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Theorem yonedalem3a 17107
Description: Lemma for yoneda 17116. (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)
yonedalem3a.m 𝑀 = (𝑓 ∈ (𝑂 Func 𝑆), 𝑥𝐵 ↦ (𝑎 ∈ (((1st𝑌)‘𝑥)(𝑂 Nat 𝑆)𝑓) ↦ ((𝑎𝑥)‘( 1𝑥))))
Assertion
Ref Expression
yonedalem3a (𝜑 → ((𝐹𝑀𝑋) = (𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹) ↦ ((𝑎𝑋)‘( 1𝑋))) ∧ (𝐹𝑀𝑋):(𝐹(1st𝑍)𝑋)⟶(𝐹(1st𝐸)𝑋)))
Distinct variable groups:   𝑓,𝑎,𝑥, 1   𝐶,𝑎,𝑓,𝑥   𝐸,𝑎,𝑓   𝐹,𝑎,𝑓,𝑥   𝐵,𝑎,𝑓,𝑥   𝑂,𝑎,𝑓,𝑥   𝑆,𝑎,𝑓,𝑥   𝑄,𝑎,𝑓,𝑥   𝑇,𝑓   𝜑,𝑎,𝑓,𝑥   𝑌,𝑎,𝑓,𝑥   𝑍,𝑎,𝑓,𝑥   𝑋,𝑎,𝑓,𝑥
Allowed substitution hints:   𝑅(𝑥,𝑓,𝑎)   𝑇(𝑥,𝑎)   𝑈(𝑥,𝑓,𝑎)   𝐸(𝑥)   𝐻(𝑥,𝑓,𝑎)   𝑀(𝑥,𝑓,𝑎)   𝑉(𝑥,𝑓,𝑎)   𝑊(𝑥,𝑓,𝑎)

Proof of Theorem yonedalem3a
StepHypRef Expression
1 yonedalem21.f . . 3 (𝜑𝐹 ∈ (𝑂 Func 𝑆))
2 yonedalem21.x . . 3 (𝜑𝑋𝐵)
3 simpr 479 . . . . . . 7 ((𝑓 = 𝐹𝑥 = 𝑋) → 𝑥 = 𝑋)
43fveq2d 6348 . . . . . 6 ((𝑓 = 𝐹𝑥 = 𝑋) → ((1st𝑌)‘𝑥) = ((1st𝑌)‘𝑋))
5 simpl 474 . . . . . 6 ((𝑓 = 𝐹𝑥 = 𝑋) → 𝑓 = 𝐹)
64, 5oveq12d 6823 . . . . 5 ((𝑓 = 𝐹𝑥 = 𝑋) → (((1st𝑌)‘𝑥)(𝑂 Nat 𝑆)𝑓) = (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹))
73fveq2d 6348 . . . . . 6 ((𝑓 = 𝐹𝑥 = 𝑋) → (𝑎𝑥) = (𝑎𝑋))
83fveq2d 6348 . . . . . 6 ((𝑓 = 𝐹𝑥 = 𝑋) → ( 1𝑥) = ( 1𝑋))
97, 8fveq12d 6350 . . . . 5 ((𝑓 = 𝐹𝑥 = 𝑋) → ((𝑎𝑥)‘( 1𝑥)) = ((𝑎𝑋)‘( 1𝑋)))
106, 9mpteq12dv 4877 . . . 4 ((𝑓 = 𝐹𝑥 = 𝑋) → (𝑎 ∈ (((1st𝑌)‘𝑥)(𝑂 Nat 𝑆)𝑓) ↦ ((𝑎𝑥)‘( 1𝑥))) = (𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹) ↦ ((𝑎𝑋)‘( 1𝑋))))
11 yonedalem3a.m . . . 4 𝑀 = (𝑓 ∈ (𝑂 Func 𝑆), 𝑥𝐵 ↦ (𝑎 ∈ (((1st𝑌)‘𝑥)(𝑂 Nat 𝑆)𝑓) ↦ ((𝑎𝑥)‘( 1𝑥))))
12 ovex 6833 . . . . 5 (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹) ∈ V
1312mptex 6642 . . . 4 (𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹) ↦ ((𝑎𝑋)‘( 1𝑋))) ∈ V
1410, 11, 13ovmpt2a 6948 . . 3 ((𝐹 ∈ (𝑂 Func 𝑆) ∧ 𝑋𝐵) → (𝐹𝑀𝑋) = (𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹) ↦ ((𝑎𝑋)‘( 1𝑋))))
151, 2, 14syl2anc 696 . 2 (𝜑 → (𝐹𝑀𝑋) = (𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹) ↦ ((𝑎𝑋)‘( 1𝑋))))
16 eqid 2752 . . . . . . 7 (𝑂 Nat 𝑆) = (𝑂 Nat 𝑆)
17 simpr 479 . . . . . . . 8 ((𝜑𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → 𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹))
1816, 17nat1st2nd 16804 . . . . . . 7 ((𝜑𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → 𝑎 ∈ (⟨(1st ‘((1st𝑌)‘𝑋)), (2nd ‘((1st𝑌)‘𝑋))⟩(𝑂 Nat 𝑆)⟨(1st𝐹), (2nd𝐹)⟩))
19 yoneda.o . . . . . . . 8 𝑂 = (oppCat‘𝐶)
20 yoneda.b . . . . . . . 8 𝐵 = (Base‘𝐶)
2119, 20oppcbas 16571 . . . . . . 7 𝐵 = (Base‘𝑂)
22 eqid 2752 . . . . . . 7 (Hom ‘𝑆) = (Hom ‘𝑆)
