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Theorem yonedalem4b 16897
Description: Lemma for yoneda 16904. (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))
yonedalem4b.p (𝜑𝑃𝐵)
yonedalem4b.g (𝜑𝐺 ∈ (𝑃(Hom ‘𝐶)𝑋))
Assertion
Ref Expression
yonedalem4b (𝜑 → ((((𝐹𝑁𝑋)‘𝐴)‘𝑃)‘𝐺) = (((𝑋(2nd𝐹)𝑃)‘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,𝐺,𝑔,𝑥,𝑦   𝑓,𝑂,𝑔,𝑢,𝑥,𝑦   𝑆,𝑓,𝑔,𝑢,𝑥,𝑦   𝑄,𝑓,𝑔,𝑢,𝑥   𝑇,𝑓,𝑔,𝑢,𝑦   𝑃,𝑓,𝑔,𝑥,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)   𝑃(𝑢)   𝑄(𝑦)   𝑅(𝑥,𝑦,𝑓,𝑔)   𝑇(𝑥)   𝑈(𝑥,𝑦,𝑢,𝑓,𝑔)   1 (𝑢)   𝐸(𝑥)   𝐺(𝑢)   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 yonedalem4b
StepHypRef Expression
1 yoneda.y . . . . 5 𝑌 = (Yon‘𝐶)
2 yoneda.b . . . . 5 𝐵 = (Base‘𝐶)
3 yoneda.1 . . . . 5 1 = (Id‘𝐶)
4 yoneda.o . . . . 5 𝑂 = (oppCat‘𝐶)
5 yoneda.s . . . . 5 𝑆 = (SetCat‘𝑈)
6 yoneda.t . . . . 5 𝑇 = (SetCat‘𝑉)
7 yoneda.q . . . . 5 𝑄 = (𝑂 FuncCat 𝑆)
8 yoneda.h . . . . 5 𝐻 = (HomF𝑄)
9 yoneda.r . . . . 5 𝑅 = ((𝑄 ×c 𝑂) FuncCat 𝑇)
10 yoneda.e . . . . 5 𝐸 = (𝑂 evalF 𝑆)
11 yoneda.z . . . . 5 𝑍 = (𝐻func ((⟨(1st𝑌), tpos (2nd𝑌)⟩ ∘func (𝑄 2ndF 𝑂)) ⟨,⟩F (𝑄 1stF 𝑂)))
12 yoneda.c . . . . 5 (𝜑𝐶 ∈ Cat)
13 yoneda.w . . . . 5 (𝜑𝑉𝑊)
14 yoneda.u . . . . 5 (𝜑 → ran (Homf𝐶) ⊆ 𝑈)
15 yoneda.v . . . . 5 (𝜑 → (ran (Homf𝑄) ∪ 𝑈) ⊆ 𝑉)
16 yonedalem21.f . . . . 5 (𝜑𝐹 ∈ (𝑂 Func 𝑆))
17 yonedalem21.x . . . . 5 (𝜑𝑋𝐵)
18 yonedalem4.n . . . . 5 𝑁 = (𝑓 ∈ (𝑂 Func 𝑆), 𝑥𝐵 ↦ (𝑢 ∈ ((1st𝑓)‘𝑥) ↦ (𝑦𝐵 ↦ (𝑔 ∈ (𝑦(Hom ‘𝐶)𝑥) ↦ (((𝑥(2nd𝑓)𝑦)‘𝑔)‘𝑢)))))
19 yonedalem4.p . . . . 5 (𝜑𝐴 ∈ ((1st𝐹)‘𝑋))
201, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19yonedalem4a 16896 . . . 4 (𝜑 → ((𝐹𝑁𝑋)‘𝐴) = (𝑦𝐵 ↦ (𝑔 ∈ (𝑦(Hom ‘𝐶)𝑋) ↦ (((𝑋(2nd𝐹)𝑦)‘𝑔)‘𝐴))))
