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Theorem yonedalem3a 18089
Description: Lemma for yoneda 18098. (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 485 . . . . . . 7 ((𝑓 = 𝐹𝑥 = 𝑋) → 𝑥 = 𝑋)
43fveq2d 6829 . . . . . 6 ((𝑓 = 𝐹𝑥 = 𝑋) → ((1st𝑌)‘𝑥) = ((1st𝑌)‘𝑋))
5 simpl 483 . . . . . 6 ((𝑓 = 𝐹𝑥 = 𝑋) → 𝑓 = 𝐹)
64, 5oveq12d 7355 . . . . 5 ((𝑓 = 𝐹𝑥 = 𝑋) → (((1st𝑌)‘𝑥)(𝑂 Nat 𝑆)𝑓) = (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹))
73fveq2d 6829 . . . . . 6 ((𝑓 = 𝐹𝑥 = 𝑋) → (𝑎𝑥) = (𝑎𝑋))
83fveq2d 6829 . . . . . 6 ((𝑓 = 𝐹𝑥 = 𝑋) → ( 1𝑥) = ( 1𝑋))
97, 8fveq12d 6832 . . . . 5 ((𝑓 = 𝐹𝑥 = 𝑋) → ((𝑎𝑥)‘( 1𝑥)) = ((𝑎𝑋)‘( 1𝑋)))
106, 9mpteq12dv 5183 . . . 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 7370 . . . . 5 (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹) ∈ V
1312mptex 7155 . . . 4 (𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹) ↦ ((𝑎𝑋)‘( 1𝑋))) ∈ V
1410, 11, 13ovmpoa 7490 . . 3 ((𝐹 ∈ (𝑂 Func 𝑆) ∧ 𝑋𝐵) → (𝐹𝑀𝑋) = (𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹) ↦ ((𝑎𝑋)‘( 1𝑋))))
151, 2, 14syl2anc 584 . 2 (𝜑 → (𝐹𝑀𝑋) = (𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹) ↦ ((𝑎𝑋)‘( 1𝑋))))
16 eqid 2736 . . . . . . 7 (𝑂 Nat 𝑆) = (𝑂 Nat 𝑆)
17 simpr 485 . . . . . . . 8 ((𝜑𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → 𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹))
1816, 17nat1st2nd 17764 . . . . . . 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 17525 . . . . . . 7 𝐵 = (Base‘𝑂)
22 eqid 2736 . . . . . . 7 (Hom ‘𝑆) = (Hom ‘𝑆)
232adantr 481 . . . . . . 7 ((𝜑𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → 𝑋𝐵)
2416, 18, 21, 22, 23natcl 17766 . . . . . 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 4135 . . . . . . . . 9 (𝜑𝑈𝑉)
2926, 28ssexd 5268 . . . . . . . 8 (𝜑𝑈 ∈ V)
3029adantr 481 . . . . . . 7 ((𝜑𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → 𝑈 ∈ V)
31 eqid 2736 . . . . . . . . . . 11 (Base‘𝑆) = (Base‘𝑆)
32 relfunc 17674 . . . . . . . . . . . 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 18078 . . . . . . . . . . . 12 (𝜑 → ((1st𝑌)‘𝑋) ∈ (𝑂 Func 𝑆))
37 1st2ndbr 7951 . . . . . . . . . . . 12 ((Rel (𝑂 Func 𝑆) ∧ ((1st𝑌)‘𝑋) ∈ (𝑂 Func 𝑆)) → (1st ‘((1st𝑌)‘𝑋))(𝑂 Func 𝑆)(2nd ‘((1st𝑌)‘𝑋)))
3832, 36, 37sylancr 587 . . . . . . . . . . 11 (𝜑 → (1st ‘((1st𝑌)‘𝑋))(𝑂 Func 𝑆)(2nd ‘((1st𝑌)‘𝑋)))
3921, 31, 38funcf1 17678 . . . . . . . . . 10 (𝜑 → (1st ‘((1st𝑌)‘𝑋)):𝐵⟶(Base‘𝑆))
4039, 2ffvelcdmd 7018 . . . . . . . . 9 (𝜑 → ((1st ‘((1st𝑌)‘𝑋))‘𝑋) ∈ (Base‘𝑆))