232adantr 472 . . . . . . 7 ((𝜑𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → 𝑋𝐵)
2416, 18, 21, 22, 23natcl 16806 . . . . . 6 ((𝜑𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → (𝑎𝑋) ∈ (((1st ‘((1st𝑌)‘𝑋))‘𝑋)(Hom ‘𝑆)((1st𝐹)‘𝑋)))
25 yoneda.s . . . . . . 7 𝑆 = (SetCat‘𝑈)
26 yoneda.w . . . . . . . . 9 (𝜑𝑉𝑊)
27 yoneda.v . . . . . . . . . 10 (𝜑 → (ran (Homf𝑄) ∪ 𝑈) ⊆ 𝑉)
2827unssbd 3926 . . . . . . . . 9 (𝜑𝑈𝑉)
2926, 28ssexd 4949 . . . . . . . 8 (𝜑𝑈 ∈ V)
3029adantr 472 . . . . . . 7 ((𝜑𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → 𝑈 ∈ V)
31 eqid 2752 . . . . . . . . . . 11 (Base‘𝑆) = (Base‘𝑆)
32 relfunc 16715 . . . . . . . . . . . 12 Rel (𝑂 Func 𝑆)
33 yoneda.y . . . . . . . . . . . . 13 𝑌 = (Yon‘𝐶)
34 yoneda.c . . . . . . . . . . . . 13 (𝜑𝐶 ∈ Cat)
35 yoneda.u . . . . . . . . . . . . 13 (𝜑 → ran (Homf𝐶) ⊆ 𝑈)
3633, 20, 34, 2, 19, 25, 29, 35yon1cl 17096 . . . . . . . . . . . 12 (𝜑 → ((1st𝑌)‘𝑋) ∈ (𝑂 Func 𝑆))
37 1st2ndbr 7376 . . . . . . . . . . . 12 ((Rel (𝑂 Func 𝑆) ∧ ((1st𝑌)‘𝑋) ∈ (𝑂 Func 𝑆)) → (1st ‘((1st𝑌)‘𝑋))(𝑂 Func 𝑆)(2nd ‘((1st𝑌)‘𝑋)))
3832, 36, 37sylancr 698 . . . . . . . . . . 11 (𝜑 → (1st ‘((1st𝑌)‘𝑋))(𝑂 Func 𝑆)(2nd ‘((1st𝑌)‘𝑋)))
3921, 31, 38funcf1 16719 . . . . . . . . . 10 (𝜑 → (1st ‘((1st𝑌)‘𝑋)):𝐵⟶(Base‘𝑆))
4039, 2ffvelrnd 6515 . . . . . . . . 9 (𝜑 → ((1st ‘((1st𝑌)‘𝑋))‘𝑋) ∈ (Base‘𝑆))
4125, 29setcbas 16921 . . . . . . . . 9 (𝜑𝑈 = (Base‘𝑆))
4240, 41eleqtrrd 2834 . . . . . . . 8 (𝜑 → ((1st ‘((1st𝑌)‘𝑋))‘𝑋) ∈ 𝑈)
4342adantr 472 . . . . . . 7 ((𝜑𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → ((1st ‘((1st𝑌)‘𝑋))‘𝑋) ∈ 𝑈)
44 1st2ndbr 7376 . . . . . . . . . . . 12 ((Rel (𝑂 Func 𝑆) ∧ 𝐹 ∈ (𝑂 Func 𝑆)) → (1st𝐹)(𝑂 Func 𝑆)(2nd𝐹))
4532, 1, 44sylancr 698 . . . . . . . . . . 11 (𝜑 → (1st𝐹)(𝑂 Func 𝑆)(2nd𝐹))
4621, 31, 45funcf1 16719 . . . . . . . . . 10 (𝜑 → (1st𝐹):𝐵⟶(Base‘𝑆))
4746, 2ffvelrnd 6515 . . . . . . . . 9 (𝜑 → ((1st𝐹)‘𝑋) ∈ (Base‘𝑆))
4847, 41eleqtrrd 2834 . . . . . . . 8 (𝜑 → ((1st𝐹)‘𝑋) ∈ 𝑈)
4948adantr 472 . . . . . . 7 ((𝜑𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → ((1st𝐹)‘𝑋) ∈ 𝑈)
5025, 30, 22, 43, 49elsetchom 16924 . . . . . 6 ((𝜑𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → ((𝑎𝑋) ∈ (((1st ‘((1st𝑌)‘𝑋))‘𝑋)(Hom ‘𝑆)((1st𝐹)‘𝑋)) ↔ (𝑎𝑋):((1st ‘((1st𝑌)‘𝑋))‘𝑋)⟶((1st𝐹)‘𝑋)))
5124, 50mpbid 222 . . . . 5 ((𝜑𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → (𝑎𝑋):((1st ‘((1st𝑌)‘𝑋))‘𝑋)⟶((1st𝐹)‘𝑋))
52 eqid 2752 . . . . . . . 8 (Hom ‘𝐶) = (Hom ‘𝐶)
53 yoneda.1 . . . . . . . 8 1 = (Id‘𝐶)
