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Theorem yonedalem4a 17301
 Description: Lemma for yoneda 17309. (Contributed by Mario Carneiro, 29-Jan-2017.)
Hypotheses
Ref Expression
yoneda.y 𝑌 = (Yon‘𝐶)
yoneda.b 𝐵 = (Base‘𝐶)
yoneda.1 1 = (Id‘𝐶)
yoneda.o 𝑂 = (oppCat‘𝐶)
yoneda.s 𝑆 = (SetCat‘𝑈)
yoneda.t 𝑇 = (SetCat‘𝑉)
yoneda.q 𝑄 = (𝑂 FuncCat 𝑆)
yoneda.h 𝐻 = (HomF𝑄)
yoneda.r 𝑅 = ((𝑄 ×c 𝑂) FuncCat 𝑇)
yoneda.e 𝐸 = (𝑂 evalF 𝑆)
yoneda.z 𝑍 = (𝐻func ((⟨(1st𝑌), tpos (2nd𝑌)⟩ ∘func (𝑄 2ndF 𝑂)) ⟨,⟩F (𝑄 1stF 𝑂)))
yoneda.c (𝜑𝐶 ∈ Cat)
yoneda.w (𝜑𝑉𝑊)
yoneda.u (𝜑 → ran (Homf𝐶) ⊆ 𝑈)
yoneda.v (𝜑 → (ran (Homf𝑄) ∪ 𝑈) ⊆ 𝑉)
yonedalem21.f (𝜑𝐹 ∈ (𝑂 Func 𝑆))
yonedalem21.x (𝜑𝑋𝐵)
yonedalem4.n 𝑁 = (𝑓 ∈ (𝑂 Func 𝑆), 𝑥𝐵 ↦ (𝑢 ∈ ((1st𝑓)‘𝑥) ↦ (𝑦𝐵 ↦ (𝑔 ∈ (𝑦(Hom ‘𝐶)𝑥) ↦ (((𝑥(2nd𝑓)𝑦)‘𝑔)‘𝑢)))))
yonedalem4.p (𝜑𝐴 ∈ ((1st𝐹)‘𝑋))
Assertion
Ref Expression
yonedalem4a (𝜑 → ((𝐹𝑁𝑋)‘𝐴) = (𝑦𝐵 ↦ (𝑔 ∈ (𝑦(Hom ‘𝐶)𝑋) ↦ (((𝑋(2nd𝐹)𝑦)‘𝑔)‘𝐴))))
Distinct variable groups:   𝑓,𝑔,𝑥,𝑦, 1   𝑢,𝑔,𝐴,𝑦   𝑢,𝑓,𝐶,𝑔,𝑥,𝑦   𝑓,𝐸,𝑔,𝑢,𝑦   𝑓,𝐹,𝑔,𝑢,𝑥,𝑦   𝐵,𝑓,𝑔,𝑢,𝑥,𝑦   𝑓,𝑂,𝑔,𝑢,𝑥,𝑦   𝑆,𝑓,𝑔,𝑢,𝑥,𝑦   𝑄,𝑓,𝑔,𝑢,𝑥   𝑇,𝑓,𝑔,𝑢,𝑦   𝜑,𝑓,𝑔,𝑢,𝑥,𝑦   𝑢,𝑅   𝑓,𝑌,𝑔,𝑢,𝑥,𝑦   𝑓,𝑍,𝑔,𝑢,𝑥,𝑦   𝑓,𝑋,𝑔,𝑢,𝑥,𝑦
Allowed substitution hints:   𝐴(𝑥,𝑓)   𝑄(𝑦)   𝑅(𝑥,𝑦,𝑓,𝑔)   𝑇(𝑥)   𝑈(𝑥,𝑦,𝑢,𝑓,𝑔)   1 (𝑢)   𝐸(𝑥)   𝐻(𝑥,𝑦,𝑢,𝑓,𝑔)   𝑁(𝑥,𝑦,𝑢,𝑓,𝑔)   𝑉(𝑥,𝑦,𝑢,𝑓,𝑔)   𝑊(𝑥,𝑦,𝑢,𝑓,𝑔)

Proof of Theorem yonedalem4a
StepHypRef Expression
1 yonedalem4.n . . . 4 𝑁 = (𝑓 ∈ (𝑂 Func 𝑆), 𝑥𝐵 ↦ (𝑢 ∈ ((1st𝑓)‘𝑥) ↦ (𝑦𝐵 ↦ (𝑔 ∈ (𝑦(Hom ‘𝐶)𝑥) ↦ (((𝑥(2nd𝑓)𝑦)‘𝑔)‘𝑢)))))
