∧ 𝑦 ∈ 𝐡) β†’ π‘₯ = 𝑋)
87oveq2d 7427 . . . . . 6 (((πœ‘ ∧ (𝑓 = 𝐹 ∧ π‘₯ = 𝑋)) ∧ 𝑦 ∈ 𝐡) β†’ (𝑦(Hom β€˜πΆ)π‘₯) = (𝑦(Hom β€˜πΆ)𝑋))
9 simplrl 773 . . . . . . . . . 10 (((πœ‘ ∧ (𝑓 = 𝐹 ∧ π‘₯ = 𝑋)) ∧ 𝑦 ∈ 𝐡) β†’ 𝑓 = 𝐹)
109fveq2d 6894 . . . . . . . . 9 (((πœ‘ ∧ (𝑓 = 𝐹 ∧ π‘₯ = 𝑋)) ∧ 𝑦 ∈ 𝐡) β†’ (2nd β€˜π‘“) = (2nd β€˜πΉ))
11 eqidd 2731 . . . . . . . . 9 (((πœ‘ ∧ (𝑓 = 𝐹 ∧ π‘₯ = 𝑋)) ∧ 𝑦 ∈ 𝐡) β†’ 𝑦 = 𝑦)
1210, 7, 11oveq123d 7432 . . . . . . . 8 (((πœ‘ ∧ (𝑓 = 𝐹 ∧ π‘₯ = 𝑋)) ∧ 𝑦 ∈ 𝐡) β†’ (π‘₯(2nd β€˜π‘“)𝑦) = (𝑋(2nd β€˜πΉ)𝑦))
1312fveq1d 6892 . . . . . . 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 6892 . . . . . 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 5238 . . . . 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 5247 . . . 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 5238 . . 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 6903 . . . . 5 ((1st β€˜πΉ)β€˜π‘‹) ∈ V
2120mptex 7226 . . . 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 7562 . 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 483 . . . . 5 ((πœ‘ ∧ 𝑒 = 𝐴) β†’ 𝑒 = 𝐴)
2524fveq2d 6894 . . . 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 5249 . . 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 5249 . 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 6904 . . . 4 𝐡 ∈ V
3130mptex 7226 . . 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 7004 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 394   = wceq 1539   ∈ wcel 2104  Vcvv 3472   βˆͺ cun 3945   βŠ† wss 3947  βŸ¨cop 4633   ↦ cmpt 5230  ran crn 5676  β€˜cfv 6542  (class class class)co 7411   ∈ cmpo 7413  1st c1st 7975  2nd c2nd 7976  tpos ctpos 8212  Basecbs 17148  Hom chom 17212  Catccat 17612  Idccid 17613  Homf chomf 17614  oppCatcoppc 17659   Func cfunc 17808   ∘func ccofu 17810   FuncCat cfuc 17897  SetCatcsetc 18029   Γ—c cxpc 18124   1stF c1stf 18125   2ndF c2ndf 18126   ⟨,⟩F cprf 18127   evalF cevlf 18166  HomFchof 18205  Yoncyon 18206
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 2701  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pr 5426
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-ov 7414  df-oprab 7415  df-mpo 7416
This theorem is referenced by:  yonedalem4b  18233  yonedalem4c  18234  yonffthlem  18239
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