0fucterm - Mathbox for Zhi Wang

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Theorem 0fucterm 49421

Description: The category of functors from an initial category is terminal. (Contributed by Zhi Wang, 17-Nov-2025.)

**Hypotheses**
Ref	Expression
0fucterm.c	⊢ (𝜑 → 𝐶 ∈ 𝑉)
0fucterm.b	⊢ (𝜑 → ∅ = (Base‘𝐶))
0fucterm.d	⊢ (𝜑 → 𝐷 ∈ Cat)
0fucterm.q	⊢ 𝑄 = (𝐶 FuncCat 𝐷)

**Assertion**
Ref	Expression
0fucterm	⊢ (𝜑 → 𝑄 ∈ TermCat)

Proof of Theorem 0fucterm Dummy variables 𝑎 𝑏 𝑓 𝑔 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		0fucterm.q	. . . . 5 ⊢ 𝑄 = (𝐶 FuncCat 𝐷)
2	1	fucbas 17931	. . . 4 ⊢ (𝐶 Func 𝐷) = (Base‘𝑄)
3	2	a1i 11	. . 3 ⊢ (𝜑 → (𝐶 Func 𝐷) = (Base‘𝑄))
4		eqid 2730	. . . . 5 ⊢ (𝐶 Nat 𝐷) = (𝐶 Nat 𝐷)
5	1, 4	fuchom 17932	. . . 4 ⊢ (𝐶 Nat 𝐷) = (Hom ‘𝑄)
6	5	a1i 11	. . 3 ⊢ (𝜑 → (𝐶 Nat 𝐷) = (Hom ‘𝑄))
7		simprl 770	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑓 ∈ (𝐶 Func 𝐷) ∧ 𝑔 ∈ (𝐶 Func 𝐷))) ∧ (𝑎 ∈ (𝑓(𝐶 Nat 𝐷)𝑔) ∧ 𝑏 ∈ (𝑓(𝐶 Nat 𝐷)𝑔))) → 𝑎 ∈ (𝑓(𝐶 Nat 𝐷)𝑔))
8	4, 7	nat1st2nd 17922	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑓 ∈ (𝐶 Func 𝐷) ∧ 𝑔 ∈ (𝐶 Func 𝐷))) ∧ (𝑎 ∈ (𝑓(𝐶 Nat 𝐷)𝑔) ∧ 𝑏 ∈ (𝑓(𝐶 Nat 𝐷)𝑔))) → 𝑎 ∈ (⟨(1^st ‘𝑓), (2^nd ‘𝑓)⟩(𝐶 Nat 𝐷)⟨(1^st ‘𝑔), (2^nd ‘𝑔)⟩))
9		eqid 2730	. . . . . . . . 9 ⊢ (Base‘𝐶) = (Base‘𝐶)
10	4, 8, 9	natfn 17925	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑓 ∈ (𝐶 Func 𝐷) ∧ 𝑔 ∈ (𝐶 Func 𝐷))) ∧ (𝑎 ∈ (𝑓(𝐶 Nat 𝐷)𝑔) ∧ 𝑏 ∈ (𝑓(𝐶 Nat 𝐷)𝑔))) → 𝑎 Fn (Base‘𝐶))
11		0fucterm.b	. . . . . . . . . 10 ⊢ (𝜑 → ∅ = (Base‘𝐶))
12	11	ad2antrr 726	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑓 ∈ (𝐶 Func 𝐷) ∧ 𝑔 ∈ (𝐶 Func 𝐷))) ∧ (𝑎 ∈ (𝑓(𝐶 Nat 𝐷)𝑔) ∧ 𝑏 ∈ (𝑓(𝐶 Nat 𝐷)𝑔))) → ∅ = (Base‘𝐶))
13	12	fneq2d 6620	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑓 ∈ (𝐶 Func 𝐷) ∧ 𝑔 ∈ (𝐶 Func 𝐷))) ∧ (𝑎 ∈ (𝑓(𝐶 Nat 𝐷)𝑔) ∧ 𝑏 ∈ (𝑓(𝐶 Nat 𝐷)𝑔))) → (𝑎 Fn ∅ ↔ 𝑎 Fn (Base‘𝐶)))
14	10, 13	mpbird 257	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑓 ∈ (𝐶 Func 𝐷) ∧ 𝑔 ∈ (𝐶 Func 𝐷))) ∧ (𝑎 ∈ (𝑓(𝐶 Nat 𝐷)𝑔) ∧ 𝑏 ∈ (𝑓(𝐶 Nat 𝐷)𝑔))) → 𝑎 Fn ∅)
15		fn0 6657	. . . . . . 7 ⊢ (𝑎 Fn ∅ ↔ 𝑎 = ∅)
16	14, 15	sylib 218	. . . . . 6 ⊢ (((𝜑 ∧ (𝑓 ∈ (𝐶 Func 𝐷) ∧ 𝑔 ∈ (𝐶 Func 𝐷))) ∧ (𝑎 ∈ (𝑓(𝐶 Nat 𝐷)𝑔) ∧ 𝑏 ∈ (𝑓(𝐶 Nat 𝐷)𝑔))) → 𝑎 = ∅)
17		simprr 772	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑓 ∈ (𝐶 Func 𝐷) ∧ 𝑔 ∈ (𝐶 Func 𝐷))) ∧ (𝑎 ∈ (𝑓(𝐶 Nat 𝐷)𝑔) ∧ 𝑏 ∈ (𝑓(𝐶 Nat 𝐷)𝑔))) → 𝑏 ∈ (𝑓(𝐶 Nat 𝐷)𝑔))
18	4, 17	nat1st2nd 17922	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑓 ∈ (𝐶 Func 𝐷) ∧ 𝑔 ∈ (𝐶 Func 𝐷))) ∧ (𝑎 ∈ (𝑓(𝐶 Nat 𝐷)𝑔) ∧ 𝑏 ∈ (𝑓(𝐶 Nat 𝐷)𝑔))) → 𝑏 ∈ (⟨(1^st ‘𝑓), (2^nd ‘𝑓)⟩(𝐶 Nat 𝐷)⟨(1^st ‘𝑔), (2^nd ‘𝑔)⟩))