21a1i 11 . . 3 (𝜑𝑁 = (𝑓 ∈ (𝑂 Func 𝑆), 𝑥𝐵 ↦ (𝑢 ∈ ((1st𝑓)‘𝑥) ↦ (𝑦𝐵 ↦ (𝑔 ∈ (𝑦(Hom ‘𝐶)𝑥) ↦ (((𝑥(2nd𝑓)𝑦)‘𝑔)‘𝑢))))))
3 simprl 761 . . . . . 6 ((𝜑 ∧ (𝑓 = 𝐹𝑥 = 𝑋)) → 𝑓 = 𝐹)
43fveq2d 6450 . . . . 5 ((𝜑 ∧ (𝑓 = 𝐹𝑥 = 𝑋)) → (1st𝑓) = (1st𝐹))
5 simprr 763 . . . . 5 ((𝜑 ∧ (𝑓 = 𝐹𝑥 = 𝑋)) → 𝑥 = 𝑋)
64, 5fveq12d 6453 . . . 4 ((𝜑 ∧ (𝑓 = 𝐹𝑥 = 𝑋)) → ((1st𝑓)‘𝑥) = ((1st𝐹)‘𝑋))
7 simplrr 768 . . . . . . 7 (((𝜑 ∧ (𝑓 = 𝐹𝑥 = 𝑋)) ∧ 𝑦𝐵) → 𝑥 = 𝑋)
87oveq2d 6938 . . . . . 6 (((𝜑 ∧ (𝑓 = 𝐹𝑥 = 𝑋)) ∧ 𝑦𝐵) → (𝑦(Hom ‘𝐶)𝑥) = (𝑦(Hom ‘𝐶)𝑋))
9 simplrl 767 . . . . . . . . . 10 (((𝜑 ∧ (𝑓 = 𝐹𝑥 = 𝑋)) ∧ 𝑦𝐵) → 𝑓 = 𝐹)
109fveq2d 6450 . . . . . . . . 9 (((𝜑 ∧ (𝑓 = 𝐹𝑥 = 𝑋)) ∧ 𝑦𝐵) → (2nd𝑓) = (2nd𝐹))
11 eqidd 2778 . . . . . . . . 9 (((𝜑 ∧ (𝑓 = 𝐹𝑥 = 𝑋)) ∧ 𝑦𝐵) → 𝑦 = 𝑦)
1210, 7, 11oveq123d 6943 . . . . . . . 8 (((𝜑 ∧ (𝑓 = 𝐹𝑥 = 𝑋)) ∧ 𝑦𝐵) → (𝑥(2nd𝑓)𝑦) = (𝑋(2nd𝐹)𝑦))
1312fveq1d 6448 . . . . . . 7 (((𝜑 ∧ (𝑓 = 𝐹𝑥 = 𝑋)) ∧ 𝑦𝐵) → ((𝑥(2nd𝑓)𝑦)‘𝑔) = ((𝑋(2nd𝐹)𝑦)‘𝑔))
1413fveq1d 6448 . . . . . 6 (((𝜑 ∧ (𝑓 = 𝐹𝑥 = 𝑋)) ∧ 𝑦𝐵) → (((𝑥(2nd𝑓)𝑦)‘𝑔)‘𝑢) = (((𝑋(2nd𝐹)𝑦)‘𝑔)‘𝑢))
158, 14mpteq12dv 4969 . . . . 5 (((𝜑 ∧ (𝑓 = 𝐹𝑥 = 𝑋)) ∧ 𝑦𝐵) → (𝑔 ∈ (𝑦(Hom ‘𝐶)𝑥) ↦ (((𝑥(2nd𝑓)𝑦)‘𝑔)‘𝑢)) = (𝑔 ∈ (𝑦(Hom ‘𝐶)𝑋) ↦ (((𝑋(2nd𝐹)𝑦)‘𝑔)‘𝑢)))
1615mpteq2dva 4979 . . . 4 ((𝜑 ∧ (𝑓 = 𝐹𝑥 = 𝑋)) → (𝑦𝐵 ↦ (𝑔 ∈ (𝑦(Hom ‘𝐶)𝑥) ↦ (((𝑥(2nd𝑓)𝑦)‘𝑔)‘𝑢))) = (𝑦𝐵 ↦ (𝑔 ∈ (𝑦(Hom ‘𝐶)𝑋) ↦ (((𝑋(2nd𝐹)𝑦)‘𝑔)‘𝑢))))
