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Theorem yonedalem3a 17519
Description: Lemma for yoneda 17528. (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 488 . . . . . . 7 ((𝑓 = 𝐹𝑥 = 𝑋) → 𝑥 = 𝑋)
43fveq2d 6653 . . . . . 6 ((𝑓 = 𝐹𝑥 = 𝑋) → ((1st𝑌)‘𝑥) = ((1st𝑌)‘𝑋))
5 simpl 486 . . . . . 6 ((𝑓 = 𝐹𝑥 = 𝑋) → 𝑓 = 𝐹)
64, 5oveq12d 7157 . . . . 5 ((𝑓 = 𝐹𝑥 = 𝑋) → (((1st𝑌)‘𝑥)(𝑂 Nat 𝑆)𝑓) = (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹))
73fveq2d 6653 . . . . . 6 ((𝑓 = 𝐹𝑥 = 𝑋) → (𝑎𝑥) = (𝑎𝑋))
83fveq2d 6653 . . . . . 6 ((𝑓 = 𝐹𝑥 = 𝑋) → ( 1𝑥) = ( 1𝑋))
97, 8fveq12d 6656 . . . . 5 ((𝑓 = 𝐹𝑥 = 𝑋) → ((𝑎𝑥)‘( 1𝑥)) = ((𝑎𝑋)‘( 1𝑋)))
106, 9mpteq12dv 5118 . . . 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 7172 . . . . 5 (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹) ∈ V
1312mptex 6967 . . . 4 (𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹) ↦ ((𝑎𝑋)‘( 1𝑋))) ∈ V
1410, 11, 13ovmpoa 7288 . . 3 ((𝐹 ∈ (𝑂 Func 𝑆) ∧ 𝑋𝐵) → (𝐹𝑀𝑋) = (𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹) ↦ ((𝑎𝑋)‘( 1𝑋))))
151, 2, 14syl2anc 587 . 2 (𝜑 → (𝐹𝑀𝑋) = (𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹) ↦ ((𝑎𝑋)‘( 1𝑋))))
16 eqid 2801 . . . . . . 7 (𝑂 Nat 𝑆) = (𝑂 Nat 𝑆)
17 simpr 488 . . . . . . . 8 ((𝜑𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → 𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹))
1816, 17nat1st2nd 17216 . . . . . . 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 16983 . . . . . . 7 𝐵 = (Base‘𝑂)
22 eqid 2801 . . . . . . 7 (Hom ‘𝑆) = (Hom ‘𝑆)
232adantr 484 . . . . . . 7 ((𝜑𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → 𝑋𝐵)
2416, 18, 21, 22, 23natcl 17218 . . . . . 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 4118 . . . . . . . . 9 (𝜑𝑈𝑉)
2926, 28ssexd 5195 . . . . . . . 8 (𝜑𝑈 ∈ V)
3029adantr 484 . . . . . . 7 ((𝜑𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → 𝑈 ∈ V)
31 eqid 2801 . . . . . . . . . . 11 (Base‘𝑆) = (Base‘𝑆)
32 relfunc 17127 . . . . . . . . . . . 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 17508 . . . . . . . . . . . 12 (𝜑 → ((1st𝑌)‘𝑋) ∈ (𝑂 Func 𝑆))
37 1st2ndbr 7727 . . . . . . . . . . . 12 ((Rel (𝑂 Func 𝑆) ∧ ((1st𝑌)‘𝑋) ∈ (𝑂 Func 𝑆)) → (1st ‘((1st𝑌)‘𝑋))(𝑂 Func 𝑆)(2nd ‘((1st𝑌)‘𝑋)))
3832, 36, 37sylancr 590 . . . . . . . . . . 11 (𝜑 → (1st ‘((1st𝑌)‘𝑋))(𝑂 Func 𝑆)(2nd ‘((1st𝑌)‘𝑋)))
3921, 31, 38funcf1 17131 . . . . . . . . . 10 (𝜑 → (1st ‘((1st𝑌)‘𝑋)):𝐵⟶(Base‘𝑆))
4039, 2ffvelrnd 6833 . . . . . . . . 9 (𝜑 → ((1st ‘((1st𝑌)‘𝑋))‘𝑋) ∈ (Base‘𝑆))
4125, 29setcbas 17333 . . . . . . . . 9 (𝜑𝑈 = (Base‘𝑆))
