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Theorem fuccofval 16840
 Description: Value of the functor category. (Contributed by Mario Carneiro, 6-Jan-2017.)
Hypotheses
Ref Expression
fucval.q 𝑄 = (𝐶 FuncCat 𝐷)
fucval.b 𝐵 = (𝐶 Func 𝐷)
fucval.n 𝑁 = (𝐶 Nat 𝐷)
fucval.a 𝐴 = (Base‘𝐶)
fucval.o · = (comp‘𝐷)
fucval.c (𝜑𝐶 ∈ Cat)
fucval.d (𝜑𝐷 ∈ Cat)
fuccofval.x = (comp‘𝑄)
Assertion
Ref Expression
fuccofval (𝜑 = (𝑣 ∈ (𝐵 × 𝐵), 𝐵(1st𝑣) / 𝑓(2nd𝑣) / 𝑔(𝑏 ∈ (𝑔𝑁), 𝑎 ∈ (𝑓𝑁𝑔) ↦ (𝑥𝐴 ↦ ((𝑏𝑥)(⟨((1st𝑓)‘𝑥), ((1st𝑔)‘𝑥)⟩ · ((1st)‘𝑥))(𝑎𝑥))))))
Distinct variable groups:   𝑣,,𝐵   𝑎,𝑏,𝑓,𝑔,,𝑣,𝑥,𝜑   𝐶,𝑎,𝑏,𝑓,𝑔,,𝑣,𝑥   𝐷,𝑎,𝑏,𝑓,𝑔,,𝑣,𝑥
Allowed substitution hints:   𝐴(𝑥,𝑣,𝑓,𝑔,,𝑎,𝑏)   𝐵(𝑥,𝑓,𝑔,𝑎,𝑏)   𝑄(𝑥,𝑣,𝑓,𝑔,,𝑎,𝑏)   (𝑥,𝑣,𝑓,𝑔,,𝑎,𝑏)   · (𝑥,𝑣,𝑓,𝑔,,𝑎,𝑏)   𝑁(𝑥,𝑣,𝑓,𝑔,,𝑎,𝑏)

Proof of Theorem fuccofval
StepHypRef Expression
1 fucval.q . . . 4 𝑄 = (𝐶 FuncCat 𝐷)
2 fucval.b . . . 4 𝐵 = (𝐶 Func 𝐷)
3 fucval.n . . . 4 𝑁 = (𝐶 Nat 𝐷)
4 fucval.a . . . 4 𝐴 = (Base‘𝐶)
5 fucval.o . . . 4 · = (comp‘𝐷)
6 fucval.c . . . 4 (𝜑𝐶 ∈ Cat)
7 fucval.d . . . 4 (𝜑𝐷 ∈ Cat)
8 eqidd 2761 . . . 4 (𝜑 → (𝑣 ∈ (𝐵 × 𝐵), 𝐵(1st𝑣) / 𝑓(2nd𝑣) / 𝑔(𝑏 ∈ (𝑔𝑁), 𝑎 ∈ (𝑓𝑁𝑔) ↦ (𝑥𝐴 ↦ ((𝑏𝑥)(⟨((1st𝑓)‘𝑥), ((1st𝑔)‘𝑥)⟩ · ((1st)‘𝑥))(𝑎𝑥))))) = (𝑣 ∈ (𝐵 × 𝐵), 𝐵(1st𝑣) / 𝑓(2nd𝑣) / 𝑔(𝑏 ∈ (𝑔𝑁), 𝑎 ∈ (𝑓𝑁𝑔) ↦ (𝑥𝐴 ↦ ((𝑏𝑥)(⟨((1st𝑓)‘𝑥), ((1st𝑔)‘𝑥)⟩ · ((1st)‘𝑥))(𝑎𝑥))))))
91, 2, 3, 4, 5, 6, 7, 8fucval 16839 . . 3 (𝜑𝑄 = {⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), 𝑁⟩, ⟨(comp‘ndx), (𝑣 ∈ (𝐵 × 𝐵), 𝐵(1st𝑣) / 𝑓(2nd𝑣) / 𝑔(𝑏 ∈ (𝑔𝑁), 𝑎 ∈ (𝑓𝑁𝑔) ↦ (𝑥𝐴 ↦ ((𝑏𝑥)(⟨((1st𝑓)‘𝑥), ((1st𝑔)‘𝑥)⟩ · ((1st)‘𝑥))(𝑎𝑥)))))⟩})
109fveq2d 6357 . 2 (𝜑 → (comp‘𝑄) = (comp‘{⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), 𝑁⟩, ⟨(comp‘ndx), (𝑣 ∈ (𝐵 × 𝐵), 𝐵(1st𝑣) / 𝑓(2nd𝑣) / 𝑔(𝑏 ∈ (𝑔𝑁), 𝑎 ∈ (𝑓𝑁𝑔) ↦ (𝑥𝐴 ↦ ((𝑏𝑥)(⟨((1st𝑓)‘𝑥), ((1st𝑔)‘𝑥)⟩ · ((1st)‘𝑥))(𝑎𝑥)))))⟩}))
11 fuccofval.x . 2 = (comp‘𝑄)
