Theorem funcsn 50360

Description: The category of one functor to a thin category is terminal. (Contributed by Zhi Wang, 17-Nov-2025.)

Hypotheses

Ref	Expression
funcsn.q	⊢ 𝑄 = (𝐶 FuncCat 𝐷)
funcsn.f	⊢ (𝜑 → 𝐹 ∈ 𝑉)
funcsn.c	⊢ (𝜑 → (𝐶 Func 𝐷) = {𝐹})
funcsn.d	⊢ (𝜑 → 𝐷 ∈ ThinCat)

Assertion

Ref	Expression
funcsn	⊢ (𝜑 → 𝑄 ∈ TermCat)

Proof of Theorem funcsn

Dummy variables 𝑎 𝑏 𝑓 𝑔 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		funcsn.q	. . . . 5 ⊢ 𝑄 = (𝐶 FuncCat 𝐷)
2	1	fucbas 18045	. . . 4 ⊢ (𝐶 Func 𝐷) = (Base‘𝑄)
3	2	a1i 11	. . 3 ⊢ (𝜑 → (𝐶 Func 𝐷) = (Base‘𝑄))
4		eqid 2766	. . . . 5 ⊢ (𝐶 Nat 𝐷) = (𝐶 Nat 𝐷)
5	1, 4	fuchom 18046	. . . 4 ⊢ (𝐶 Nat 𝐷) = (Hom ‘𝑄)
6	5	a1i 11	. . 3 ⊢ (𝜑 → (𝐶 Nat 𝐷) = (Hom ‘𝑄))
7		simprl 783	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑎 ∈ (𝑓(𝐶 Nat 𝐷)𝑔) ∧ 𝑏 ∈ (𝑓(𝐶 Nat 𝐷)𝑔))) → 𝑎 ∈ (𝑓(𝐶 Nat 𝐷)𝑔))
8	4, 7	nat1st2nd 18036	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑎 ∈ (𝑓(𝐶 Nat 𝐷)𝑔) ∧ 𝑏 ∈ (𝑓(𝐶 Nat 𝐷)𝑔))) → 𝑎 ∈ (⟨(1st ‘𝑓), (2nd ‘𝑓)⟩(𝐶 Nat 𝐷)⟨(1st ‘𝑔), (2nd ‘𝑔)⟩))
9		eqid 2766	. . . . . . . 8 ⊢ (Base‘𝐶) = (Base‘𝐶)
10	4, 8, 9	natfn 18039	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑎 ∈ (𝑓(𝐶 Nat 𝐷)𝑔) ∧ 𝑏 ∈ (𝑓(𝐶 Nat 𝐷)𝑔))) → 𝑎 Fn (Base‘𝐶))
11		simprr 785	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑎 ∈ (𝑓(𝐶 Nat 𝐷)𝑔) ∧ 𝑏 ∈ (𝑓(𝐶 Nat 𝐷)𝑔))) → 𝑏 ∈ (𝑓(𝐶 Nat 𝐷)𝑔))
12	4, 11	nat1st2nd 18036	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑎 ∈ (𝑓(𝐶 Nat 𝐷)𝑔) ∧ 𝑏 ∈ (𝑓(𝐶 Nat 𝐷)𝑔))) → 𝑏 ∈ (⟨(1st ‘𝑓), (2nd ‘𝑓)⟩(𝐶 Nat 𝐷)⟨(1st ‘𝑔), (2nd ‘𝑔)⟩))
13	4, 12, 9	natfn 18039	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑎 ∈ (𝑓(𝐶 Nat 𝐷)𝑔) ∧ 𝑏 ∈ (𝑓(𝐶 Nat 𝐷)𝑔))) → 𝑏 Fn (Base‘𝐶))
14		eqid 2766	. . . . . . . . . 10 ⊢ (Base‘𝐷) = (Base‘𝐷)
15	4, 8	natrcl2 50043	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑎 ∈ (𝑓(𝐶 Nat 𝐷)𝑔) ∧ 𝑏 ∈ (𝑓(𝐶 Nat 𝐷)𝑔))) → (1st ‘𝑓)(𝐶 Func 𝐷)(2nd ‘𝑓))
