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Theorem yonedalem3a 16835
Description: Lemma for yoneda 16844. (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 477 . . . . . . 7 ((𝑓 = 𝐹𝑥 = 𝑋) → 𝑥 = 𝑋)
43fveq2d 6152 . . . . . 6 ((𝑓 = 𝐹𝑥 = 𝑋) → ((1st𝑌)‘𝑥) = ((1st𝑌)‘𝑋))
5 simpl 473 . . . . . 6 ((𝑓 = 𝐹𝑥 = 𝑋) → 𝑓 = 𝐹)
64, 5oveq12d 6622 . . . . 5 ((𝑓 = 𝐹𝑥 = 𝑋) → (((1st𝑌)‘𝑥)(𝑂 Nat 𝑆)𝑓) = (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹))
73fveq2d 6152 . . . . . 6 ((𝑓 = 𝐹𝑥 = 𝑋) → (𝑎𝑥) = (𝑎𝑋))
83fveq2d 6152 . . . . . 6 ((𝑓 = 𝐹𝑥 = 𝑋) → ( 1𝑥) = ( 1𝑋))
97, 8fveq12d 6154 . . . . 5 ((𝑓 = 𝐹𝑥 = 𝑋) → ((𝑎𝑥)‘( 1𝑥)) = ((𝑎𝑋)‘( 1𝑋)))
106, 9mpteq12dv 4693 . . . 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 6632 . . . . 5 (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹) ∈ V
1312mptex 6440 . . . 4 (𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹) ↦ ((𝑎𝑋)‘( 1𝑋))) ∈ V
1410, 11, 13ovmpt2a 6744 . . 3 ((𝐹 ∈ (𝑂 Func 𝑆) ∧ 𝑋𝐵) → (𝐹𝑀𝑋) = (𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹) ↦ ((𝑎𝑋)‘( 1𝑋))))
151, 2, 14syl2anc 692 . 2 (𝜑 → (𝐹𝑀𝑋) = (𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹) ↦ ((𝑎𝑋)‘( 1𝑋))))
16 eqid 2621 . . . . . . 7 (𝑂 Nat 𝑆) = (𝑂 Nat 𝑆)
17 simpr 477 . . . . . . . 8 ((𝜑𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → 𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹))
1816, 17nat1st2nd 16532 . . . . . . 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 16299 . . . . . . 7 𝐵 = (Base‘𝑂)
22 eqid 2621 . . . . . . 7 (Hom ‘𝑆) = (Hom ‘𝑆)
232adantr 481 . . . . . . 7 ((𝜑𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → 𝑋𝐵)
2416, 18, 21, 22, 23natcl 16534 . . . . . 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 3769 . . . . . . . . 9 (𝜑𝑈𝑉)
2926, 28ssexd 4765 . . . . . . . 8 (𝜑𝑈 ∈ V)
3029adantr 481 . . . . . . 7 ((𝜑𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → 𝑈 ∈ V)
31 eqid 2621 . . . . . . . . . . 11 (Base‘𝑆) = (Base‘𝑆)
32 relfunc 16443 . . . . . . . . . . . 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 16824 . . . . . . . . . . . 12 (𝜑 → ((1st𝑌)‘𝑋) ∈ (𝑂 Func 𝑆))
37 1st2ndbr 7162 . . . . . . . . . . . 12 ((Rel (𝑂 Func 𝑆) ∧ ((1st𝑌)‘𝑋) ∈ (𝑂 Func 𝑆)) → (1st ‘((1st𝑌)‘𝑋))(𝑂 Func 𝑆)(2nd ‘((1st𝑌)‘𝑋)))
3832, 36, 37sylancr 694 . . . . . . . . . . 11 (𝜑 → (1st ‘((1st𝑌)‘𝑋))(𝑂 Func 𝑆)(2nd ‘((1st𝑌)‘𝑋)))
3921, 31, 38funcf1 16447 . . . . . . . . . 10 (𝜑 → (1st ‘((1st𝑌)‘𝑋)):𝐵⟶(Base‘𝑆))
4039, 2ffvelrnd 6316 . . . . . . . . 9 (𝜑 → ((1st ‘((1st𝑌)‘𝑋))‘𝑋) ∈ (Base‘𝑆))
4125, 29setcbas 16649 . . . . . . . . 9 (𝜑𝑈 = (Base‘𝑆))
4240, 41eleqtrrd 2701 . . . . . . . 8 (𝜑 → ((1st ‘((1st𝑌)‘𝑋))‘𝑋) ∈ 𝑈)
4342adantr 481 . . . . . . 7 ((𝜑𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → ((1st ‘((1st𝑌)‘𝑋))‘𝑋) ∈ 𝑈)
44 1st2ndbr 7162 . . . . . . . . . . . 12 ((Rel (𝑂 Func 𝑆) ∧ 𝐹 ∈ (𝑂 Func 𝑆)) → (1st𝐹)(𝑂 Func 𝑆)(2nd𝐹))
4532, 1, 44sylancr 694 . . . . . . . . . . 11 (𝜑 → (1st𝐹)(𝑂 Func 𝑆)(2nd𝐹))
4621, 31, 45funcf1 16447 . . . . . . . . . 10 (𝜑 → (1st𝐹):𝐵⟶(Base‘𝑆))
4746, 2ffvelrnd 6316 . . . . . . . . 9 (𝜑 → ((1st𝐹)‘𝑋) ∈ (Base‘𝑆))
4847, 41eleqtrrd 2701 . . . . . . . 8 (𝜑 → ((1st𝐹)‘𝑋) ∈ 𝑈)
4948adantr 481 . . . . . . 7 ((𝜑𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → ((1st𝐹)‘𝑋) ∈ 𝑈)
5025, 30, 22, 43, 49elsetchom 16652 . . . . . 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 2621 . . . . . . . 8 (Hom ‘𝐶) = (Hom ‘𝐶)
53 yoneda.1 . . . . . . . 8 1 = (Id‘𝐶)
