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Theorem yonedalem3a 17353
Description: Lemma for yoneda 17362. (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 6542 . . . . . 6 ((𝑓 = 𝐹𝑥 = 𝑋) → ((1st𝑌)‘𝑥) = ((1st𝑌)‘𝑋))
5 simpl 483 . . . . . 6 ((𝑓 = 𝐹𝑥 = 𝑋) → 𝑓 = 𝐹)
64, 5oveq12d 7034 . . . . 5 ((𝑓 = 𝐹𝑥 = 𝑋) → (((1st𝑌)‘𝑥)(𝑂 Nat 𝑆)𝑓) = (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹))
73fveq2d 6542 . . . . . 6 ((𝑓 = 𝐹𝑥 = 𝑋) → (𝑎𝑥) = (𝑎𝑋))
83fveq2d 6542 . . . . . 6 ((𝑓 = 𝐹𝑥 = 𝑋) → ( 1𝑥) = ( 1𝑋))
97, 8fveq12d 6545 . . . . 5 ((𝑓 = 𝐹𝑥 = 𝑋) → ((𝑎𝑥)‘( 1𝑥)) = ((𝑎𝑋)‘( 1𝑋)))
106, 9mpteq12dv 5045 . . . 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 7048 . . . . 5 (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹) ∈ V
1312mptex 6852 . . . 4 (𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹) ↦ ((𝑎𝑋)‘( 1𝑋))) ∈ V
1410, 11, 13ovmpoa 7161 . . 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 2795 . . . . . . 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 17050 . . . . . . 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 16817 . . . . . . 7 𝐵 = (Base‘𝑂)
22 eqid 2795 . . . . . . 7 (Hom ‘𝑆) = (Hom ‘𝑆)
232adantr 481 . . . . . . 7 ((𝜑𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → 𝑋𝐵)
2416, 18, 21, 22, 23natcl 17052 . . . . . 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 4085 . . . . . . . . 9 (𝜑𝑈𝑉)
2926, 28ssexd 5119 . . . . . . . 8 (𝜑𝑈 ∈ V)
3029adantr 481 . . . . . . 7 ((𝜑𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → 𝑈 ∈ V)
31 eqid 2795 . . . . . . . . . . 11 (Base‘𝑆) = (Base‘𝑆)
32 relfunc 16961 . . . . . . . . . . . 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 17342 . . . . . . . . . . . 12 (𝜑 → ((1st𝑌)‘𝑋) ∈ (𝑂 Func 𝑆))
37 1st2ndbr 7597 . . . . . . . . . . . 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 16965 . . . . . . . . . 10 (𝜑 → (1st ‘((1st𝑌)‘𝑋)):𝐵⟶(Base‘𝑆))
4039, 2ffvelrnd 6717 . . . . . . . . 9 (𝜑 → ((1st ‘((1st𝑌)‘𝑋))‘𝑋) ∈ (Base‘𝑆))
4125, 29setcbas 17167 . . . . . . . . 9 (𝜑𝑈 = (Base‘𝑆))
4240, 41eleqtrrd 2886 . . . . . . . 8 (𝜑 → ((1st ‘((1st𝑌)‘𝑋))‘𝑋) ∈ 𝑈)
4342adantr 481 . . . . . . 7 ((𝜑𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → ((1st ‘((1st𝑌)‘𝑋))‘𝑋) ∈ 𝑈)
44 1st2ndbr 7597 . . . . . . . . . . . 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 16965 . . . . . . . . . 10 (𝜑 → (1st𝐹):𝐵⟶(Base‘𝑆))
4746, 2ffvelrnd 6717 . . . . . . . . 9 (𝜑 → ((1st𝐹)‘𝑋) ∈ (Base‘𝑆))
4847, 41eleqtrrd 2886 . . . . . . . 8 (𝜑 → ((1st𝐹)‘𝑋) ∈ 𝑈)
4948adantr 481 . . . . . . 7 ((𝜑𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → ((1st𝐹)‘𝑋) ∈ 𝑈)
5025, 30, 22, 43, 49elsetchom 17170 . . . . . 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 233 . . . . 5 ((𝜑𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → (𝑎𝑋):((1st ‘((1st𝑌)‘𝑋))‘𝑋)⟶((1st𝐹)‘𝑋))
52 eqid 2795 . . . . . . . 8 (Hom ‘𝐶) = (Hom ‘𝐶)
53 yoneda.1 . . . . . . . 8 1 = (Id‘𝐶)
5420, 52, 53, 34, 2catidcl 16782 . . . . . . 7 (𝜑 → ( 1𝑋) ∈ (𝑋(Hom ‘𝐶)𝑋))
5533, 20, 34, 2, 52, 2yon11 17343 . . . . . . 7 (𝜑 → ((1st ‘((1st𝑌)‘𝑋))‘𝑋) = (𝑋(Hom ‘𝐶)𝑋))
5654, 55eleqtrrd 2886 . . . . . 6 (𝜑 → ( 1𝑋) ∈ ((1st ‘((1st𝑌)‘𝑋))‘𝑋))
5756adantr 481 . . . . 5 ((𝜑𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → ( 1𝑋) ∈ ((1st ‘((1st𝑌)‘𝑋))‘𝑋))
5851, 57ffvelrnd 6717 . . . 4 ((𝜑𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → ((𝑎𝑋)‘( 1𝑋)) ∈ ((1st𝐹)‘𝑋))