16	9, 14, 15	funcf1 17948	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑎 ∈ (𝑓(𝐶 Nat 𝐷)𝑔) ∧ 𝑏 ∈ (𝑓(𝐶 Nat 𝐷)𝑔))) → (1st ‘𝑓):(Base‘𝐶)⟶(Base‘𝐷))
17	16	ffvelcdmda 7086	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑎 ∈ (𝑓(𝐶 Nat 𝐷)𝑔) ∧ 𝑏 ∈ (𝑓(𝐶 Nat 𝐷)𝑔))) ∧ 𝑥 ∈ (Base‘𝐶)) → ((1st ‘𝑓)‘𝑥) ∈ (Base‘𝐷))
18	4, 8	natrcl3 50044	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑎 ∈ (𝑓(𝐶 Nat 𝐷)𝑔) ∧ 𝑏 ∈ (𝑓(𝐶 Nat 𝐷)𝑔))) → (1st ‘𝑔)(𝐶 Func 𝐷)(2nd ‘𝑔))
19	9, 14, 18	funcf1 17948	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑎 ∈ (𝑓(𝐶 Nat 𝐷)𝑔) ∧ 𝑏 ∈ (𝑓(𝐶 Nat 𝐷)𝑔))) → (1st ‘𝑔):(Base‘𝐶)⟶(Base‘𝐷))
20	19	ffvelcdmda 7086	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑎 ∈ (𝑓(𝐶 Nat 𝐷)𝑔) ∧ 𝑏 ∈ (𝑓(𝐶 Nat 𝐷)𝑔))) ∧ 𝑥 ∈ (Base‘𝐶)) → ((1st ‘𝑔)‘𝑥) ∈ (Base‘𝐷))
21	8	adantr 486	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑎 ∈ (𝑓(𝐶 Nat 𝐷)𝑔) ∧ 𝑏 ∈ (𝑓(𝐶 Nat 𝐷)𝑔))) ∧ 𝑥 ∈ (Base‘𝐶)) → 𝑎 ∈ (⟨(1st ‘𝑓), (2nd ‘𝑓)⟩(𝐶 Nat 𝐷)⟨(1st ‘𝑔), (2nd ‘𝑔)⟩))
22		eqid 2766	. . . . . . . . 9 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
23		simpr 490	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑎 ∈ (𝑓(𝐶 Nat 𝐷)𝑔) ∧ 𝑏 ∈ (𝑓(𝐶 Nat 𝐷)𝑔))) ∧ 𝑥 ∈ (Base‘𝐶)) → 𝑥 ∈ (Base‘𝐶))
24	4, 21, 9, 22, 23	natcl 18038	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑎 ∈ (𝑓(𝐶 Nat 𝐷)𝑔) ∧ 𝑏 ∈ (𝑓(𝐶 Nat 𝐷)𝑔))) ∧ 𝑥 ∈ (Base‘𝐶)) → (𝑎‘𝑥) ∈ (((1st ‘𝑓)‘𝑥)(Hom ‘𝐷)((1st ‘𝑔)‘𝑥)))
25	12	adantr 486	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑎 ∈ (𝑓(𝐶 Nat 𝐷)𝑔) ∧ 𝑏 ∈ (𝑓(𝐶 Nat 𝐷)𝑔))) ∧ 𝑥 ∈ (Base‘𝐶)) → 𝑏 ∈ (⟨(1st ‘𝑓), (2nd ‘𝑓)⟩(𝐶 Nat 𝐷)⟨(1st ‘𝑔), (2nd ‘𝑔)⟩))
26	4, 25, 9, 22, 23	natcl 18038	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑎 ∈ (𝑓(𝐶 Nat 𝐷)𝑔) ∧ 𝑏 ∈ (𝑓(𝐶 Nat 𝐷)𝑔))) ∧ 𝑥 ∈ (Base‘𝐶)) → (𝑏‘𝑥) ∈ (((1st ‘𝑓)‘𝑥)(Hom ‘𝐷)((1st ‘𝑔)‘𝑥)))
27		funcsn.d	. . . . . . . . 9 ⊢ (𝜑 → 𝐷 ∈ ThinCat)
28	27	ad2antrr 739	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑎 ∈ (𝑓(𝐶 Nat 𝐷)𝑔) ∧ 𝑏 ∈ (𝑓(𝐶 Nat 𝐷)𝑔))) ∧ 𝑥 ∈ (Base‘𝐶)) → 𝐷 ∈ ThinCat)