176, 16mpteq12dv 4969 . . 3 ((𝜑 ∧ (𝑓 = 𝐹𝑥 = 𝑋)) → (𝑢 ∈ ((1st𝑓)‘𝑥) ↦ (𝑦𝐵 ↦ (𝑔 ∈ (𝑦(Hom ‘𝐶)𝑥) ↦ (((𝑥(2nd𝑓)𝑦)‘𝑔)‘𝑢)))) = (𝑢 ∈ ((1st𝐹)‘𝑋) ↦ (𝑦𝐵 ↦ (𝑔 ∈ (𝑦(Hom ‘𝐶)𝑋) ↦ (((𝑋(2nd𝐹)𝑦)‘𝑔)‘𝑢)))))
18 yonedalem21.f . . 3 (𝜑𝐹 ∈ (𝑂 Func 𝑆))
19 yonedalem21.x . . 3 (𝜑𝑋𝐵)
20 fvex 6459 . . . . 5 ((1st𝐹)‘𝑋) ∈ V
2120mptex 6758 . . . 4 (𝑢 ∈ ((1st𝐹)‘𝑋) ↦ (𝑦𝐵 ↦ (𝑔 ∈ (𝑦(Hom ‘𝐶)𝑋) ↦ (((𝑋(2nd𝐹)𝑦)‘𝑔)‘𝑢)))) ∈ V
2221a1i 11 . . 3 (𝜑 → (𝑢 ∈ ((1st𝐹)‘𝑋) ↦ (𝑦𝐵 ↦ (𝑔 ∈ (𝑦(Hom ‘𝐶)𝑋) ↦ (((𝑋(2nd𝐹)𝑦)‘𝑔)‘𝑢)))) ∈ V)
232, 17, 18, 19, 22ovmpt2d 7065 . 2 (𝜑 → (𝐹𝑁𝑋) = (𝑢 ∈ ((1st𝐹)‘𝑋) ↦ (𝑦𝐵 ↦ (𝑔 ∈ (𝑦(Hom ‘𝐶)𝑋) ↦ (((𝑋(2nd𝐹)𝑦)‘𝑔)‘𝑢)))))
24 simpr 479 . . . . 5 ((𝜑𝑢 = 𝐴) → 𝑢 = 𝐴)
2524fveq2d 6450 . . . 4 ((𝜑𝑢 = 𝐴) → (((𝑋(2nd𝐹)𝑦)‘𝑔)‘𝑢) = (((𝑋(2nd𝐹)𝑦)‘𝑔)‘𝐴))
2625mpteq2dv 4980 . . 3 ((𝜑𝑢 = 𝐴) → (𝑔 ∈ (𝑦(Hom ‘𝐶)𝑋) ↦ (((𝑋(2nd𝐹)𝑦)‘𝑔)‘𝑢)) = (𝑔 ∈ (𝑦(Hom ‘𝐶)𝑋) ↦ (((𝑋(2nd𝐹)𝑦)‘𝑔)‘𝐴)))
2726mpteq2dv 4980 . 2 ((𝜑𝑢 = 𝐴) → (𝑦𝐵 ↦ (𝑔 ∈ (𝑦(Hom ‘𝐶)𝑋) ↦ (((𝑋(2nd𝐹)𝑦)‘𝑔)‘𝑢))) = (𝑦𝐵 ↦ (𝑔 ∈ (𝑦(Hom ‘𝐶)𝑋) ↦ (((𝑋(2nd𝐹)𝑦)‘𝑔)‘𝐴))))
28 yonedalem4.p . 2 (𝜑𝐴 ∈ ((1st𝐹)‘𝑋))
29 yoneda.b . . . . 5 𝐵 = (Base‘𝐶)
3029fvexi 6460 . . . 4 𝐵 ∈ V
3130mptex 6758 . . 3 (𝑦𝐵 ↦ (𝑔 ∈ (𝑦(Hom ‘𝐶)𝑋) ↦ (((𝑋(2nd𝐹)𝑦)‘𝑔)‘𝐴))) ∈ V
3231a1i 11 . 2 (𝜑 → (𝑦𝐵 ↦ (𝑔 ∈ (𝑦(Hom ‘𝐶)𝑋) ↦ (((𝑋(2nd𝐹)𝑦)‘𝑔)‘𝐴))) ∈ V)
3323, 27, 28, 32fvmptd 6548 1 (𝜑 → ((𝐹𝑁𝑋)‘𝐴) = (𝑦𝐵 ↦ (𝑔 ∈ (𝑦(Hom ‘𝐶)𝑋) ↦ (((𝑋(2nd𝐹)𝑦)‘𝑔)‘𝐴))))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 386   = wceq 1601   ∈ wcel 2106  Vcvv 3397   ∪ cun 3789   ⊆ wss 3791  ⟨cop 4403   ↦ cmpt 4965  ran crn 5356  ‘cfv 6135  (class class class)co 6922   ↦ cmpt2 6924  1st c1st 7443  2nd c2nd 7444  tpos ctpos 7633  Basecbs 16255  Hom chom 16349  Catccat 16710  Idccid 16711  Homf chomf 16712  oppCatcoppc 16756   Func cfunc 16899   ∘func ccofu 16901   FuncCat cfuc 16987  SetCatcsetc 17110   ×c cxpc 17194   1stF c1stf 17195   2ndF c2ndf 17196   ⟨,⟩F cprf 17197   evalF cevlf 17235  HomFchof 17274  Yoncyon 17275 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2054  ax-9 2115  ax-10 2134  ax-11 2149  ax-12 2162  ax-13 2333  ax-ext 2753  ax-rep 5006  ax-sep 5017  ax-nul 5025  ax-pr 5138 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2550  df-eu 2586  df-clab 2763  df-cleq 2769  df-clel 2773  df-nfc 2920  df-ne 2969  df-ral 3094  df-rex 3095  df-reu 3096  df-rab 3098  df-v 3399  df-sbc 3652  df-csb 3751  df-dif 3794  df-un 3796  df-in 3798  df-ss 3805  df-nul 4141  df-if 4307  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4672  df-iun 4755  df-br 4887  df-opab 4949  df-mpt 4966  df-id 5261  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-res 5367  df-ima 5368  df-iota 6099  df-fun 6137  df-fn 6138  df-f 6139  df-f1 6140  df-fo 6141  df-f1o 6142  df-fv 6143  df-ov 6925  df-oprab 6926  df-mpt2 6927 This theorem is referenced by:  yonedalem4b  17302  yonedalem4c  17303  yonffthlem  17308
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