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Theorem yonedalem1 17034
Description: Lemma for yoneda 17045. (Contributed by Mario Carneiro, 28-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)
Assertion
Ref Expression
yonedalem1 (𝜑 → (𝑍 ∈ ((𝑄 ×c 𝑂) Func 𝑇) ∧ 𝐸 ∈ ((𝑄 ×c 𝑂) Func 𝑇)))

Proof of Theorem yonedalem1
StepHypRef Expression
1 yoneda.z . . 3 𝑍 = (𝐻func ((⟨(1st𝑌), tpos (2nd𝑌)⟩ ∘func (𝑄 2ndF 𝑂)) ⟨,⟩F (𝑄 1stF 𝑂)))
2 eqid 2724 . . . . 5 ((⟨(1st𝑌), tpos (2nd𝑌)⟩ ∘func (𝑄 2ndF 𝑂)) ⟨,⟩F (𝑄 1stF 𝑂)) = ((⟨(1st𝑌), tpos (2nd𝑌)⟩ ∘func (𝑄 2ndF 𝑂)) ⟨,⟩F (𝑄 1stF 𝑂))
3 eqid 2724 . . . . 5 ((oppCat‘𝑄) ×c 𝑄) = ((oppCat‘𝑄) ×c 𝑄)
4 eqid 2724 . . . . . . 7 (𝑄 ×c 𝑂) = (𝑄 ×c 𝑂)
5 yoneda.q . . . . . . . 8 𝑄 = (𝑂 FuncCat 𝑆)
6 yoneda.c . . . . . . . . 9 (𝜑𝐶 ∈ Cat)
7 yoneda.o . . . . . . . . . 10 𝑂 = (oppCat‘𝐶)
87oppccat 16504 . . . . . . . . 9 (𝐶 ∈ Cat → 𝑂 ∈ Cat)
96, 8syl 17 . . . . . . . 8 (𝜑𝑂 ∈ Cat)
10 yoneda.w . . . . . . . . . 10 (𝜑𝑉𝑊)
11 yoneda.v . . . . . . . . . . 11 (𝜑 → (ran (Homf𝑄) ∪ 𝑈) ⊆ 𝑉)
1211unssbd 3899 . . . . . . . . . 10 (𝜑𝑈𝑉)
1310, 12ssexd 4913 . . . . . . . . 9 (𝜑𝑈 ∈ V)
14 yoneda.s . . . . . . . . . 10 𝑆 = (SetCat‘𝑈)
1514setccat 16857 . . . . . . . . 9 (𝑈 ∈ V → 𝑆 ∈ Cat)
1613, 15syl 17 . . . . . . . 8 (𝜑𝑆 ∈ Cat)
175, 9, 16fuccat 16752 . . . . . . 7 (𝜑𝑄 ∈ Cat)
18 eqid 2724 . . . . . . 7 (𝑄 2ndF 𝑂) = (𝑄 2ndF 𝑂)
194, 17, 9, 182ndfcl 16960 . . . . . 6 (𝜑 → (𝑄 2ndF 𝑂) ∈ ((𝑄 ×c 𝑂) Func 𝑂))
20 eqid 2724 . . . . . . . 8 (oppCat‘𝑄) = (oppCat‘𝑄)
21 relfunc 16644 . . . . . . . . 9 Rel (𝐶 Func 𝑄)
22 yoneda.y . . . . . . . . . 10 𝑌 = (Yon‘𝐶)
23 yoneda.u . . . . . . . . . 10 (𝜑 → ran (Homf𝐶) ⊆ 𝑈)
2422, 6, 7, 14, 5, 13, 23yoncl 17024 . . . . . . . . 9 (𝜑𝑌 ∈ (𝐶 Func 𝑄))
25 1st2ndbr 7336 . . . . . . . . 9 ((Rel (𝐶 Func 𝑄) ∧ 𝑌 ∈ (𝐶 Func 𝑄)) → (1st𝑌)(𝐶 Func 𝑄)(2nd𝑌))
2621, 24, 25sylancr 698 . . . . . . . 8 (𝜑 → (1st𝑌)(𝐶 Func 𝑄)(2nd𝑌))
277, 20, 26funcoppc 16657 . . . . . . 7 (𝜑 → (1st𝑌)(𝑂 Func (oppCat‘𝑄))tpos (2nd𝑌))
28 df-br 4761 . . . . . . 7 ((1st𝑌)(𝑂 Func (oppCat‘𝑄))tpos (2nd𝑌) ↔ ⟨(1st𝑌), tpos (2nd𝑌)⟩ ∈ (𝑂 Func (oppCat‘𝑄)))
2927, 28sylib 208 . . . . . 6 (𝜑 → ⟨(1st𝑌), tpos (2nd𝑌)⟩ ∈ (𝑂 Func (oppCat‘𝑄)))