29	17, 20, 24, 26, 14, 22, 28	thincmo2 50245	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑎 ∈ (𝑓(𝐶 Nat 𝐷)𝑔) ∧ 𝑏 ∈ (𝑓(𝐶 Nat 𝐷)𝑔))) ∧ 𝑥 ∈ (Base‘𝐶)) → (𝑎‘𝑥) = (𝑏‘𝑥))
30	10, 13, 29	eqfnfvd 7035	. . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ∈ (𝑓(𝐶 Nat 𝐷)𝑔) ∧ 𝑏 ∈ (𝑓(𝐶 Nat 𝐷)𝑔))) → 𝑎 = 𝑏)
31	30	ralrimivva 3211	. . . . 5 ⊢ (𝜑 → ∀𝑎 ∈ (𝑓(𝐶 Nat 𝐷)𝑔)∀𝑏 ∈ (𝑓(𝐶 Nat 𝐷)𝑔)𝑎 = 𝑏)
32		moel 3392	. . . . 5 ⊢ (∃*𝑎 𝑎 ∈ (𝑓(𝐶 Nat 𝐷)𝑔) ↔ ∀𝑎 ∈ (𝑓(𝐶 Nat 𝐷)𝑔)∀𝑏 ∈ (𝑓(𝐶 Nat 𝐷)𝑔)𝑎 = 𝑏)
33	31, 32	sylibr 237	. . . 4 ⊢ (𝜑 → ∃*𝑎 𝑎 ∈ (𝑓(𝐶 Nat 𝐷)𝑔))
34	33	adantr 486	. . 3 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝐶 Func 𝐷) ∧ 𝑔 ∈ (𝐶 Func 𝐷))) → ∃*𝑎 𝑎 ∈ (𝑓(𝐶 Nat 𝐷)𝑔))
35		funcsn.f	. . . . . . . 8 ⊢ (𝜑 → 𝐹 ∈ 𝑉)
36		snidg 4631	. . . . . . . 8 ⊢ (𝐹 ∈ 𝑉 → 𝐹 ∈ {𝐹})
37	35, 36	syl 18	. . . . . . 7 ⊢ (𝜑 → 𝐹 ∈ {𝐹})
38		funcsn.c	. . . . . . 7 ⊢ (𝜑 → (𝐶 Func 𝐷) = {𝐹})
39	37, 38	eleqtrrd 2869	. . . . . 6 ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))
40	39	func1st2nd 49895	. . . . 5 ⊢ (𝜑 → (1st ‘𝐹)(𝐶 Func 𝐷)(2nd ‘𝐹))
41	40	funcrcl2 49898	. . . 4 ⊢ (𝜑 → 𝐶 ∈ Cat)
42	27	thinccd 50242	. . . 4 ⊢ (𝜑 → 𝐷 ∈ Cat)
43	1, 41, 42	fuccat 18055	. . 3 ⊢ (𝜑 → 𝑄 ∈ Cat)
44	3, 6, 34, 43	isthincd 50255	. 2 ⊢ (𝜑 → 𝑄 ∈ ThinCat)
45		sneq 4604	. . . 4 ⊢ (𝑓 = 𝐹 → {𝑓} = {𝐹})
46	45	eqeq2d 2777	. . 3 ⊢ (𝑓 = 𝐹 → ((𝐶 Func 𝐷) = {𝑓} ↔ (𝐶 Func 𝐷) = {𝐹}))
47	35, 38, 46	spcedv 3560	. 2 ⊢ (𝜑 → ∃𝑓(𝐶 Func 𝐷) = {𝑓})
48	2	istermc 50293	. 2 ⊢ (𝑄 ∈ TermCat ↔ (𝑄 ∈ ThinCat ∧ ∃𝑓(𝐶 Func 𝐷) = {𝑓}))
49	44, 47, 48	sylanbrc 595	1 ⊢ (𝜑 → 𝑄 ∈ TermCat)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570  ∃wex 1812   ∈ wcel 2146  ∃*wmo 2568  ∀wral 3082  {csn 4594  ⟨cop 4600  ‘cfv 6543  (class class class)co 7423  1^st c1st 7993  2^nd c2nd 7994  Basecbs 17294  Hom chom 17346   Func cfunc 17936   Nat cnat 18026   FuncCat cfuc 18027  ThinCatcthinc 50236  TermCatctermc 50291