5420, 52, 53, 34, 2catidcl 16536 . . . . . . 7 (𝜑 → ( 1𝑋) ∈ (𝑋(Hom ‘𝐶)𝑋))
5533, 20, 34, 2, 52, 2yon11 17097 . . . . . . 7 (𝜑 → ((1st ‘((1st𝑌)‘𝑋))‘𝑋) = (𝑋(Hom ‘𝐶)𝑋))
5654, 55eleqtrrd 2834 . . . . . 6 (𝜑 → ( 1𝑋) ∈ ((1st ‘((1st𝑌)‘𝑋))‘𝑋))
5756adantr 472 . . . . 5 ((𝜑𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → ( 1𝑋) ∈ ((1st ‘((1st𝑌)‘𝑋))‘𝑋))
5851, 57ffvelrnd 6515 . . . 4 ((𝜑𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → ((𝑎𝑋)‘( 1𝑋)) ∈ ((1st𝐹)‘𝑋))
59 eqid 2752 . . . 4 (𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹) ↦ ((𝑎𝑋)‘( 1𝑋))) = (𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹) ↦ ((𝑎𝑋)‘( 1𝑋)))
6058, 59fmptd 6540 . . 3 (𝜑 → (𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹) ↦ ((𝑎𝑋)‘( 1𝑋))):(((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)⟶((1st𝐹)‘𝑋))
61 yoneda.t . . . . 5 𝑇 = (SetCat‘𝑉)
62 yoneda.q . . . . 5 𝑄 = (𝑂 FuncCat 𝑆)
63 yoneda.h . . . . 5 𝐻 = (HomF𝑄)
64 yoneda.r . . . . 5 𝑅 = ((𝑄 ×c 𝑂) FuncCat 𝑇)
65 yoneda.e . . . . 5 𝐸 = (𝑂 evalF 𝑆)
66 yoneda.z . . . . 5 𝑍 = (𝐻func ((⟨(1st𝑌), tpos (2nd𝑌)⟩ ∘func (𝑄 2ndF 𝑂)) ⟨,⟩F (𝑄 1stF 𝑂)))
6733, 20, 53, 19, 25, 61, 62, 63, 64, 65, 66, 34, 26, 35, 27, 1, 2yonedalem21 17106 . . . 4 (𝜑 → (𝐹(1st𝑍)𝑋) = (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹))
6819oppccat 16575 . . . . . 6 (𝐶 ∈ Cat → 𝑂 ∈ Cat)
6934, 68syl 17 . . . . 5 (𝜑𝑂 ∈ Cat)
7025setccat 16928 . . . . . 6 (𝑈 ∈ V → 𝑆 ∈ Cat)
7129, 70syl 17 . . . . 5 (𝜑𝑆 ∈ Cat)
7265, 69, 71, 21, 1, 2evlf1 17053 . . . 4 (𝜑 → (𝐹(1st𝐸)𝑋) = ((1st𝐹)‘𝑋))
7315, 67, 72feq123d 6187 . . 3 (𝜑 → ((𝐹𝑀𝑋):(𝐹(1st𝑍)𝑋)⟶(𝐹(1st𝐸)𝑋) ↔ (𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹) ↦ ((𝑎𝑋)‘( 1𝑋))):(((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)⟶((1st𝐹)‘𝑋)))
7460, 73mpbird 247 . 2 (𝜑 → (𝐹𝑀𝑋):(𝐹(1st𝑍)𝑋)⟶(𝐹(1st𝐸)𝑋))
7515, 74jca 555 1 (𝜑 → ((𝐹𝑀𝑋) = (𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹) ↦ ((𝑎𝑋)‘( 1𝑋))) ∧ (𝐹𝑀𝑋):(𝐹(1st𝑍)𝑋)⟶(𝐹(1st𝐸)𝑋)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1624  wcel 2131  Vcvv 3332  cun 3705  wss 3707  cop 4319   class class class wbr 4796  cmpt 4873  ran crn 5259  Rel wrel 5263  wf 6037  cfv 6041  (class class class)co 6805  cmpt2 6807  1st c1st 7323  2nd c2nd 7324  tpos ctpos 7512  Basecbs 16051  Hom chom 16146  Catccat 16518  Idccid 16519  Homf chomf 16520  oppCatcoppc 16564   Func cfunc 16707  func ccofu 16709   Nat cnat 16794   FuncCat cfuc 16795  SetCatcsetc 16918   ×c cxpc 17001   1stF c1stf 17002   2ndF c2ndf 17003   ⟨,⟩F cprf 17004   evalF cevlf 17042  HomFchof 17081  Yoncyon 17082