5958fmpttd 6742 . . 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 17352 . . . 4 (𝜑 → (𝐹(1st𝑍)𝑋) = (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹))
6719oppccat 16821 . . . . . 6 (𝐶 ∈ Cat → 𝑂 ∈ Cat)
6834, 67syl 17 . . . . 5 (𝜑𝑂 ∈ Cat)
6925setccat 17174 . . . . . 6 (𝑈 ∈ V → 𝑆 ∈ Cat)
7029, 69syl 17 . . . . 5 (𝜑𝑆 ∈ Cat)
7164, 68, 70, 21, 1, 2evlf1 17299 . . . 4 (𝜑 → (𝐹(1st𝐸)𝑋) = ((1st𝐹)‘𝑋))
7215, 66, 71feq123d 6371 . . 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 258 . 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 1522  wcel 2081  Vcvv 3437  cun 3857  wss 3859  cop 4478   class class class wbr 4962  cmpt 5041  ran crn 5444  Rel wrel 5448  wf 6221  cfv 6225  (class class class)co 7016  cmpo 7018  1st c1st 7543  2nd c2nd 7544  tpos ctpos 7742  Basecbs 16312  Hom chom 16405  Catccat 16764  Idccid 16765  Homf chomf 16766  oppCatcoppc 16810   Func cfunc 16953  func ccofu 16955   Nat cnat 17040   FuncCat cfuc 17041  SetCatcsetc 17164   ×c cxpc 17247   1stF c1stf 17248   2ndF c2ndf 17249   ⟨,⟩F cprf 17250   evalF cevlf 17288  HomFchof 17327  Yoncyon 17328
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-rep 5081  ax-sep 5094  ax-nul 5101  ax-pow 5157  ax-pr 5221  ax-un 7319  ax-cnex 10439  ax-resscn 10440  ax-1cn 10441  ax-icn 10442  ax-addcl 10443  ax-addrcl 10444  ax-mulcl 10445  ax-mulrcl 10446  ax-mulcom 10447  ax-addass 10448  ax-mulass 10449  ax-distr 10450  ax-i2m1 10451  ax-1ne0 10452  ax-1rid 10453  ax-rnegex 10454  ax-rrecex 10455  ax-cnre 10456  ax-pre-lttri 10457  ax-pre-lttrn 10458  ax-pre-ltadd 10459  ax-pre-mulgt0 10460
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3or 1081  df-3an 1082  df-tru 1525  df-fal 1535  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ne 2985  df-nel 3091  df-ral 3110  df-rex 3111  df-reu 3112  df-rmo 3113  df-rab 3114  df-v 3439  df-sbc 3707  df-csb 3812  df-dif 3862  df-un 3864  df-in 3866  df-ss 3874  df-pss 3876  df-nul 4212  df-if 4382  df-pw 4455  df-sn 4473  df-pr 4475  df-tp 4477  df-op 4479  df-uni 4746  df-int 4783  df-iun 4827  df-br 4963  df-opab 5025  df-mpt 5042  df-tr 5064  df-id 5348  df-eprel 5353  df-po 5362  df-so 5363  df-fr 5402  df-we 5404  df-xp 5449  df-rel 5450  df-cnv 5451  df-co 5452  df-dm 5453  df-rn 5454  df-res 5455  df-ima 5456  df-pred 6023  df-ord 6069  df-on 6070  df-lim 6071  df-suc 6072  df-iota 6189  df-fun 6227  df-fn 6228  df-f 6229  df-f1 6230  df-fo 6231  df-f1o 6232  df-fv 6233  df-riota 6977  df-ov 7019  df-oprab 7020  df-mpo 7021  df-om 7437  df-1st 7545  df-2nd 7546  df-tpos 7743  df-wrecs 7798  df-recs 7860  df-rdg 7898  df-1o 7953  df-oadd 7957  df-er 8139  df-map 8258  df-ixp 8311  df-en 8358  df-dom 8359  df-sdom 8360  df-fin 8361  df-pnf 10523  df-mnf 10524  df-xr 10525  df-ltxr 10526  df-le 10527  df-sub 10719  df-neg 10720  df-nn 11487  df-2 11548  df-3 11549  df-4 11550  df-5 11551  df-6 11552  df-7 11553  df-8 11554  df-9 11555  df-n0 11746  df-z 11830  df-dec 11948  df-uz 12094  df-fz 12743  df-struct 16314  df-ndx 16315  df-slot 16316  df-base 16318  df-sets 16319  df-hom 16418  df-cco 16419  df-cat 16768  df-cid 16769  df-homf 16770  df-comf 16771  df-oppc 16811  df-func 16957  df-cofu 16959  df-nat 17042  df-fuc 17043  df-setc 17165  df-xpc 17251  df-1stf 17252  df-2ndf 17253  df-prf 17254  df-evlf 17292  df-curf 17293  df-hof 17329  df-yon 17330
This theorem is referenced by:  yonedalem3b  17358  yonedalem3  17359  yonedainv  17360
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