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Theorem yonedalem4a 17122
Description: Lemma for yoneda 17130. (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 746 . . . . . 6 ((𝜑 ∧ (𝑓 = 𝐹𝑥 = 𝑋)) → 𝑓 = 𝐹)
43fveq2d 6336 . . . . 5 ((𝜑 ∧ (𝑓 = 𝐹𝑥 = 𝑋)) → (1st𝑓) = (1st𝐹))
5 simprr 748 . . . . 5 ((𝜑 ∧ (𝑓 = 𝐹𝑥 = 𝑋)) → 𝑥 = 𝑋)
64, 5fveq12d 6338 . . . 4 ((𝜑 ∧ (𝑓 = 𝐹𝑥 = 𝑋)) → ((1st𝑓)‘𝑥) = ((1st𝐹)‘𝑋))
7 simplrr 755 . . . . . . 7 (((𝜑 ∧ (𝑓 = 𝐹𝑥 = 𝑋)) ∧ 𝑦𝐵) → 𝑥 = 𝑋)
87oveq2d 6808 . . . . . 6 (((𝜑 ∧ (𝑓 = 𝐹𝑥 = 𝑋)) ∧ 𝑦𝐵) → (𝑦(Hom ‘𝐶)𝑥) = (𝑦(Hom ‘𝐶)𝑋))
9 simplrl 754 . . . . . . . . . 10 (((𝜑 ∧ (𝑓 = 𝐹𝑥 = 𝑋)) ∧ 𝑦𝐵) → 𝑓 = 𝐹)
109fveq2d 6336 . . . . . . . . 9 (((𝜑 ∧ (𝑓 = 𝐹𝑥 = 𝑋)) ∧ 𝑦𝐵) → (2nd𝑓) = (2nd𝐹))
11 eqidd 2771 . . . . . . . . 9 (((𝜑 ∧ (𝑓 = 𝐹𝑥 = 𝑋)) ∧ 𝑦𝐵) → 𝑦 = 𝑦)
1210, 7, 11oveq123d 6813 . . . . . . . 8 (((𝜑 ∧ (𝑓 = 𝐹𝑥 = 𝑋)) ∧ 𝑦𝐵) → (𝑥(2nd𝑓)𝑦) = (𝑋(2nd𝐹)𝑦))
1312fveq1d 6334 . . . . . . 7 (((𝜑 ∧ (𝑓 = 𝐹𝑥 = 𝑋)) ∧ 𝑦𝐵) → ((𝑥(2nd𝑓)𝑦)‘𝑔) = ((𝑋(2nd𝐹)𝑦)‘𝑔))
1413fveq1d 6334 . . . . . 6 (((𝜑 ∧ (𝑓 = 𝐹𝑥 = 𝑋)) ∧ 𝑦𝐵) → (((𝑥(2nd𝑓)𝑦)‘𝑔)‘𝑢) = (((𝑋(2nd𝐹)𝑦)‘𝑔)‘𝑢))
158, 14mpteq12dv 4865 . . . . 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 4876 . . . 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 4865 . . 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 6342 . . . . 5 ((1st𝐹)‘𝑋) ∈ V
2120mptex 6629 . . . 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 6934 . 2 (𝜑 → (𝐹𝑁𝑋) = (𝑢 ∈ ((1st𝐹)‘𝑋) ↦ (𝑦𝐵 ↦ (𝑔 ∈ (𝑦(Hom ‘𝐶)𝑋) ↦ (((𝑋(2nd𝐹)𝑦)‘𝑔)‘𝑢)))))
24 simpr 471 . . . . 5 ((𝜑𝑢 = 𝐴) → 𝑢 = 𝐴)
2524fveq2d 6336 . . . 4 ((𝜑𝑢 = 𝐴) → (((𝑋(2nd𝐹)𝑦)‘𝑔)‘𝑢) = (((𝑋(2nd𝐹)𝑦)‘𝑔)‘𝐴))
2625mpteq2dv 4877 . . 3 ((𝜑𝑢 = 𝐴) → (𝑔 ∈ (𝑦(Hom ‘𝐶)𝑋) ↦ (((𝑋(2nd𝐹)𝑦)‘𝑔)‘𝑢)) = (𝑔 ∈ (𝑦(Hom ‘𝐶)𝑋) ↦ (((𝑋(2nd𝐹)𝑦)‘𝑔)‘𝐴)))
2726mpteq2dv 4877 . 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 6342 . . . . 5 (Base‘𝐶) ∈ V
3129, 30eqeltri 2845 . . . 4 𝐵 ∈ V
3231mptex 6629 . . 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 6430 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 382   = wceq 1630  wcel 2144  Vcvv 3349  cun 3719  wss 3721  cop 4320  cmpt 4861  ran crn 5250  cfv 6031  (class class class)co 6792  cmpt2 6794  1st c1st 7312  2nd c2nd 7313  tpos ctpos 7502  Basecbs 16063  Hom chom 16159  Catccat 16531  Idccid 16532  Homf chomf 16533  oppCatcoppc 16577   Func cfunc 16720  func ccofu 16722   FuncCat cfuc 16808  SetCatcsetc 16931   ×c cxpc 17015   1stF c1stf 17016   2ndF c2ndf 17017   ⟨,⟩F cprf 17018   evalF cevlf 17056  HomFchof 17095  Yoncyon 17096
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-rep 4902  ax-sep 4912  ax-nul 4920  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-ral 3065  df-rex 3066  df-reu 3067  df-rab 3069  df-v 3351  df-sbc 3586  df-csb 3681  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-sn 4315  df-pr 4317  df-op 4321  df-uni 4573  df-iun 4654  df-br 4785  df-opab 4845  df-mpt 4862  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-ov 6795  df-oprab 6796  df-mpt2 6797
This theorem is referenced by:  yonedalem4b  17123  yonedalem4c  17124  yonffthlem  17129
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