4240, 41eleqtrrd 2896 . . . . . . . 8 (𝜑 → ((1st ‘((1st𝑌)‘𝑋))‘𝑋) ∈ 𝑈)
4342adantr 484 . . . . . . 7 ((𝜑𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → ((1st ‘((1st𝑌)‘𝑋))‘𝑋) ∈ 𝑈)
44 1st2ndbr 7727 . . . . . . . . . . . 12 ((Rel (𝑂 Func 𝑆) ∧ 𝐹 ∈ (𝑂 Func 𝑆)) → (1st𝐹)(𝑂 Func 𝑆)(2nd𝐹))
4532, 1, 44sylancr 590 . . . . . . . . . . 11 (𝜑 → (1st𝐹)(𝑂 Func 𝑆)(2nd𝐹))
4621, 31, 45funcf1 17131 . . . . . . . . . 10 (𝜑 → (1st𝐹):𝐵⟶(Base‘𝑆))
4746, 2ffvelrnd 6833 . . . . . . . . 9 (𝜑 → ((1st𝐹)‘𝑋) ∈ (Base‘𝑆))
4847, 41eleqtrrd 2896 . . . . . . . 8 (𝜑 → ((1st𝐹)‘𝑋) ∈ 𝑈)
4948adantr 484 . . . . . . 7 ((𝜑𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → ((1st𝐹)‘𝑋) ∈ 𝑈)
5025, 30, 22, 43, 49elsetchom 17336 . . . . . 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 235 . . . . 5 ((𝜑𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → (𝑎𝑋):((1st ‘((1st𝑌)‘𝑋))‘𝑋)⟶((1st𝐹)‘𝑋))
52 eqid 2801 . . . . . . . 8 (Hom ‘𝐶) = (Hom ‘𝐶)
53 yoneda.1 . . . . . . . 8 1 = (Id‘𝐶)
5420, 52, 53, 34, 2catidcl 16948 . . . . . . 7 (𝜑 → ( 1𝑋) ∈ (𝑋(Hom ‘𝐶)𝑋))
5533, 20, 34, 2, 52, 2yon11 17509 . . . . . . 7 (𝜑 → ((1st ‘((1st𝑌)‘𝑋))‘𝑋) = (𝑋(Hom ‘𝐶)𝑋))
5654, 55eleqtrrd 2896 . . . . . 6 (𝜑 → ( 1𝑋) ∈ ((1st ‘((1st𝑌)‘𝑋))‘𝑋))
5756adantr 484 . . . . 5 ((𝜑𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → ( 1𝑋) ∈ ((1st ‘((1st𝑌)‘𝑋))‘𝑋))
5851, 57ffvelrnd 6833 . . . 4 ((𝜑𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → ((𝑎𝑋)‘( 1𝑋)) ∈ ((1st𝐹)‘𝑋))
5958fmpttd 6860 . . 3 (𝜑 → (𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹) ↦ ((𝑎𝑋)‘( 1𝑋))):(((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)⟶((1st𝐹)‘𝑋))
60 yoneda.t . . . . 5 𝑇 = (SetCat‘𝑉)
61 yoneda.q . . . . 5 𝑄 = (𝑂 FuncCat 𝑆)
62 yoneda.h . . . . 5 𝐻 = (HomF𝑄)
63 yoneda.r . . . . 5 𝑅 = ((𝑄 ×c 𝑂) FuncCat 𝑇)
64 yoneda.e . . . . 5 𝐸 = (𝑂 evalF 𝑆)
65 yoneda.z . . . . 5 𝑍 = (𝐻func ((⟨(1st𝑌), tpos (2nd𝑌)⟩ ∘func (𝑄 2ndF 𝑂)) ⟨,⟩F (𝑄 1stF 𝑂)))
6633, 20, 53, 19, 25, 60, 61, 62, 63, 64, 65, 34, 26, 35, 27, 1, 2yonedalem21 17518 . . . 4 (𝜑 → (𝐹(1st𝑍)𝑋) = (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹))
6719oppccat 16987 . . . . . 6 (𝐶 ∈ Cat → 𝑂 ∈ Cat)
6834, 67syl 17 . . . . 5 (𝜑𝑂 ∈ Cat)
6925setccat 17340 . . . . . 6 (𝑈 ∈ V → 𝑆 ∈ Cat)
7029, 69syl 17 . . . . 5 (𝜑𝑆 ∈ Cat)
7164, 68, 70, 21, 1, 2evlf1 17465 . . . 4 (𝜑 → (𝐹(1st𝐸)𝑋) = ((1st𝐹)‘𝑋))
7215, 66, 71feq123d 6480 . . 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)))
7359, 72mpbird 260 . 2 (𝜑 → (𝐹𝑀𝑋):(𝐹(1st𝑍)𝑋)⟶(𝐹(1st𝐸)𝑋))
