Theorem funcsn 49419

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 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	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑎 ∈ (𝑓(𝐶 Nat 𝐷)𝑔) ∧ 𝑏 ∈ (𝑓(𝐶 Nat 𝐷)𝑔))) → 𝑎 ∈ (𝑓(𝐶 Nat 𝐷)𝑔))
8	4, 7	nat1st2nd 17922	. . . . . . . 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 2730	. . . . . . . 8 ⊢ (Base‘𝐶) = (Base‘𝐶)
10	4, 8, 9	natfn 17925	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑎 ∈ (𝑓(𝐶 Nat 𝐷)𝑔) ∧ 𝑏 ∈ (𝑓(𝐶 Nat 𝐷)𝑔))) → 𝑎 Fn (Base‘𝐶))
11		simprr 772	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑎 ∈ (𝑓(𝐶 Nat 𝐷)𝑔) ∧ 𝑏 ∈ (𝑓(𝐶 Nat 𝐷)𝑔))) → 𝑏 ∈ (𝑓(𝐶 Nat 𝐷)𝑔))
12	4, 11	nat1st2nd 17922	. . . . . . . 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 17925	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑎 ∈ (𝑓(𝐶 Nat 𝐷)𝑔) ∧ 𝑏 ∈ (𝑓(𝐶 Nat 𝐷)𝑔))) → 𝑏 Fn (Base‘𝐶))
14		eqid 2730	. . . . . . . . . 10 ⊢ (Base‘𝐷) = (Base‘𝐷)
15	4, 8	natrcl2 49128	. . . . . . . . . 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 17834	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑎 ∈ (𝑓(𝐶 Nat 𝐷)𝑔) ∧ 𝑏 ∈ (𝑓(𝐶 Nat 𝐷)𝑔))) → (1st ‘𝑓):(Base‘𝐶)⟶(Base‘𝐷))
17	16	ffvelcdmda 7063	. . . . . . . 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 49129	. . . . . . . . . 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 17834	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑎 ∈ (𝑓(𝐶 Nat 𝐷)𝑔) ∧ 𝑏 ∈ (𝑓(𝐶 Nat 𝐷)𝑔))) → (1st ‘𝑔):(Base‘𝐶)⟶(Base‘𝐷))
20	19	ffvelcdmda 7063	. . . . . . . 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 480	. . . . . . . . 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 2730	. . . . . . . . 9 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
23		simpr 484	. . . . . . . . 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 17924	. . . . . . . 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 480	. . . . . . . . 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 17924	. . . . . . . 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 726	. . . . . . . 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 49304	. . . . . . 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 7013	. . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ∈ (𝑓(𝐶 Nat 𝐷)𝑔) ∧ 𝑏 ∈ (𝑓(𝐶 Nat 𝐷)𝑔))) → 𝑎 = 𝑏)
31	30	ralrimivva 3182	. . . . 5 ⊢ (𝜑 → ∀𝑎 ∈ (𝑓(𝐶 Nat 𝐷)𝑔)∀𝑏 ∈ (𝑓(𝐶 Nat 𝐷)𝑔)𝑎 = 𝑏)
32		moel 3379	. . . . 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 234	. . . 4 ⊢ (𝜑 → ∃*𝑎 𝑎 ∈ (𝑓(𝐶 Nat 𝐷)𝑔))
34	33	adantr 480	. . 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 4632	. . . . . . . 8 ⊢ (𝐹 ∈ 𝑉 → 𝐹 ∈ {𝐹})
37	35, 36	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝐹 ∈ {𝐹})
38		funcsn.c	. . . . . . 7 ⊢ (𝜑 → (𝐶 Func 𝐷) = {𝐹})
39	37, 38	eleqtrrd 2832	. . . . . 6 ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))
40	39	func1st2nd 48993	. . . . 5 ⊢ (𝜑 → (1st ‘𝐹)(𝐶 Func 𝐷)(2nd ‘𝐹))
41	40	funcrcl2 48996	. . . 4 ⊢ (𝜑 → 𝐶 ∈ Cat)
42	27	thinccd 49301	. . . 4 ⊢ (𝜑 → 𝐷 ∈ Cat)
43	1, 41, 42	fuccat 17941	. . 3 ⊢ (𝜑 → 𝑄 ∈ Cat)
44	3, 6, 34, 43	isthincd 49314	. 2 ⊢ (𝜑 → 𝑄 ∈ ThinCat)
45		sneq 4607	. . . 4 ⊢ (𝑓 = 𝐹 → {𝑓} = {𝐹})
46	45	eqeq2d 2741	. . 3 ⊢ (𝑓 = 𝐹 → ((𝐶 Func 𝐷) = {𝑓} ↔ (𝐶 Func 𝐷) = {𝐹}))
47	35, 38, 46	spcedv 3573	. 2 ⊢ (𝜑 → ∃𝑓(𝐶 Func 𝐷) = {𝑓})
48	2	istermc 49352	. 2 ⊢ (𝑄 ∈ TermCat ↔ (𝑄 ∈ ThinCat ∧ ∃𝑓(𝐶 Func 𝐷) = {𝑓}))
49	44, 47, 48	sylanbrc 583	1 ⊢ (𝜑 → 𝑄 ∈ TermCat)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540  ∃wex 1779   ∈ wcel 2109  ∃*wmo 2532  ∀wral 3046  {csn 4597  ⟨cop 4603  ‘cfv 6519  (class class class)co 7394  1^st c1st 7975  2^nd c2nd 7976  Basecbs 17185  Hom chom 17237   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:  fucterm  49420