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Theorem yonedalem4a 16738
 Description: Lemma for yoneda 16746. (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 790 . . . . . 6 ((𝜑 ∧ (𝑓 = 𝐹𝑥 = 𝑋)) → 𝑓 = 𝐹)
43fveq2d 6107 . . . . 5 ((𝜑 ∧ (𝑓 = 𝐹𝑥 = 𝑋)) → (1st𝑓) = (1st𝐹))
5 simprr 792 . . . . 5 ((𝜑 ∧ (𝑓 = 𝐹𝑥 = 𝑋)) → 𝑥 = 𝑋)
64, 5fveq12d 6109 . . . 4 ((𝜑 ∧ (𝑓 = 𝐹𝑥 = 𝑋)) → ((1st𝑓)‘𝑥) = ((1st𝐹)‘𝑋))
7 simplrr 797 . . . . . . 7 (((𝜑 ∧ (𝑓 = 𝐹𝑥 = 𝑋)) ∧ 𝑦𝐵) → 𝑥 = 𝑋)
87oveq2d 6565 . . . . . 6 (((𝜑 ∧ (𝑓 = 𝐹𝑥 = 𝑋)) ∧ 𝑦𝐵) → (𝑦(Hom ‘𝐶)𝑥) = (𝑦(Hom ‘𝐶)𝑋))
9 simplrl 796 . . . . . . . . . 10 (((𝜑 ∧ (𝑓 = 𝐹𝑥 = 𝑋)) ∧ 𝑦𝐵) → 𝑓 = 𝐹)
109fveq2d 6107 . . . . . . . . 9 (((𝜑 ∧ (𝑓 = 𝐹𝑥 = 𝑋)) ∧ 𝑦𝐵) → (2nd𝑓) = (2nd𝐹))
11 eqidd 2611 . . . . . . . . 9 (((𝜑 ∧ (𝑓 = 𝐹𝑥 = 𝑋)) ∧ 𝑦𝐵) → 𝑦 = 𝑦)
1210, 7, 11oveq123d 6570 . . . . . . . 8 (((𝜑 ∧ (𝑓 = 𝐹𝑥 = 𝑋)) ∧ 𝑦𝐵) → (𝑥(2nd𝑓)𝑦) = (𝑋(2nd𝐹)𝑦))
1312fveq1d 6105 . . . . . . 7 (((𝜑 ∧ (𝑓 = 𝐹𝑥 = 𝑋)) ∧ 𝑦𝐵) → ((𝑥(2nd𝑓)𝑦)‘𝑔) = ((𝑋(2nd𝐹)𝑦)‘𝑔))
1413fveq1d 6105 . . . . . 6 (((𝜑 ∧ (𝑓 = 𝐹𝑥 = 𝑋)) ∧ 𝑦𝐵) → (((𝑥(2nd𝑓)𝑦)‘𝑔)‘𝑢) = (((𝑋(2nd𝐹)𝑦)‘𝑔)‘𝑢))
158, 14mpteq12dv 4663 . . . . 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 4672 . . . 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 4663 . . 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 6113 . . . . 5 ((1st𝐹)‘𝑋) ∈ V
2120mptex 6390 . . . 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 6686 . 2 (𝜑 → (𝐹𝑁𝑋) = (𝑢 ∈ ((1st𝐹)‘𝑋) ↦ (𝑦𝐵 ↦ (𝑔 ∈ (𝑦(Hom ‘𝐶)𝑋) ↦ (((𝑋(2nd𝐹)𝑦)‘𝑔)‘𝑢)))))
24 simpr 476 . . . . 5 ((𝜑𝑢 = 𝐴) → 𝑢 = 𝐴)
2524fveq2d 6107 . . . 4 ((𝜑𝑢 = 𝐴) → (((𝑋(2nd𝐹)𝑦)‘𝑔)‘𝑢) = (((𝑋(2nd𝐹)𝑦)‘𝑔)‘𝐴))
2625mpteq2dv 4673 . . 3 ((𝜑𝑢 = 𝐴) → (𝑔 ∈ (𝑦(Hom ‘𝐶)𝑋) ↦ (((𝑋(2nd𝐹)𝑦)‘𝑔)‘𝑢)) = (𝑔 ∈ (𝑦(Hom ‘𝐶)𝑋) ↦ (((𝑋(2nd𝐹)𝑦)‘𝑔)‘𝐴)))
2726mpteq2dv 4673 . 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)
30 fvex 6113 . . . . 5 (Base‘𝐶) ∈ V
3129, 30eqeltri 2684 . . . 4 𝐵 ∈ V
3231mptex 6390 . . 3 (𝑦𝐵 ↦ (𝑔 ∈ (𝑦(Hom ‘𝐶)𝑋) ↦ (((𝑋(2nd𝐹)𝑦)‘𝑔)‘𝐴))) ∈ V
3332a1i 11 . 2 (𝜑 → (𝑦𝐵 ↦ (𝑔 ∈ (𝑦(Hom ‘𝐶)𝑋) ↦ (((𝑋(2nd𝐹)𝑦)‘𝑔)‘𝐴))) ∈ V)
3423, 27, 28, 33fvmptd 6197 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 383   = wceq 1475   ∈ wcel 1977  Vcvv 3173   ∪ cun 3538   ⊆ wss 3540  ⟨cop 4131   ↦ cmpt 4643  ran crn 5039  ‘cfv 5804  (class class class)co 6549   ↦ cmpt2 6551  1st c1st 7057  2nd c2nd 7058  tpos ctpos 7238  Basecbs 15695  Hom chom 15779  Catccat 16148  Idccid 16149  Homf chomf 16150  oppCatcoppc 16194   Func cfunc 16337   ∘func ccofu 16339   FuncCat cfuc 16425  SetCatcsetc 16548   ×c cxpc 16631   1stF c1stf 16632   2ndF c2ndf 16633   ⟨,⟩F cprf 16634   evalF cevlf 16672  HomFchof 16711  Yoncyon 16712 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554 This theorem is referenced by:  yonedalem4b  16739  yonedalem4c  16740  yonffthlem  16745
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