2120fveq1d 6180 . . 3 (𝜑 → (((𝐹𝑁𝑋)‘𝐴)‘𝑃) = ((𝑦𝐵 ↦ (𝑔 ∈ (𝑦(Hom ‘𝐶)𝑋) ↦ (((𝑋(2nd𝐹)𝑦)‘𝑔)‘𝐴)))‘𝑃))
2221fveq1d 6180 . 2 (𝜑 → ((((𝐹𝑁𝑋)‘𝐴)‘𝑃)‘𝐺) = (((𝑦𝐵 ↦ (𝑔 ∈ (𝑦(Hom ‘𝐶)𝑋) ↦ (((𝑋(2nd𝐹)𝑦)‘𝑔)‘𝐴)))‘𝑃)‘𝐺))
23 eqidd 2621 . . 3 (𝜑 → (𝑦𝐵 ↦ (𝑔 ∈ (𝑦(Hom ‘𝐶)𝑋) ↦ (((𝑋(2nd𝐹)𝑦)‘𝑔)‘𝐴))) = (𝑦𝐵 ↦ (𝑔 ∈ (𝑦(Hom ‘𝐶)𝑋) ↦ (((𝑋(2nd𝐹)𝑦)‘𝑔)‘𝐴))))
24 yonedalem4b.p . . . 4 (𝜑𝑃𝐵)
25 ovex 6663 . . . . . 6 (𝑦(Hom ‘𝐶)𝑋) ∈ V
2625mptex 6471 . . . . 5 (𝑔 ∈ (𝑦(Hom ‘𝐶)𝑋) ↦ (((𝑋(2nd𝐹)𝑦)‘𝑔)‘𝐴)) ∈ V
2726a1i 11 . . . 4 ((𝜑𝑦 = 𝑃) → (𝑔 ∈ (𝑦(Hom ‘𝐶)𝑋) ↦ (((𝑋(2nd𝐹)𝑦)‘𝑔)‘𝐴)) ∈ V)
28 yonedalem4b.g . . . . . . 7 (𝜑𝐺 ∈ (𝑃(Hom ‘𝐶)𝑋))
2928adantr 481 . . . . . 6 ((𝜑𝑦 = 𝑃) → 𝐺 ∈ (𝑃(Hom ‘𝐶)𝑋))
30 simpr 477 . . . . . . 7 ((𝜑𝑦 = 𝑃) → 𝑦 = 𝑃)
3130oveq1d 6650 . . . . . 6 ((𝜑𝑦 = 𝑃) → (𝑦(Hom ‘𝐶)𝑋) = (𝑃(Hom ‘𝐶)𝑋))
3229, 31eleqtrrd 2702 . . . . 5 ((𝜑𝑦 = 𝑃) → 𝐺 ∈ (𝑦(Hom ‘𝐶)𝑋))
33 fvexd 6190 . . . . 5 (((𝜑𝑦 = 𝑃) ∧ 𝑔 = 𝐺) → (((𝑋(2nd𝐹)𝑦)‘𝑔)‘𝐴) ∈ V)
34 simplr 791 . . . . . . . 8 (((𝜑𝑦 = 𝑃) ∧ 𝑔 = 𝐺) → 𝑦 = 𝑃)
3534oveq2d 6651 . . . . . . 7 (((𝜑𝑦 = 𝑃) ∧ 𝑔 = 𝐺) → (𝑋(2nd𝐹)𝑦) = (𝑋(2nd𝐹)𝑃))
36 simpr 477 . . . . . . 7 (((𝜑𝑦 = 𝑃) ∧ 𝑔 = 𝐺) → 𝑔 = 𝐺)
3735, 36fveq12d 6184 . . . . . 6 (((𝜑𝑦 = 𝑃) ∧ 𝑔 = 𝐺) → ((𝑋(2nd𝐹)𝑦)‘𝑔) = ((𝑋(2nd𝐹)𝑃)‘𝐺))
3837fveq1d 6180 . . . . 5 (((𝜑𝑦 = 𝑃) ∧ 𝑔 = 𝐺) → (((𝑋(2nd𝐹)𝑦)‘𝑔)‘𝐴) = (((𝑋(2nd𝐹)𝑃)‘𝐺)‘𝐴))
3932, 33, 38fvmptdv2 6284 . . . 4 ((𝜑𝑦 = 𝑃) → (((𝑦𝐵 ↦ (𝑔 ∈ (𝑦(Hom ‘𝐶)𝑋) ↦ (((𝑋(2nd𝐹)𝑦)‘𝑔)‘𝐴)))‘𝑃) = (𝑔 ∈ (𝑦(Hom ‘𝐶)𝑋) ↦ (((𝑋(2nd𝐹)𝑦)‘𝑔)‘𝐴)) → (((𝑦𝐵 ↦ (𝑔 ∈ (𝑦(Hom ‘𝐶)𝑋) ↦ (((𝑋(2nd𝐹)𝑦)‘𝑔)‘𝐴)))‘𝑃)‘𝐺) = (((𝑋(2nd𝐹)𝑃)‘𝐺)‘𝐴)))
40 nfmpt1 4738 . . . 4 𝑦(𝑦𝐵 ↦ (𝑔 ∈ (𝑦(Hom ‘𝐶)𝑋) ↦ (((𝑋(2nd𝐹)𝑦)‘𝑔)‘𝐴)))
41 nffvmpt1 6186 . . . . . 6 𝑦((𝑦𝐵 ↦ (𝑔 ∈ (𝑦(Hom ‘𝐶)𝑋) ↦ (((𝑋(2nd𝐹)𝑦)‘𝑔)‘𝐴)))‘𝑃)