3019, 29cofucl 16670 . . . . 5 (𝜑 → (⟨(1st𝑌), tpos (2nd𝑌)⟩ ∘func (𝑄 2ndF 𝑂)) ∈ ((𝑄 ×c 𝑂) Func (oppCat‘𝑄)))
31 eqid 2724 . . . . . 6 (𝑄 1stF 𝑂) = (𝑄 1stF 𝑂)
324, 17, 9, 311stfcl 16959 . . . . 5 (𝜑 → (𝑄 1stF 𝑂) ∈ ((𝑄 ×c 𝑂) Func 𝑄))
332, 3, 30, 32prfcl 16965 . . . 4 (𝜑 → ((⟨(1st𝑌), tpos (2nd𝑌)⟩ ∘func (𝑄 2ndF 𝑂)) ⟨,⟩F (𝑄 1stF 𝑂)) ∈ ((𝑄 ×c 𝑂) Func ((oppCat‘𝑄) ×c 𝑄)))
34 yoneda.h . . . . 5 𝐻 = (HomF𝑄)
35 yoneda.t . . . . 5 𝑇 = (SetCat‘𝑉)
3611unssad 3898 . . . . 5 (𝜑 → ran (Homf𝑄) ⊆ 𝑉)
3734, 20, 35, 17, 10, 36hofcl 17021 . . . 4 (𝜑𝐻 ∈ (((oppCat‘𝑄) ×c 𝑄) Func 𝑇))
3833, 37cofucl 16670 . . 3 (𝜑 → (𝐻func ((⟨(1st𝑌), tpos (2nd𝑌)⟩ ∘func (𝑄 2ndF 𝑂)) ⟨,⟩F (𝑄 1stF 𝑂))) ∈ ((𝑄 ×c 𝑂) Func 𝑇))
391, 38syl5eqel 2807 . 2 (𝜑𝑍 ∈ ((𝑄 ×c 𝑂) Func 𝑇))
4035, 14, 10, 12funcsetcres2 16865 . . 3 (𝜑 → ((𝑄 ×c 𝑂) Func 𝑆) ⊆ ((𝑄 ×c 𝑂) Func 𝑇))
41 yoneda.e . . . 4 𝐸 = (𝑂 evalF 𝑆)
4241, 5, 9, 16evlfcl 16984 . . 3 (𝜑𝐸 ∈ ((𝑄 ×c 𝑂) Func 𝑆))
4340, 42sseldd 3710 . 2 (𝜑𝐸 ∈ ((𝑄 ×c 𝑂) Func 𝑇))
4439, 43jca 555 1 (𝜑 → (𝑍 ∈ ((𝑄 ×c 𝑂) Func 𝑇) ∧ 𝐸 ∈ ((𝑄 ×c 𝑂) Func 𝑇)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1596  wcel 2103  Vcvv 3304  cun 3678  wss 3680  cop 4291   class class class wbr 4760  ran crn 5219  Rel wrel 5223  cfv 6001  (class class class)co 6765  1st c1st 7283  2nd c2nd 7284  tpos ctpos 7471  Basecbs 15980  Catccat 16447  Idccid 16448  Homf chomf 16449  oppCatcoppc 16493   Func cfunc 16636  func ccofu 16638   FuncCat cfuc 16724  SetCatcsetc 16847   ×c cxpc 16930   1stF c1stf 16931   2ndF c2ndf 16932   ⟨,⟩F cprf 16933   evalF cevlf 16971  HomFchof 17010  Yoncyon 17011
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-8 2105  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704  ax-rep 4879  ax-sep 4889  ax-nul 4897  ax-pow 4948  ax-pr 5011  ax-un 7066  ax-cnex 10105  ax-resscn 10106  ax-1cn 10107  ax-icn 10108  ax-addcl 10109  ax-addrcl 10110  ax-mulcl 10111  ax-mulrcl 10112  ax-mulcom 10113  ax-addass 10114  ax-mulass 10115  ax-distr 10116  ax-i2m1 10117  ax-1ne0 10118  ax-1rid 10119  ax-rnegex 10120  ax-rrecex 10121  ax-cnre 10122  ax-pre-lttri 10123  ax-pre-lttrn 10124  ax-pre-ltadd 10125  ax-pre-mulgt0 10126