7415, 73jca 515 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 399   = wceq 1538  wcel 2112  Vcvv 3444  cun 3882  wss 3884  cop 4534   class class class wbr 5033  cmpt 5113  ran crn 5524  Rel wrel 5528  wf 6324  cfv 6328  (class class class)co 7139  cmpo 7141  1st c1st 7673  2nd c2nd 7674  tpos ctpos 7878  Basecbs 16478  Hom chom 16571  Catccat 16930  Idccid 16931  Homf chomf 16932  oppCatcoppc 16976   Func cfunc 17119  func ccofu 17121   Nat cnat 17206   FuncCat cfuc 17207  SetCatcsetc 17330   ×c cxpc 17413   1stF c1stf 17414   2ndF c2ndf 17415   ⟨,⟩F cprf 17416   evalF cevlf 17454  HomFchof 17493  Yoncyon 17494
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445  ax-cnex 10586  ax-resscn 10587  ax-1cn 10588  ax-icn 10589  ax-addcl 10590  ax-addrcl 10591  ax-mulcl 10592  ax-mulrcl 10593  ax-mulcom 10594  ax-addass 10595  ax-mulass 10596  ax-distr 10597  ax-i2m1 10598  ax-1ne0 10599  ax-1rid 10600  ax-rnegex 10601  ax-rrecex 10602  ax-cnre 10603  ax-pre-lttri 10604  ax-pre-lttrn 10605  ax-pre-ltadd 10606  ax-pre-mulgt0 10607
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-nel 3095  df-ral 3114  df-rex 3115  df-reu 3116  df-rmo 3117  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-pss 3903  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-tp 4533  df-op 4535  df-uni 4804  df-int 4842  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-tr 5140  df-id 5428  df-eprel 5433  df-po 5442  df-so 5443  df-fr 5482  df-we 5484  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-pred 6120  df-ord 6166  df-on 6167  df-lim 6168  df-suc 6169  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-riota 7097  df-ov 7142  df-oprab 7143  df-mpo 7144  df-om 7565  df-1st 7675  df-2nd 7676  df-tpos 7879  df-wrecs 7934  df-recs 7995  df-rdg 8033  df-1o 8089  df-oadd 8093  df-er 8276  df-map 8395  df-ixp 8449  df-en 8497  df-dom 8498  df-sdom 8499  df-fin 8500  df-pnf 10670  df-mnf 10671  df-xr 10672  df-ltxr 10673  df-le 10674  df-sub 10865  df-neg 10866  df-nn 11630  df-2 11692  df-3 11693  df-4 11694  df-5 11695  df-6 11696  df-7 11697  df-8 11698  df-9 11699  df-n0 11890  df-z 11974  df-dec 12091  df-uz 12236  df-fz 12890  df-struct 16480  df-ndx 16481  df-slot 16482  df-base 16484  df-sets 16485  df-hom 16584  df-cco 16585  df-cat 16934  df-cid 16935  df-homf 16936  df-comf 16937  df-oppc 16977  df-func 17123  df-cofu 17125  df-nat 17208  df-fuc 17209  df-setc 17331  df-xpc 17417  df-1stf 17418  df-2ndf 17419  df-prf 17420  df-evlf 17458  df-curf 17459  df-hof 17495  df-yon 17496
This theorem is referenced by:  yonedalem3b  17524  yonedalem3  17525  yonedainv  17526
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