5420, 52, 53, 34, 2catidcl 16264 . . . . . . 7 (𝜑 → ( 1𝑋) ∈ (𝑋(Hom ‘𝐶)𝑋))
5533, 20, 34, 2, 52, 2yon11 16825 . . . . . . 7 (𝜑 → ((1st ‘((1st𝑌)‘𝑋))‘𝑋) = (𝑋(Hom ‘𝐶)𝑋))
5654, 55eleqtrrd 2701 . . . . . 6 (𝜑 → ( 1𝑋) ∈ ((1st ‘((1st𝑌)‘𝑋))‘𝑋))
5756adantr 481 . . . . 5 ((𝜑𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → ( 1𝑋) ∈ ((1st ‘((1st𝑌)‘𝑋))‘𝑋))
5851, 57ffvelrnd 6316 . . . 4 ((𝜑𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → ((𝑎𝑋)‘( 1𝑋)) ∈ ((1st𝐹)‘𝑋))
59 eqid 2621 . . . 4 (𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹) ↦ ((𝑎𝑋)‘( 1𝑋))) = (𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹) ↦ ((𝑎𝑋)‘( 1𝑋)))
6058, 59fmptd 6340 . . 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 16834 . . . 4 (𝜑 → (𝐹(1st𝑍)𝑋) = (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹))
6819oppccat 16303 . . . . . 6 (𝐶 ∈ Cat → 𝑂 ∈ Cat)
6934, 68syl 17 . . . . 5 (𝜑𝑂 ∈ Cat)
7025setccat 16656 . . . . . 6 (𝑈 ∈ V → 𝑆 ∈ Cat)
7129, 70syl 17 . . . . 5 (𝜑𝑆 ∈ Cat)
7265, 69, 71, 21, 1, 2evlf1 16781 . . . 4 (𝜑 → (𝐹(1st𝐸)𝑋) = ((1st𝐹)‘𝑋))
7315, 67, 72feq123d 5991 . . 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 554 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 384   = wceq 1480  wcel 1987  Vcvv 3186  cun 3553  wss 3555  cop 4154   class class class wbr 4613  cmpt 4673  ran crn 5075  Rel wrel 5079  wf 5843  cfv 5847  (class class class)co 6604  cmpt2 6606  1st c1st 7111  2nd c2nd 7112  tpos ctpos 7296  Basecbs 15781  Hom chom 15873  Catccat 16246  Idccid 16247  Homf chomf 16248  oppCatcoppc 16292   Func cfunc 16435  func ccofu 16437   Nat cnat 16522   FuncCat cfuc 16523  SetCatcsetc 16646   ×c cxpc 16729   1stF c1stf 16730   2ndF c2ndf 16731   ⟨,⟩F cprf 16732   evalF cevlf 16770  HomFchof 16809  Yoncyon 16810
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902  ax-cnex 9936  ax-resscn 9937  ax-1cn 9938  ax-icn 9939  ax-addcl 9940  ax-addrcl 9941  ax-mulcl 9942  ax-mulrcl 9943  ax-mulcom 9944  ax-addass 9945  ax-mulass 9946  ax-distr 9947  ax-i2m1 9948  ax-1ne0 9949  ax-1rid 9950  ax-rnegex 9951  ax-rrecex 9952  ax-cnre 9953  ax-pre-lttri 9954  ax-pre-lttrn 9955  ax-pre-ltadd 9956  ax-pre-mulgt0 9957
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-fal 1486  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-int 4441  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-riota 6565  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-om 7013  df-1st 7113  df-2nd 7114  df-tpos 7297  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-1o 7505  df-oadd 7509  df-er 7687  df-map 7804  df-ixp 7853  df-en 7900  df-dom 7901  df-sdom 7902  df-fin 7903  df-pnf 10020  df-mnf 10021  df-xr 10022  df-ltxr 10023  df-le 10024  df-sub 10212  df-neg 10213  df-nn 10965  df-2 11023  df-3 11024  df-4 11025  df-5 11026  df-6 11027  df-7 11028  df-8 11029  df-9 11030  df-n0 11237  df-z 11322  df-dec 11438  df-uz 11632  df-fz 12269  df-struct 15783  df-ndx 15784  df-slot 15785  df-base 15786  df-sets 15787  df-hom 15887  df-cco 15888  df-cat 16250  df-cid 16251  df-homf 16252  df-comf 16253  df-oppc 16293  df-func 16439  df-cofu 16441  df-nat 16524  df-fuc 16525  df-setc 16647  df-xpc 16733  df-1stf 16734  df-2ndf 16735  df-prf 16736  df-evlf 16774  df-curf 16775  df-hof 16811  df-yon 16812
This theorem is referenced by:  yonedalem3b  16840  yonedalem3  16841  yonedainv  16842
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