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Theorem fuchom 17957

Description: The morphisms in the functor category are natural transformations. (Contributed by Mario Carneiro, 6-Jan-2017.) (Proof shortened by AV, 14-Oct-2024.)

**Hypotheses**
Ref	Expression
fucbas.q	⊢ 𝑄 = (𝐶 FuncCat 𝐷)
fuchom.n	⊢ 𝑁 = (𝐶 Nat 𝐷)

**Assertion**
Ref	Expression
fuchom	⊢ 𝑁 = (Hom ‘𝑄)

Proof of Theorem fuchom 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 2727	. . . . 5 ⊢ (𝐶 Func 𝐷) = (𝐶 Func 𝐷)
3		fuchom.n	. . . . 5 ⊢ 𝑁 = (𝐶 Nat 𝐷)
4		eqid 2727	. . . . 5 ⊢ (Base‘𝐶) = (Base‘𝐶)
5		eqid 2727	. . . . 5 ⊢ (comp‘𝐷) = (comp‘𝐷)
6		simpl 481	. . . . 5 ⊢ ((𝐶 ∈ Cat ∧ 𝐷 ∈ Cat) → 𝐶 ∈ Cat)
7		simpr 483	. . . . 5 ⊢ ((𝐶 ∈ Cat ∧ 𝐷 ∈ Cat) → 𝐷 ∈ Cat)
8		eqid 2727	. . . . . 6 ⊢ (comp‘𝑄) = (comp‘𝑄)
9	1, 2, 3, 4, 5, 6, 7, 8	fuccofval 17955	. . . . 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 ∈ (𝑔𝑁ℎ), 𝑎 ∈ (𝑓𝑁𝑔) ↦ (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 17954	. . . 4 ⊢ ((𝐶 ∈ Cat ∧ 𝐷 ∈ Cat) → 𝑄 = {⟨(Base‘ndx), (𝐶 Func 𝐷)⟩, ⟨(Hom ‘ndx), 𝑁⟩, ⟨(comp‘ndx), (comp‘𝑄)⟩})
11		catstr 17953	. . . 4 ⊢ {⟨(Base‘ndx), (𝐶 Func 𝐷)⟩, ⟨(Hom ‘ndx), 𝑁⟩, ⟨(comp‘ndx), (comp‘𝑄)⟩} Struct ⟨1, ;15⟩
12		homid 17398	. . . 4 ⊢ Hom = Slot (Hom ‘ndx)
13		snsstp2 4823	. . . 4 ⊢ {⟨(Hom ‘ndx), 𝑁⟩} ⊆ {⟨(Base‘ndx), (𝐶 Func 𝐷)⟩, ⟨(Hom ‘ndx), 𝑁⟩, ⟨(comp‘ndx), (comp‘𝑄)⟩}
14	3	ovexi 7458	. . . . 5 ⊢ 𝑁 ∈ V
15	14	a1i 11	. . . 4 ⊢ ((𝐶 ∈ Cat ∧ 𝐷 ∈ Cat) → 𝑁 ∈ V)
16		eqid 2727	. . . 4 ⊢ (Hom ‘𝑄) = (Hom ‘𝑄)
17	10, 11, 12, 13, 15, 16	strfv3 17179	. . 3 ⊢ ((𝐶 ∈ Cat ∧ 𝐷 ∈ Cat) → (Hom ‘𝑄) = 𝑁)
18	17	eqcomd 2733	. 2 ⊢ ((𝐶 ∈ Cat ∧ 𝐷 ∈ Cat) → 𝑁 = (Hom ‘𝑄))
19	12	str0 17163	. . 3 ⊢ ∅ = (Hom ‘∅)
20	3	natffn 17944	. . . . 5 ⊢ 𝑁 Fn ((𝐶 Func 𝐷) × (𝐶 Func 𝐷))
21		funcrcl 17854	. . . . . . . . . 10 ⊢ (𝑓 ∈ (𝐶 Func 𝐷) → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
22	21	con3i 154	. . . . . . . . 9 ⊢ (¬ (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat) → ¬ 𝑓 ∈ (𝐶 Func 𝐷))
23	22	eq0rdv 4406	. . . . . . . 8 ⊢ (¬ (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat) → (𝐶 Func 𝐷) = ∅)
24	23	xpeq2d 5710	. . . . . . 7 ⊢ (¬ (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat) → ((𝐶 Func 𝐷) × (𝐶 Func 𝐷)) = ((𝐶 Func 𝐷) × ∅))
25		xp0 6165	. . . . . . 7 ⊢ ((𝐶 Func 𝐷) × ∅) = ∅
26	24, 25	eqtrdi 2783	. . . . . 6 ⊢ (¬ (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat) → ((𝐶 Func 𝐷) × (𝐶 Func 𝐷)) = ∅)
27	26	fneq2d 6651	. . . . 5 ⊢ (¬ (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat) → (𝑁 Fn ((𝐶 Func 𝐷) × (𝐶 Func 𝐷)) ↔ 𝑁 Fn ∅))
28	20, 27	mpbii 232	. . . 4 ⊢ (¬ (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat) → 𝑁 Fn ∅)
29		fn0 6689	. . . 4 ⊢ (𝑁 Fn ∅ ↔ 𝑁 = ∅)
30	28, 29	sylib 217	. . 3 ⊢ (¬ (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat) → 𝑁 = ∅)
31		fnfuc 17940	. . . . . . 7 ⊢ FuncCat Fn (Cat × Cat)
32	31	fndmi 6661	. . . . . 6 ⊢ dom FuncCat = (Cat × Cat)
33	32	ndmov 7609	. . . . 5 ⊢ (¬ (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat) → (𝐶 FuncCat 𝐷) = ∅)
34	1, 33	eqtrid 2779	. . . 4 ⊢ (¬ (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat) → 𝑄 = ∅)
35	34	fveq2d 6904	. . 3 ⊢ (¬ (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat) → (Hom ‘𝑄) = (Hom ‘∅))
36	19, 30, 35	3eqtr4a 2793	. 2 ⊢ (¬ (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat) → 𝑁 = (Hom ‘𝑄))
37	18, 36	pm2.61i 182	1 ⊢ 𝑁 = (Hom ‘𝑄)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 ∧ wa 394 = wceq 1533 ∈ wcel 2098 Vcvv 3471 ∅c0 4324 {ctp 4634 ⟨cop 4636 × cxp 5678 Fn wfn 6546 ‘cfv 6551 (class class class)co 7424 1c1 11145 5c5 12306 cdc 12713 ndxcnx 17167 Basecbs 17185 Hom chom 17249 compcco 17250 Catccat 17649 Func cfunc 17845 Nat cnat 17936 FuncCat cfuc 17937