12 ovex 6842 . . . . . 6 (𝐶 Func 𝐷) ∈ V
132, 12eqeltri 2835 . . . . 5 𝐵 ∈ V
1413, 13xpex 7128 . . . 4 (𝐵 × 𝐵) ∈ V
1514, 13mpt2ex 7416 . . 3 (𝑣 ∈ (𝐵 × 𝐵), 𝐵(1st𝑣) / 𝑓(2nd𝑣) / 𝑔(𝑏 ∈ (𝑔𝑁), 𝑎 ∈ (𝑓𝑁𝑔) ↦ (𝑥𝐴 ↦ ((𝑏𝑥)(⟨((1st𝑓)‘𝑥), ((1st𝑔)‘𝑥)⟩ · ((1st)‘𝑥))(𝑎𝑥))))) ∈ V
16 catstr 16838 . . . 4 {⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), 𝑁⟩, ⟨(comp‘ndx), (𝑣 ∈ (𝐵 × 𝐵), 𝐵(1st𝑣) / 𝑓(2nd𝑣) / 𝑔(𝑏 ∈ (𝑔𝑁), 𝑎 ∈ (𝑓𝑁𝑔) ↦ (𝑥𝐴 ↦ ((𝑏𝑥)(⟨((1st𝑓)‘𝑥), ((1st𝑔)‘𝑥)⟩ · ((1st)‘𝑥))(𝑎𝑥)))))⟩} Struct ⟨1, 15⟩
17 ccoid 16299 . . . 4 comp = Slot (comp‘ndx)
18 snsstp3 4494 . . . 4 {⟨(comp‘ndx), (𝑣 ∈ (𝐵 × 𝐵), 𝐵(1st𝑣) / 𝑓(2nd𝑣) / 𝑔(𝑏 ∈ (𝑔𝑁), 𝑎 ∈ (𝑓𝑁𝑔) ↦ (𝑥𝐴 ↦ ((𝑏𝑥)(⟨((1st𝑓)‘𝑥), ((1st𝑔)‘𝑥)⟩ · ((1st)‘𝑥))(𝑎𝑥)))))⟩} ⊆ {⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), 𝑁⟩, ⟨(comp‘ndx), (𝑣 ∈ (𝐵 × 𝐵), 𝐵(1st𝑣) / 𝑓(2nd𝑣) / 𝑔(𝑏 ∈ (𝑔𝑁), 𝑎 ∈ (𝑓𝑁𝑔) ↦ (𝑥𝐴 ↦ ((𝑏𝑥)(⟨((1st𝑓)‘𝑥), ((1st𝑔)‘𝑥)⟩ · ((1st)‘𝑥))(𝑎𝑥)))))⟩}
1916, 17, 18strfv 16129 . . 3 ((𝑣 ∈ (𝐵 × 𝐵), 𝐵(1st𝑣) / 𝑓(2nd𝑣) / 𝑔(𝑏 ∈ (𝑔𝑁), 𝑎 ∈ (𝑓𝑁𝑔) ↦ (𝑥𝐴 ↦ ((𝑏𝑥)(⟨((1st𝑓)‘𝑥), ((1st𝑔)‘𝑥)⟩ · ((1st)‘𝑥))(𝑎𝑥))))) ∈ V → (𝑣 ∈ (𝐵 × 𝐵), 𝐵(1st𝑣) / 𝑓(2nd𝑣) / 𝑔(𝑏 ∈ (𝑔𝑁), 𝑎 ∈ (𝑓𝑁𝑔) ↦ (𝑥𝐴 ↦ ((𝑏𝑥)(⟨((1st𝑓)‘𝑥), ((1st𝑔)‘𝑥)⟩ · ((1st)‘𝑥))(𝑎𝑥))))) = (comp‘{⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), 𝑁⟩, ⟨(comp‘ndx), (𝑣 ∈ (𝐵 × 𝐵), 𝐵(1st𝑣) / 𝑓(2nd𝑣) / 𝑔(𝑏 ∈ (𝑔𝑁), 𝑎 ∈ (𝑓𝑁𝑔) ↦ (𝑥𝐴 ↦ ((𝑏𝑥)(⟨((1st𝑓)‘𝑥), ((1st𝑔)‘𝑥)⟩ · ((1st)‘𝑥))(𝑎𝑥)))))⟩}))
2015, 19ax-mp 5 . 2 (𝑣 ∈ (𝐵 × 𝐵), 𝐵(1st𝑣) / 𝑓(2nd𝑣) / 𝑔(𝑏 ∈ (𝑔𝑁), 𝑎 ∈ (𝑓𝑁𝑔) ↦ (𝑥𝐴 ↦ ((𝑏𝑥)(⟨((1st𝑓)‘𝑥), ((1st𝑔)‘𝑥)⟩ · ((1st)‘𝑥))(𝑎𝑥))))) = (comp‘{⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), 𝑁⟩, ⟨(comp‘ndx), (𝑣 ∈ (𝐵 × 𝐵), 𝐵(1st𝑣) / 𝑓(2nd𝑣) / 𝑔(𝑏 ∈ (𝑔𝑁), 𝑎 ∈ (𝑓𝑁𝑔) ↦ (𝑥𝐴 ↦ ((𝑏𝑥)(⟨((1st𝑓)‘𝑥), ((1st𝑔)‘𝑥)⟩ · ((1st)‘𝑥))(𝑎𝑥)))))⟩})
