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Theorem fucbas 16841

Description: The objects of the functor category are functors from 𝐶 to 𝐷. (Contributed by Mario Carneiro, 6-Jan-2017.) (Revised by Mario Carneiro, 12-Jan-2017.)

**Hypothesis**
Ref	Expression
fucbas.q	⊢ 𝑄 = (𝐶 FuncCat 𝐷)

**Assertion**
Ref	Expression
fucbas	⊢ (𝐶 Func 𝐷) = (Base‘𝑄)

Proof of Theorem fucbas Dummy variables 𝑎 𝑏 𝑓 𝑔 ℎ 𝑣 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fucbas.q	. . . . 5 ⊢ 𝑄 = (𝐶 FuncCat 𝐷)
2		eqid 2760	. . . . 5 ⊢ (𝐶 Func 𝐷) = (𝐶 Func 𝐷)
3		eqid 2760	. . . . 5 ⊢ (𝐶 Nat 𝐷) = (𝐶 Nat 𝐷)
4		eqid 2760	. . . . 5 ⊢ (Base‘𝐶) = (Base‘𝐶)
5		eqid 2760	. . . . 5 ⊢ (comp‘𝐷) = (comp‘𝐷)
6		simpl 474	. . . . 5 ⊢ ((𝐶 ∈ Cat ∧ 𝐷 ∈ Cat) → 𝐶 ∈ Cat)
7		simpr 479	. . . . 5 ⊢ ((𝐶 ∈ Cat ∧ 𝐷 ∈ Cat) → 𝐷 ∈ Cat)
8		eqid 2760	. . . . . 6 ⊢ (comp‘𝑄) = (comp‘𝑄)
9	1, 2, 3, 4, 5, 6, 7, 8	fuccofval 16840	. . . . 5 ⊢ ((𝐶 ∈ Cat ∧ 𝐷 ∈ Cat) → (comp‘𝑄) = (𝑣 ∈ ((𝐶 Func 𝐷) × (𝐶 Func 𝐷)), ℎ ∈ (𝐶 Func 𝐷) ↦ ⦋(1^st ‘𝑣) / 𝑓⦌⦋(2^nd ‘𝑣) / 𝑔⦌(𝑏 ∈ (𝑔(𝐶 Nat 𝐷)ℎ), 𝑎 ∈ (𝑓(𝐶 Nat 𝐷)𝑔) ↦ (𝑥 ∈ (Base‘𝐶) ↦ ((𝑏‘𝑥)(⟨((1^st ‘𝑓)‘𝑥), ((1^st ‘𝑔)‘𝑥)⟩(comp‘𝐷)((1^st ‘ℎ)‘𝑥))(𝑎‘𝑥))))))
10	1, 2, 3, 4, 5, 6, 7, 9	fucval 16839	. . . 4 ⊢ ((𝐶 ∈ Cat ∧ 𝐷 ∈ Cat) → 𝑄 = {⟨(Base‘ndx), (𝐶 Func 𝐷)⟩, ⟨(Hom ‘ndx), (𝐶 Nat 𝐷)⟩, ⟨(comp‘ndx), (comp‘𝑄)⟩})
11		catstr 16838	. . . 4 ⊢ {⟨(Base‘ndx), (𝐶 Func 𝐷)⟩, ⟨(Hom ‘ndx), (𝐶 Nat 𝐷)⟩, ⟨(comp‘ndx), (comp‘𝑄)⟩} Struct ⟨1, ;15⟩
12		baseid 16141	. . . 4 ⊢ Base = Slot (Base‘ndx)
13		snsstp1 4492	. . . 4 ⊢ {⟨(Base‘ndx), (𝐶 Func 𝐷)⟩} ⊆ {⟨(Base‘ndx), (𝐶 Func 𝐷)⟩, ⟨(Hom ‘ndx), (𝐶 Nat 𝐷)⟩, ⟨(comp‘ndx), (comp‘𝑄)⟩}
14		ovexd 6844	. . . 4 ⊢ ((𝐶 ∈ Cat ∧ 𝐷 ∈ Cat) → (𝐶 Func 𝐷) ∈ V)
15		eqid 2760	. . . 4 ⊢ (Base‘𝑄) = (Base‘𝑄)
16	10, 11, 12, 13, 14, 15	strfv3 16130	. . 3 ⊢ ((𝐶 ∈ Cat ∧ 𝐷 ∈ Cat) → (Base‘𝑄) = (𝐶 Func 𝐷))
17	16	eqcomd 2766	. 2 ⊢ ((𝐶 ∈ Cat ∧ 𝐷 ∈ Cat) → (𝐶 Func 𝐷) = (Base‘𝑄))
18		base0 16134	. . 3 ⊢ ∅ = (Base‘∅)
19		funcrcl 16744	. . . . 5 ⊢ (𝑓 ∈ (𝐶 Func 𝐷) → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
20	19	con3i 150	. . . 4 ⊢ (¬ (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat) → ¬ 𝑓 ∈ (𝐶 Func 𝐷))
21	20	eq0rdv 4122	. . 3 ⊢ (¬ (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat) → (𝐶 Func 𝐷) = ∅)
22		fnfuc 16826	. . . . . . 7 ⊢ FuncCat Fn (Cat × Cat)
23		fndm 6151	. . . . . . 7 ⊢ ( FuncCat Fn (Cat × Cat) → dom FuncCat = (Cat × Cat))
24	22, 23	ax-mp 5	. . . . . 6 ⊢ dom FuncCat = (Cat × Cat)
25	24	ndmov 6984	. . . . 5 ⊢ (¬ (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat) → (𝐶 FuncCat 𝐷) = ∅)
26	1, 25	syl5eq 2806	. . . 4 ⊢ (¬ (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat) → 𝑄 = ∅)
27	26	fveq2d 6357	. . 3 ⊢ (¬ (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat) → (Base‘𝑄) = (Base‘∅))
28	18, 21, 27	3eqtr4a 2820	. 2 ⊢ (¬ (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat) → (𝐶 Func 𝐷) = (Base‘𝑄))
29	17, 28	pm2.61i 176	1 ⊢ (𝐶 Func 𝐷) = (Base‘𝑄)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 ∧ wa 383 = wceq 1632 ∈ wcel 2139 Vcvv 3340 ∅c0 4058 {ctp 4325 ⟨cop 4327 × cxp 5264 dom cdm 5266 Fn wfn 6044 ‘cfv 6049 (class class class)co 6814 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-func 16739 df-fuc 16825

This theorem is referenced by: fuccatid 16850 fucsect 16853 fucinv 16854 fuciso 16856 evlfcllem 17082 evlfcl 17083 curfcl 17093 uncf1 17097 uncf2 17098 curfuncf 17099 diag1cl 17103 curf2ndf 17108 yon1cl 17124 oyon1cl 17132 yonedalem21 17134 yonedalem22 17139 yonedalem3b 17140 yonedalem3 17141 yonedainv 17142 yonffthlem 17143 yoneda 17144 yoniso 17146

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