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2698 ax-rep 5287 ax-sep 5301 ax-nul 5308 ax-pow 5367 ax-pr 5431 ax-un 7744 ax-cnex 11200 ax-resscn 11201 ax-1cn 11202 ax-icn 11203 ax-addcl 11204 ax-addrcl 11205 ax-mulcl 11206 ax-mulrcl 11207 ax-mulcom 11208 ax-addass 11209 ax-mulass 11210 ax-distr 11211 ax-i2m1 11212 ax-1ne0 11213 ax-1rid 11214 ax-rnegex 11215 ax-rrecex 11216 ax-cnre 11217 ax-pre-lttri 11218 ax-pre-lttrn 11219 ax-pre-ltadd 11220 ax-pre-mulgt0 11221

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2937 df-nel 3043 df-ral 3058 df-rex 3067 df-reu 3373 df-rab 3429 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4325 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-tp 4635 df-op 4637 df-uni 4911 df-iun 5000 df-br 5151 df-opab 5213 df-mpt 5234 df-tr 5268 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5635 df-we 5637 df-xp 5686 df-rel 5687 df-cnv 5688 df-co 5689 df-dm 5690 df-rn 5691 df-res 5692 df-ima 5693 df-pred 6308 df-ord 6375 df-on 6376 df-lim 6377 df-suc 6378 df-iota 6503 df-fun 6553 df-fn 6554 df-f 6555 df-f1 6556 df-fo 6557 df-f1o 6558 df-fv 6559 df-riota 7380 df-ov 7427 df-oprab 7428 df-mpo 7429 df-om 7875 df-1st 7997 df-2nd 7998 df-frecs 8291 df-wrecs 8322 df-recs 8396 df-rdg 8435 df-1o 8491 df-er 8729 df-ixp 8921 df-en 8969 df-dom 8970 df-sdom 8971 df-fin 8972 df-pnf 11286 df-mnf 11287 df-xr 11288 df-ltxr 11289 df-le 11290 df-sub 11482 df-neg 11483 df-nn 12249 df-2 12311 df-3 12312 df-4 12313 df-5 12314 df-6 12315 df-7 12316 df-8 12317 df-9 12318 df-n0 12509 df-z 12595 df-dec 12714 df-uz 12859 df-fz 13523 df-struct 17121 df-slot 17156 df-ndx 17168 df-base 17186 df-hom 17262 df-cco 17263 df-func 17849 df-nat 17938 df-fuc 17939

This theorem is referenced by: fuccatid 17966 fucsect 17969 fuciso 17972 evlfcllem 18218 evlfcl 18219 curfcl 18229 uncf2 18234 curfuncf 18235 diag2cl 18243 curf2ndf 18244 yonedalem21 18270 yonedalem22 18275 yonedalem3b 18276 yonedalem3 18277 yonffthlem 18279

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