Theorem funcsn 49782

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 17887	. . . 4 ⊢ (𝐶 Func 𝐷) = (Base‘𝑄)
3	2	a1i 11	. . 3 ⊢ (𝜑 → (𝐶 Func 𝐷) = (Base‘𝑄))
4		eqid 2736	. . . . 5 ⊢ (𝐶 Nat 𝐷) = (𝐶 Nat 𝐷)
5	1, 4	fuchom 17888	. . . 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 17878	. . . . . . . 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 2736	. . . . . . . 8 ⊢ (Base‘𝐶) = (Base‘𝐶)
10	4, 8, 9	natfn 17881	. . . . . . 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 17878	. . . . . . . 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 17881	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑎 ∈ (𝑓(𝐶 Nat 𝐷)𝑔) ∧ 𝑏 ∈ (𝑓(𝐶 Nat 𝐷)𝑔))) → 𝑏 Fn (Base‘𝐶))
14		eqid 2736	. . . . . . . . . 10 ⊢ (Base‘𝐷) = (Base‘𝐷)
15	4, 8	natrcl2 49465	. . . . . . . . . 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 17790	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑎 ∈ (𝑓(𝐶 Nat 𝐷)𝑔) ∧ 𝑏 ∈ (𝑓(𝐶 Nat 𝐷)𝑔))) → (1st ‘𝑓):(Base‘𝐶)⟶(Base‘𝐷))
17	16	ffvelcdmda 7029	. . . . . . . 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 49466	. . . . . . . . . 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 17790	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑎 ∈ (𝑓(𝐶 Nat 𝐷)𝑔) ∧ 𝑏 ∈ (𝑓(𝐶 Nat 𝐷)𝑔))) → (1st ‘𝑔):(Base‘𝐶)⟶(Base‘𝐷))
20	19	ffvelcdmda 7029	. . . . . . . 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 2736	. . . . . . . . 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 17880	. . . . . . . 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 17880	. . . . . . . 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 49667	. . . . . . 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 6979	. . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ∈ (𝑓(𝐶 Nat 𝐷)𝑔) ∧ 𝑏 ∈ (𝑓(𝐶 Nat 𝐷)𝑔))) → 𝑎 = 𝑏)
31	30	ralrimivva 3179	. . . . 5 ⊢ (𝜑 → ∀𝑎 ∈ (𝑓(𝐶 Nat 𝐷)𝑔)∀𝑏 ∈ (𝑓(𝐶 Nat 𝐷)𝑔)𝑎 = 𝑏)
32		moel 3370	. . . . 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 4617	. . . . . . . 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 2839	. . . . . 6 ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))
40	39	func1st2nd 49317	. . . . 5 ⊢ (𝜑 → (1st ‘𝐹)(𝐶 Func 𝐷)(2nd ‘𝐹))
41	40	funcrcl2 49320	. . . 4 ⊢ (𝜑 → 𝐶 ∈ Cat)
42	27	thinccd 49664	. . . 4 ⊢ (𝜑 → 𝐷 ∈ Cat)
43	1, 41, 42	fuccat 17897	. . 3 ⊢ (𝜑 → 𝑄 ∈ Cat)
44	3, 6, 34, 43	isthincd 49677	. 2 ⊢ (𝜑 → 𝑄 ∈ ThinCat)
45		sneq 4590	. . . 4 ⊢ (𝑓 = 𝐹 → {𝑓} = {𝐹})
46	45	eqeq2d 2747	. . 3 ⊢ (𝑓 = 𝐹 → ((𝐶 Func 𝐷) = {𝑓} ↔ (𝐶 Func 𝐷) = {𝐹}))
47	35, 38, 46	spcedv 3552	. 2 ⊢ (𝜑 → ∃𝑓(𝐶 Func 𝐷) = {𝑓})
48	2	istermc 49715	. 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 1541  ∃wex 1780   ∈ wcel 2113  ∃*wmo 2537  ∀wral 3051  {csn 4580  ⟨cop 4586  ‘cfv 6492  (class class class)co 7358  1^st c1st 7931  2^nd c2nd 7932  Basecbs 17136  Hom chom 17188   Func cfunc 17778   Nat cnat 17868   FuncCat cfuc 17869  ThinCatcthinc 49658  TermCatctermc 49713

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-rep 5224  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680  ax-cnex 11082  ax-resscn 11083  ax-1cn 11084  ax-icn 11085  ax-addcl 11086  ax-addrcl 11087  ax-mulcl 11088  ax-mulrcl 11089  ax-mulcom 11090  ax-addass 11091  ax-mulass 11092  ax-distr 11093  ax-i2m1 11094  ax-1ne0 11095  ax-1rid 11096  ax-rnegex 11097  ax-rrecex 11098  ax-cnre 11099  ax-pre-lttri 11100  ax-pre-lttrn 11101  ax-pre-ltadd 11102  ax-pre-mulgt0 11103

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3350  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-pss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-tp 4585  df-op 4587  df-uni 4864  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-tr 5206  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-riota 7315  df-ov 7361  df-oprab 7362  df-mpo 7363  df-om 7809  df-1st 7933  df-2nd 7934  df-frecs 8223  df-wrecs 8254  df-recs 8303  df-rdg 8341  df-1o 8397  df-er 8635  df-map 8765  df-ixp 8836  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-pnf 11168  df-mnf 11169  df-xr 11170  df-ltxr 11171  df-le 11172  df-sub 11366  df-neg 11367  df-nn 12146  df-2 12208  df-3 12209  df-4 12210  df-5 12211  df-6 12212  df-7 12213  df-8 12214  df-9 12215  df-n0 12402  df-z 12489  df-dec 12608  df-uz 12752  df-fz 13424  df-struct 17074  df-slot 17109  df-ndx 17121  df-base 17137  df-hom 17201  df-cco 17202  df-cat 17591  df-cid 17592  df-func 17782  df-nat 17870  df-fuc 17871  df-thinc 49659  df-termc 49714

This theorem is referenced by:  fucterm  49783