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-8 2133  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732  ax-rep 4915  ax-sep 4925  ax-nul 4933  ax-pow 4984  ax-pr 5047  ax-un 7106  ax-cnex 10176  ax-resscn 10177  ax-1cn 10178  ax-icn 10179  ax-addcl 10180  ax-addrcl 10181  ax-mulcl 10182  ax-mulrcl 10183  ax-mulcom 10184  ax-addass 10185  ax-mulass 10186  ax-distr 10187  ax-i2m1 10188  ax-1ne0 10189  ax-1rid 10190  ax-rnegex 10191  ax-rrecex 10192  ax-cnre 10193  ax-pre-lttri 10194  ax-pre-lttrn 10195  ax-pre-ltadd 10196  ax-pre-mulgt0 10197
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1627  df-fal 1630  df-ex 1846  df-nf 1851  df-sb 2039  df-eu 2603  df-mo 2604  df-clab 2739  df-cleq 2745  df-clel 2748  df-nfc 2883  df-ne 2925  df-nel 3028  df-ral 3047  df-rex 3048  df-reu 3049  df-rmo 3050  df-rab 3051  df-v 3334  df-sbc 3569  df-csb 3667  df-dif 3710  df-un 3712  df-in 3714  df-ss 3721  df-pss 3723  df-nul 4051  df-if 4223  df-pw 4296  df-sn 4314  df-pr 4316  df-tp 4318  df-op 4320  df-uni 4581  df-int 4620  df-iun 4666  df-br 4797  df-opab 4857  df-mpt 4874  df-tr 4897  df-id 5166  df-eprel 5171  df-po 5179  df-so 5180  df-fr 5217  df-we 5219  df-xp 5264  df-rel 5265  df-cnv 5266  df-co 5267  df-dm 5268  df-rn 5269  df-res 5270  df-ima 5271  df-pred 5833  df-ord 5879  df-on 5880  df-lim 5881  df-suc 5882  df-iota 6004  df-fun 6043  df-fn 6044  df-f 6045  df-f1 6046  df-fo 6047  df-f1o 6048  df-fv 6049  df-riota 6766  df-ov 6808  df-oprab 6809  df-mpt2 6810  df-om 7223  df-1st 7325  df-2nd 7326  df-tpos 7513  df-wrecs 7568  df-recs 7629  df-rdg 7667  df-1o 7721  df-oadd 7725  df-er 7903  df-map 8017  df-ixp 8067  df-en 8114  df-dom 8115  df-sdom 8116  df-fin 8117  df-pnf 10260  df-mnf 10261  df-xr 10262  df-ltxr 10263  df-le 10264  df-sub 10452  df-neg 10453  df-nn 11205  df-2 11263  df-3 11264  df-4 11265  df-5 11266  df-6 11267  df-7 11268  df-8 11269  df-9 11270  df-n0 11477  df-z 11562  df-dec 11678  df-uz 11872  df-fz 12512  df-struct 16053  df-ndx 16054  df-slot 16055  df-base 16057  df-sets 16058  df-hom 16160  df-cco 16161  df-cat 16522  df-cid 16523  df-homf 16524  df-comf 16525  df-oppc 16565  df-func 16711  df-cofu 16713  df-nat 16796  df-fuc 16797  df-setc 16919  df-xpc 17005  df-1stf 17006  df-2ndf 17007  df-prf 17008  df-evlf 17046  df-curf 17047  df-hof 17083  df-yon 17084
This theorem is referenced by:  yonedalem3b  17112  yonedalem3  17113  yonedainv  17114
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