42 nfcv 2762 . . . . . 6 𝑦𝐺
4341, 42nffv 6185 . . . . 5 𝑦(((𝑦𝐵 ↦ (𝑔 ∈ (𝑦(Hom ‘𝐶)𝑋) ↦ (((𝑋(2nd𝐹)𝑦)‘𝑔)‘𝐴)))‘𝑃)‘𝐺)
4443nfeq1 2775 . . . 4 𝑦(((𝑦𝐵 ↦ (𝑔 ∈ (𝑦(Hom ‘𝐶)𝑋) ↦ (((𝑋(2nd𝐹)𝑦)‘𝑔)‘𝐴)))‘𝑃)‘𝐺) = (((𝑋(2nd𝐹)𝑃)‘𝐺)‘𝐴)
4524, 27, 39, 40, 44fvmptdf 6282 . . 3 (𝜑 → ((𝑦𝐵 ↦ (𝑔 ∈ (𝑦(Hom ‘𝐶)𝑋) ↦ (((𝑋(2nd𝐹)𝑦)‘𝑔)‘𝐴))) = (𝑦𝐵 ↦ (𝑔 ∈ (𝑦(Hom ‘𝐶)𝑋) ↦ (((𝑋(2nd𝐹)𝑦)‘𝑔)‘𝐴))) → (((𝑦𝐵 ↦ (𝑔 ∈ (𝑦(Hom ‘𝐶)𝑋) ↦ (((𝑋(2nd𝐹)𝑦)‘𝑔)‘𝐴)))‘𝑃)‘𝐺) = (((𝑋(2nd𝐹)𝑃)‘𝐺)‘𝐴)))
4623, 45mpd 15 . 2 (𝜑 → (((𝑦𝐵 ↦ (𝑔 ∈ (𝑦(Hom ‘𝐶)𝑋) ↦ (((𝑋(2nd𝐹)𝑦)‘𝑔)‘𝐴)))‘𝑃)‘𝐺) = (((𝑋(2nd𝐹)𝑃)‘𝐺)‘𝐴))
4722, 46eqtrd 2654 1 (𝜑 → ((((𝐹𝑁𝑋)‘𝐴)‘𝑃)‘𝐺) = (((𝑋(2nd𝐹)𝑃)‘𝐺)‘𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1481  wcel 1988  Vcvv 3195  cun 3565  wss 3567  cop 4174  cmpt 4720  ran crn 5105  cfv 5876  (class class class)co 6635  cmpt2 6637  1st c1st 7151  2nd c2nd 7152  tpos ctpos 7336  Basecbs 15838  Hom chom 15933  Catccat 16306  Idccid 16307  Homf chomf 16308  oppCatcoppc 16352   Func cfunc 16495  func ccofu 16497   FuncCat cfuc 16583  SetCatcsetc 16706   ×c cxpc 16789   1stF c1stf 16790   2ndF c2ndf 16791   ⟨,⟩F cprf 16792   evalF cevlf 16830  HomFchof 16869  Yoncyon 16870
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-rep 4762  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-reu 2916  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-iun 4513  df-br 4645  df-opab 4704  df-mpt 4721  df-id 5014  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-ov 6638  df-oprab 6639  df-mpt2 6640
This theorem is referenced by:  yonedalem4c  16898  yonedainv  16902
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