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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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