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1599  df-fal 1602  df-ex 1818  df-nf 1823  df-sb 2011  df-eu 2575  df-mo 2576  df-clab 2711  df-cleq 2717  df-clel 2720  df-nfc 2855  df-ne 2897  df-nel 3000  df-ral 3019  df-rex 3020  df-reu 3021  df-rmo 3022  df-rab 3023  df-v 3306  df-sbc 3542  df-csb 3640  df-dif 3683  df-un 3685  df-in 3687  df-ss 3694  df-pss 3696  df-nul 4024  df-if 4195  df-pw 4268  df-sn 4286  df-pr 4288  df-tp 4290  df-op 4292  df-uni 4545  df-int 4584  df-iun 4630  df-br 4761  df-opab 4821  df-mpt 4838  df-tr 4861  df-id 5128  df-eprel 5133  df-po 5139  df-so 5140  df-fr 5177  df-we 5179  df-xp 5224  df-rel 5225  df-cnv 5226  df-co 5227  df-dm 5228  df-rn 5229  df-res 5230  df-ima 5231  df-pred 5793  df-ord 5839  df-on 5840  df-lim 5841  df-suc 5842  df-iota 5964  df-fun 6003  df-fn 6004  df-f 6005  df-f1 6006  df-fo 6007  df-f1o 6008  df-fv 6009  df-riota 6726  df-ov 6768  df-oprab 6769  df-mpt2 6770  df-om 7183  df-1st 7285  df-2nd 7286  df-tpos 7472  df-wrecs 7527  df-recs 7588  df-rdg 7626  df-1o 7680  df-oadd 7684  df-er 7862  df-map 7976  df-pm 7977  df-ixp 8026  df-en 8073  df-dom 8074  df-sdom 8075  df-fin 8076  df-pnf 10189  df-mnf 10190  df-xr 10191  df-ltxr 10192  df-le 10193  df-sub 10381  df-neg 10382  df-nn 11134  df-2 11192  df-3 11193  df-4 11194  df-5 11195  df-6 11196  df-7 11197  df-8 11198  df-9 11199  df-n0 11406  df-z 11491  df-dec 11607  df-uz 11801  df-fz 12441  df-struct 15982  df-ndx 15983  df-slot 15984  df-base 15986  df-sets 15987  df-ress 15988  df-hom 16089  df-cco 16090  df-cat 16451  df-cid 16452  df-homf 16453  df-comf 16454  df-oppc 16494  df-ssc 16592  df-resc 16593  df-subc 16594  df-func 16640  df-cofu 16642  df-nat 16725  df-fuc 16726  df-setc 16848  df-xpc 16934  df-1stf 16935  df-2ndf 16936  df-prf 16937  df-evlf 16975  df-curf 16976  df-hof 17012  df-yon 17013
This theorem is referenced by:  yonedalem3b  17041  yonedalem3  17042  yonedainv  17043  yonffthlem  17044  yoneda  17045
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