4125, 29setcbas 17890 . . . . . . . . 9 (𝜑𝑈 = (Base‘𝑆))
4240, 41eleqtrrd 2840 . . . . . . . 8 (𝜑 → ((1st ‘((1st𝑌)‘𝑋))‘𝑋) ∈ 𝑈)
4342adantr 481 . . . . . . 7 ((𝜑𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → ((1st ‘((1st𝑌)‘𝑋))‘𝑋) ∈ 𝑈)
44 1st2ndbr 7951 . . . . . . . . . . . 12 ((Rel (𝑂 Func 𝑆) ∧ 𝐹 ∈ (𝑂 Func 𝑆)) → (1st𝐹)(𝑂 Func 𝑆)(2nd𝐹))
4532, 1, 44sylancr 587 . . . . . . . . . . 11 (𝜑 → (1st𝐹)(𝑂 Func 𝑆)(2nd𝐹))
4621, 31, 45funcf1 17678 . . . . . . . . . 10 (𝜑 → (1st𝐹):𝐵⟶(Base‘𝑆))
4746, 2ffvelcdmd 7018 . . . . . . . . 9 (𝜑 → ((1st𝐹)‘𝑋) ∈ (Base‘𝑆))
4847, 41eleqtrrd 2840 . . . . . . . 8 (𝜑 → ((1st𝐹)‘𝑋) ∈ 𝑈)
4948adantr 481 . . . . . . 7 ((𝜑𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → ((1st𝐹)‘𝑋) ∈ 𝑈)
5025, 30, 22, 43, 49elsetchom 17893 . . . . . 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 231 . . . . 5 ((𝜑𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → (𝑎𝑋):((1st ‘((1st𝑌)‘𝑋))‘𝑋)⟶((1st𝐹)‘𝑋))
52 eqid 2736 . . . . . . . 8 (Hom ‘𝐶) = (Hom ‘𝐶)
53 yoneda.1 . . . . . . . 8 1 = (Id‘𝐶)
5420, 52, 53, 34, 2catidcl 17488 . . . . . . 7 (𝜑 → ( 1𝑋) ∈ (𝑋(Hom ‘𝐶)𝑋))
5533, 20, 34, 2, 52, 2yon11 18079 . . . . . . 7 (𝜑 → ((1st ‘((1st𝑌)‘𝑋))‘𝑋) = (𝑋(Hom ‘𝐶)𝑋))
5654, 55eleqtrrd 2840 . . . . . 6 (𝜑 → ( 1𝑋) ∈ ((1st ‘((1st𝑌)‘𝑋))‘𝑋))
5756adantr 481 . . . . 5 ((𝜑𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → ( 1𝑋) ∈ ((1st ‘((1st𝑌)‘𝑋))‘𝑋))
5851, 57ffvelcdmd 7018 . . . 4 ((𝜑𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → ((𝑎𝑋)‘( 1𝑋)) ∈ ((1st𝐹)‘𝑋))
5958fmpttd 7045 . . 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 18088 . . . 4 (𝜑 → (𝐹(1st𝑍)𝑋) = (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹))
6719oppccat 17530 . . . . . 6 (𝐶 ∈ Cat → 𝑂 ∈ Cat)
6834, 67syl 17 . . . . 5 (𝜑𝑂 ∈ Cat)
6925setccat 17897 . . . . . 6 (𝑈 ∈ V → 𝑆 ∈ Cat)
7029, 69syl 17 . . . . 5 (𝜑𝑆 ∈ Cat)
7164, 68, 70, 21, 1, 2evlf1 18035 . . . 4 (𝜑 → (𝐹(1st𝐸)𝑋) = ((1st𝐹)‘𝑋))
7215, 66, 71feq123d 6640 . . 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 256 . 2 (𝜑 → (𝐹𝑀𝑋):(𝐹(1st𝑍)𝑋)⟶(𝐹(1st𝐸)𝑋))
7415, 73jca 512 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 396   = wceq 1540  wcel 2105  Vcvv 3441  cun 3896  wss 3898  cop 4579   class class class wbr 5092  cmpt 5175  ran crn 5621  Rel wrel 5625  wf 6475  cfv 6479  (class class class)co 7337  cmpo 7339  1st c1st 7897  2nd c2nd 7898  tpos ctpos 8111  Basecbs 17009  Hom chom 17070  Catccat 17470  Idccid 17471  Homf chomf 17472  oppCatcoppc 17517   Func cfunc 17666  func ccofu 17668   Nat cnat 17754   FuncCat cfuc 17755  SetCatcsetc 17887   ×c cxpc 17982   1stF c1stf 17983   2ndF c2ndf 17984   ⟨,⟩F cprf 17985   evalF cevlf 18024  HomFchof 18063  Yoncyon 18064