19	4, 18, 9	natfn 17925	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑓 ∈ (𝐶 Func 𝐷) ∧ 𝑔 ∈ (𝐶 Func 𝐷))) ∧ (𝑎 ∈ (𝑓(𝐶 Nat 𝐷)𝑔) ∧ 𝑏 ∈ (𝑓(𝐶 Nat 𝐷)𝑔))) → 𝑏 Fn (Base‘𝐶))
20	12	fneq2d 6620	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑓 ∈ (𝐶 Func 𝐷) ∧ 𝑔 ∈ (𝐶 Func 𝐷))) ∧ (𝑎 ∈ (𝑓(𝐶 Nat 𝐷)𝑔) ∧ 𝑏 ∈ (𝑓(𝐶 Nat 𝐷)𝑔))) → (𝑏 Fn ∅ ↔ 𝑏 Fn (Base‘𝐶)))
21	19, 20	mpbird 257	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑓 ∈ (𝐶 Func 𝐷) ∧ 𝑔 ∈ (𝐶 Func 𝐷))) ∧ (𝑎 ∈ (𝑓(𝐶 Nat 𝐷)𝑔) ∧ 𝑏 ∈ (𝑓(𝐶 Nat 𝐷)𝑔))) → 𝑏 Fn ∅)
22		fn0 6657	. . . . . . 7 ⊢ (𝑏 Fn ∅ ↔ 𝑏 = ∅)
23	21, 22	sylib 218	. . . . . 6 ⊢ (((𝜑 ∧ (𝑓 ∈ (𝐶 Func 𝐷) ∧ 𝑔 ∈ (𝐶 Func 𝐷))) ∧ (𝑎 ∈ (𝑓(𝐶 Nat 𝐷)𝑔) ∧ 𝑏 ∈ (𝑓(𝐶 Nat 𝐷)𝑔))) → 𝑏 = ∅)
24	16, 23	eqtr4d 2768	. . . . 5 ⊢ (((𝜑 ∧ (𝑓 ∈ (𝐶 Func 𝐷) ∧ 𝑔 ∈ (𝐶 Func 𝐷))) ∧ (𝑎 ∈ (𝑓(𝐶 Nat 𝐷)𝑔) ∧ 𝑏 ∈ (𝑓(𝐶 Nat 𝐷)𝑔))) → 𝑎 = 𝑏)
25	24	ralrimivva 3182	. . . 4 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝐶 Func 𝐷) ∧ 𝑔 ∈ (𝐶 Func 𝐷))) → ∀𝑎 ∈ (𝑓(𝐶 Nat 𝐷)𝑔)∀𝑏 ∈ (𝑓(𝐶 Nat 𝐷)𝑔)𝑎 = 𝑏)
26		moel 3379	. . . 4 ⊢ (∃*𝑎 𝑎 ∈ (𝑓(𝐶 Nat 𝐷)𝑔) ↔ ∀𝑎 ∈ (𝑓(𝐶 Nat 𝐷)𝑔)∀𝑏 ∈ (𝑓(𝐶 Nat 𝐷)𝑔)𝑎 = 𝑏)
27	25, 26	sylibr 234	. . 3 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝐶 Func 𝐷) ∧ 𝑔 ∈ (𝐶 Func 𝐷))) → ∃*𝑎 𝑎 ∈ (𝑓(𝐶 Nat 𝐷)𝑔))
28		0fucterm.c	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ 𝑉)
29		0catg 17655	. . . . 5 ⊢ ((𝐶 ∈ 𝑉 ∧ ∅ = (Base‘𝐶)) → 𝐶 ∈ Cat)
30	28, 11, 29	syl2anc 584	. . . 4 ⊢ (𝜑 → 𝐶 ∈ Cat)
31		0fucterm.d	. . . 4 ⊢ (𝜑 → 𝐷 ∈ Cat)
32	1, 30, 31	fuccat 17941	. . 3 ⊢ (𝜑 → 𝑄 ∈ Cat)
33	3, 6, 27, 32	isthincd 49314	. 2 ⊢ (𝜑 → 𝑄 ∈ ThinCat)
34		opex 5432	. . . 4 ⊢ ⟨∅, ∅⟩ ∈ V
35	34	a1i 11	. . 3 ⊢ (𝜑 → ⟨∅, ∅⟩ ∈ V)
36	28, 11, 31	0funcg 49002	. . 3 ⊢ (𝜑 → (𝐶 Func 𝐷) = {⟨∅, ∅⟩})
37		sneq 4607	. . . 4 ⊢ (𝑓 = ⟨∅, ∅⟩ → {𝑓} = {⟨∅, ∅⟩})
38	37	eqeq2d 2741	. . 3 ⊢ (𝑓 = ⟨∅, ∅⟩ → ((𝐶 Func 𝐷) = {𝑓} ↔ (𝐶 Func 𝐷) = {⟨∅, ∅⟩}))
39	35, 36, 38	spcedv 3573	. 2 ⊢ (𝜑 → ∃𝑓(𝐶 Func 𝐷) = {𝑓})
40	2	istermc 49352	. 2 ⊢ (𝑄 ∈ TermCat ↔ (𝑄 ∈ ThinCat ∧ ∃𝑓(𝐶 Func 𝐷) = {𝑓}))
41	33, 39, 40	sylanbrc 583	1 ⊢ (𝜑 → 𝑄 ∈ TermCat)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∃wex 1779 ∈ wcel 2109 ∃*wmo 2532 ∀wral 3046 Vcvv 3455 ∅c0 4304 {csn 4597 ⟨cop 4603 Fn wfn 6514 ‘cfv 6519 (class class class)co 7394 1^st c1st 7975 2^nd c2nd 7976 Basecbs 17185 Hom chom 17237 Catccat 17631 Func cfunc 17822 Nat cnat 17912 FuncCat cfuc 17913 ThinCatcthinc 49295 TermCatctermc 49350