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2738  ax-rep 5243  ax-sep 5262  ax-nul 5274  ax-pow 5341  ax-pr 5409  ax-un 7745  ax-cnex 11174  ax-resscn 11175  ax-1cn 11176  ax-icn 11177  ax-addcl 11178  ax-addrcl 11179  ax-mulcl 11180  ax-mulrcl 11181  ax-mulcom 11182  ax-addass 11183  ax-mulass 11184  ax-distr 11185  ax-i2m1 11186  ax-1ne0 11187  ax-1rid 11188  ax-rnegex 11189  ax-rrecex 11190  ax-cnre 11191  ax-pre-lttri 11192  ax-pre-lttrn 11193  ax-pre-ltadd 11194  ax-pre-mulgt0 11195

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2570  df-eu 2600  df-clab 2745  df-cleq 2758  df-clel 2841  df-nfc 2915  df-ne 2962  df-nel 3068  df-ral 3083  df-rex 3093  df-rmo 3372  df-reu 3373  df-rab 3420  df-v 3460  df-sbc 3748  df-csb 3857  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-pss 3928  df-nul 4290  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-tp 4599  df-op 4601  df-uni 4878  df-iun 4963  df-br 5115  df-opab 5179  df-mpt 5198  df-tr 5224  df-id 5561  df-eprel 5566  df-po 5574  df-so 5575  df-fr 5619  df-we 5621  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-pred 6309  df-ord 6370  df-on 6371  df-lim 6372  df-suc 6373  df-iota 6499  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7380  df-ov 7426  df-oprab 7427  df-mpo 7428  df-om 7872  df-1st 7995  df-2nd 7996  df-frecs 8287  df-wrecs 8318  df-recs 8367  df-rdg 8406  df-1o 8462  df-er 8703  df-map 8835  df-ixp 8905  df-en 8953  df-dom 8954  df-sdom 8955  df-fin 8956  df-pnf 11263  df-mnf 11264  df-xr 11265  df-ltxr 11266  df-le 11267  df-sub 11461  df-neg 11462  df-nn 12252  df-2 12321  df-3 12322  df-4 12323  df-5 12324  df-6 12325  df-7 12326  df-8 12327  df-9 12328  df-n0 12523  df-z 12610  df-dec 12730  df-uz 12881  df-fz 13554  df-struct 17232  df-slot 17267  df-ndx 17279  df-base 17295  df-hom 17359  df-cco 17360  df-cat 17749  df-cid 17750  df-func 17940  df-nat 18028  df-fuc 18029  df-thinc 50237  df-termc 50292

This theorem is used by:  fucterm  50361