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2707  ax-rep 5229  ax-sep 5243  ax-nul 5250  ax-pow 5308  ax-pr 5372  ax-un 7650  ax-cnex 11028  ax-resscn 11029  ax-1cn 11030  ax-icn 11031  ax-addcl 11032  ax-addrcl 11033  ax-mulcl 11034  ax-mulrcl 11035  ax-mulcom 11036  ax-addass 11037  ax-mulass 11038  ax-distr 11039  ax-i2m1 11040  ax-1ne0 11041  ax-1rid 11042  ax-rnegex 11043  ax-rrecex 11044  ax-cnre 11045  ax-pre-lttri 11046  ax-pre-lttrn 11047  ax-pre-ltadd 11048  ax-pre-mulgt0 11049
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3349  df-reu 3350  df-rab 3404  df-v 3443  df-sbc 3728  df-csb 3844  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3917  df-nul 4270  df-if 4474  df-pw 4549  df-sn 4574  df-pr 4576  df-tp 4578  df-op 4580  df-uni 4853  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5176  df-tr 5210  df-id 5518  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5575  df-we 5577  df-xp 5626  df-rel 5627  df-cnv 5628  df-co 5629  df-dm 5630  df-rn 5631  df-res 5632  df-ima 5633  df-pred 6238  df-ord 6305  df-on 6306  df-lim 6307  df-suc 6308  df-iota 6431  df-fun 6481  df-fn 6482  df-f 6483  df-f1 6484  df-fo 6485  df-f1o 6486  df-fv 6487  df-riota 7293  df-ov 7340  df-oprab 7341  df-mpo 7342  df-om 7781  df-1st 7899  df-2nd 7900  df-tpos 8112  df-frecs 8167  df-wrecs 8198  df-recs 8272  df-rdg 8311  df-1o 8367  df-er 8569  df-map 8688  df-ixp 8757  df-en 8805  df-dom 8806  df-sdom 8807  df-fin 8808  df-pnf 11112  df-mnf 11113  df-xr 11114  df-ltxr 11115  df-le 11116  df-sub 11308  df-neg 11309  df-nn 12075  df-2 12137  df-3 12138  df-4 12139  df-5 12140  df-6 12141  df-7 12142  df-8 12143  df-9 12144  df-n0 12335  df-z 12421  df-dec 12539  df-uz 12684  df-fz 13341  df-struct 16945  df-sets 16962  df-slot 16980  df-ndx 16992  df-base 17010  df-hom 17083  df-cco 17084  df-cat 17474  df-cid 17475  df-homf 17476  df-comf 17477  df-oppc 17518  df-func 17670  df-cofu 17672  df-nat 17756  df-fuc 17757  df-setc 17888  df-xpc 17986  df-1stf 17987  df-2ndf 17988  df-prf 17989  df-evlf 18028  df-curf 18029  df-hof 18065  df-yon 18066
This theorem is referenced by:  yonedalem3b  18094  yonedalem3  18095  yonedainv  18096
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