MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  yonedalem4a Structured version   Visualization version   GIF version

Theorem yonedalem4a 18164
Description: Lemma for yoneda 18172. (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 769 . . . . . 6 ((𝜑 ∧ (𝑓 = 𝐹𝑥 = 𝑋)) → 𝑓 = 𝐹)
43fveq2d 6846 . . . . 5 ((𝜑 ∧ (𝑓 = 𝐹𝑥 = 𝑋)) → (1st𝑓) = (1st𝐹))
5 simprr 771 . . . . 5 ((𝜑 ∧ (𝑓 = 𝐹𝑥 = 𝑋)) → 𝑥 = 𝑋)
64, 5fveq12d 6849 . . . 4 ((𝜑 ∧ (𝑓 = 𝐹𝑥 = 𝑋)) → ((1st𝑓)‘𝑥) = ((1st𝐹)‘𝑋))
7 simplrr 776 . . . . . . 7 (((𝜑 ∧ (𝑓 = 𝐹𝑥 = 𝑋)) ∧ 𝑦𝐵) → 𝑥 = 𝑋)
87oveq2d 7373 . . . . . 6 (((𝜑 ∧ (𝑓 = 𝐹𝑥 = 𝑋)) ∧ 𝑦𝐵) → (𝑦(Hom ‘𝐶)𝑥) = (𝑦(Hom ‘𝐶)𝑋))
9 simplrl 775 . . . . . . . . . 10 (((𝜑 ∧ (𝑓 = 𝐹𝑥 = 𝑋)) ∧ 𝑦𝐵) → 𝑓 = 𝐹)
109fveq2d 6846 . . . . . . . . 9 (((𝜑 ∧ (𝑓 = 𝐹𝑥 = 𝑋)) ∧ 𝑦𝐵) → (2nd𝑓) = (2nd𝐹))
11 eqidd 2737 . . . . . . . . 9 (((𝜑 ∧ (𝑓 = 𝐹𝑥 = 𝑋)) ∧ 𝑦𝐵) → 𝑦 = 𝑦)
1210, 7, 11oveq123d 7378 . . . . . . . 8 (((𝜑 ∧ (𝑓 = 𝐹𝑥 = 𝑋)) ∧ 𝑦𝐵) → (𝑥(2nd𝑓)𝑦) = (𝑋(2nd𝐹)𝑦))
1312fveq1d 6844 . . . . . . 7 (((𝜑 ∧ (𝑓 = 𝐹𝑥 = 𝑋)) ∧ 𝑦𝐵) → ((𝑥(2nd𝑓)𝑦)‘𝑔) = ((𝑋(2nd𝐹)𝑦)‘𝑔))
1413fveq1d 6844 . . . . . 6 (((𝜑 ∧ (𝑓 = 𝐹𝑥 = 𝑋)) ∧ 𝑦𝐵) → (((𝑥(2nd𝑓)𝑦)‘𝑔)‘𝑢) = (((𝑋(2nd𝐹)𝑦)‘𝑔)‘𝑢))
158, 14mpteq12dv 5196 . . . . 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 5205 . . . 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 5196 . . 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 6855 . . . . 5 ((1st𝐹)‘𝑋) ∈ V
2120mptex 7173 . . . 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, 22ovmpod 7507 . 2 (𝜑 → (𝐹𝑁𝑋) = (𝑢 ∈ ((1st𝐹)‘𝑋) ↦ (𝑦𝐵 ↦ (𝑔 ∈ (𝑦(Hom ‘𝐶)𝑋) ↦ (((𝑋(2nd𝐹)𝑦)‘𝑔)‘𝑢)))))
24 simpr 485 . . . . 5 ((𝜑𝑢 = 𝐴) → 𝑢 = 𝐴)
2524fveq2d 6846 . . . 4 ((𝜑𝑢 = 𝐴) → (((𝑋(2nd𝐹)𝑦)‘𝑔)‘𝑢) = (((𝑋(2nd𝐹)𝑦)‘𝑔)‘𝐴))
2625mpteq2dv 5207 . . 3 ((𝜑𝑢 = 𝐴) → (𝑔 ∈ (𝑦(Hom ‘𝐶)𝑋) ↦ (((𝑋(2nd𝐹)𝑦)‘𝑔)‘𝑢)) = (𝑔 ∈ (𝑦(Hom ‘𝐶)𝑋) ↦ (((𝑋(2nd𝐹)𝑦)‘𝑔)‘𝐴)))
2726mpteq2dv 5207 . 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 6856 . . . 4 𝐵 ∈ V
3130mptex 7173 . . 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 6955 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 396   = wceq 1541  wcel 2106  Vcvv 3445  cun 3908  wss 3910  cop 4592  cmpt 5188  ran crn 5634  cfv 6496  (class class class)co 7357  cmpo 7359  1st c1st 7919  2nd c2nd 7920  tpos ctpos 8156  Basecbs 17083  Hom chom 17144  Catccat 17544  Idccid 17545  Homf chomf 17546  oppCatcoppc 17591   Func cfunc 17740  func ccofu 17742   FuncCat cfuc 17829  SetCatcsetc 17961   ×c cxpc 18056   1stF c1stf 18057   2ndF c2ndf 18058   ⟨,⟩F cprf 18059   evalF cevlf 18098  HomFchof 18137  Yoncyon 18138
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 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pr 5384
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 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-ov 7360  df-oprab 7361  df-mpo 7362
This theorem is referenced by:  yonedalem4b  18165  yonedalem4c  18166  yonffthlem  18171
  Copyright terms: Public domain W3C validator