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2702 ax-rep 5242 ax-sep 5259 ax-nul 5269 ax-pow 5328 ax-pr 5395 ax-un 7718 ax-cnex 11142 ax-resscn 11143 ax-1cn 11144 ax-icn 11145 ax-addcl 11146 ax-addrcl 11147 ax-mulcl 11148 ax-mulrcl 11149 ax-mulcom 11150 ax-addass 11151 ax-mulass 11152 ax-distr 11153 ax-i2m1 11154 ax-1ne0 11155 ax-1rid 11156 ax-rnegex 11157 ax-rrecex 11158 ax-cnre 11159 ax-pre-lttri 11160 ax-pre-lttrn 11161 ax-pre-ltadd 11162 ax-pre-mulgt0 11163

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2880 df-ne 2928 df-nel 3032 df-ral 3047 df-rex 3056 df-rmo 3357 df-reu 3358 df-rab 3412 df-v 3457 df-sbc 3762 df-csb 3871 df-dif 3925 df-un 3927 df-in 3929 df-ss 3939 df-pss 3942 df-nul 4305 df-if 4497 df-pw 4573 df-sn 4598 df-pr 4600 df-tp 4602 df-op 4604 df-uni 4880 df-iun 4965 df-br 5116 df-opab 5178 df-mpt 5197 df-tr 5223 df-id 5541 df-eprel 5546 df-po 5554 df-so 5555 df-fr 5599 df-we 5601 df-xp 5652 df-rel 5653 df-cnv 5654 df-co 5655 df-dm 5656 df-rn 5657 df-res 5658 df-ima 5659 df-pred 6282 df-ord 6343 df-on 6344 df-lim 6345 df-suc 6346 df-iota 6472 df-fun 6521 df-fn 6522 df-f 6523 df-f1 6524 df-fo 6525 df-f1o 6526 df-fv 6527 df-riota 7351 df-ov 7397 df-oprab 7398 df-mpo 7399 df-om 7851 df-1st 7977 df-2nd 7978 df-frecs 8269 df-wrecs 8300 df-recs 8349 df-rdg 8387 df-1o 8443 df-er 8682 df-map 8805 df-ixp 8875 df-en 8923 df-dom 8924 df-sdom 8925 df-fin 8926 df-pnf 11228 df-mnf 11229 df-xr 11230 df-ltxr 11231 df-le 11232 df-sub 11425 df-neg 11426 df-nn 12198 df-2 12260 df-3 12261 df-4 12262 df-5 12263 df-6 12264 df-7 12265 df-8 12266 df-9 12267 df-n0 12459 df-z 12546 df-dec 12666 df-uz 12810 df-fz 13482 df-struct 17123 df-slot 17158 df-ndx 17170 df-base 17186 df-hom 17250 df-cco 17251 df-cat 17635 df-cid 17636 df-func 17826 df-nat 17914 df-fuc 17915 df-thinc 49296 df-termc 49351

This theorem is referenced by: initocmd 49538

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