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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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