2110, 11, 203eqtr4g 2819 1 (𝜑 = (𝑣 ∈ (𝐵 × 𝐵), 𝐵(1st𝑣) / 𝑓(2nd𝑣) / 𝑔(𝑏 ∈ (𝑔𝑁), 𝑎 ∈ (𝑓𝑁𝑔) ↦ (𝑥𝐴 ↦ ((𝑏𝑥)(⟨((1st𝑓)‘𝑥), ((1st𝑔)‘𝑥)⟩ · ((1st)‘𝑥))(𝑎𝑥))))))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1632   ∈ wcel 2139  Vcvv 3340  ⦋csb 3674  {ctp 4325  ⟨cop 4327   ↦ cmpt 4881   × cxp 5264  ‘cfv 6049  (class class class)co 6814   ↦ cmpt2 6816  1st c1st 7332  2nd c2nd 7333  1c1 10149  5c5 11285  ;cdc 11705  ndxcnx 16076  Basecbs 16079  Hom chom 16174  compcco 16175  Catccat 16546   Func cfunc 16735   Nat cnat 16822   FuncCat cfuc 16823 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115  ax-cnex 10204  ax-resscn 10205  ax-1cn 10206  ax-icn 10207  ax-addcl 10208  ax-addrcl 10209  ax-mulcl 10210  ax-mulrcl 10211  ax-mulcom 10212  ax-addass 10213  ax-mulass 10214  ax-distr 10215  ax-i2m1 10216  ax-1ne0 10217  ax-1rid 10218  ax-rnegex 10219  ax-rrecex 10220  ax-cnre 10221  ax-pre-lttri 10222  ax-pre-lttrn 10223  ax-pre-ltadd 10224  ax-pre-mulgt0 10225 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-nel 3036  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-int 4628  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-riota 6775  df-ov 6817  df-oprab 6818  df-mpt2 6819  df-om 7232  df-1st 7334  df-2nd 7335  df-wrecs 7577  df-recs 7638  df-rdg 7676  df-1o 7730  df-oadd 7734  df-er 7913  df-en 8124  df-dom 8125  df-sdom 8126  df-fin 8127  df-pnf 10288  df-mnf 10289  df-xr 10290  df-ltxr 10291  df-le 10292  df-sub 10480  df-neg 10481  df-nn 11233  df-2 11291  df-3 11292  df-4 11293  df-5 11294  df-6 11295  df-7 11296  df-8 11297  df-9 11298  df-n0 11505  df-z 11590  df-dec 11706  df-uz 11900  df-fz 12540  df-struct 16081  df-ndx 16082  df-slot 16083  df-base 16085  df-hom 16188  df-cco 16189  df-fuc 16825 This theorem is referenced by:  fucbas  16841  fuchom  16842  fucco  16843
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