Theorem funcsn 49530

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 17925	. . . 4 ⊢ (𝐶 Func 𝐷) = (Base‘𝑄)
3	2	a1i 11	. . 3 ⊢ (𝜑 → (𝐶 Func 𝐷) = (Base‘𝑄))
4		eqid 2729	. . . . 5 ⊢ (𝐶 Nat 𝐷) = (𝐶 Nat 𝐷)
5	1, 4	fuchom 17926	. . . 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 17916	. . . . . . . 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 2729	. . . . . . . 8 ⊢ (Base‘𝐶) = (Base‘𝐶)
10	4, 8, 9	natfn 17919	. . . . . . 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 17916	. . . . . . . 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 17919	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑎 ∈ (𝑓(𝐶 Nat 𝐷)𝑔) ∧ 𝑏 ∈ (𝑓(𝐶 Nat 𝐷)𝑔))) → 𝑏 Fn (Base‘𝐶))
14		eqid 2729	. . . . . . . . . 10 ⊢ (Base‘𝐷) = (Base‘𝐷)
15	4, 8	natrcl2 49213	. . . . . . . . . 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 17828	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑎 ∈ (𝑓(𝐶 Nat 𝐷)𝑔) ∧ 𝑏 ∈ (𝑓(𝐶 Nat 𝐷)𝑔))) → (1st ‘𝑓):(Base‘𝐶)⟶(Base‘𝐷))
17	16	ffvelcdmda 7056	. . . . . . . 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 49214	. . . . . . . . . 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 17828	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑎 ∈ (𝑓(𝐶 Nat 𝐷)𝑔) ∧ 𝑏 ∈ (𝑓(𝐶 Nat 𝐷)𝑔))) → (1st ‘𝑔):(Base‘𝐶)⟶(Base‘𝐷))
20	19	ffvelcdmda 7056	. . . . . . . 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 2729	. . . . . . . . 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 17918	. . . . . . . 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 17918	. . . . . . . 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 49415	. . . . . . 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 7006	. . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ∈ (𝑓(𝐶 Nat 𝐷)𝑔) ∧ 𝑏 ∈ (𝑓(𝐶 Nat 𝐷)𝑔))) → 𝑎 = 𝑏)
31	30	ralrimivva 3180	. . . . 5 ⊢ (𝜑 → ∀𝑎 ∈ (𝑓(𝐶 Nat 𝐷)𝑔)∀𝑏 ∈ (𝑓(𝐶 Nat 𝐷)𝑔)𝑎 = 𝑏)
32		moel 3376	. . . . 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 4624	. . . . . . . 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 2831	. . . . . 6 ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))
40	39	func1st2nd 49065	. . . . 5 ⊢ (𝜑 → (1st ‘𝐹)(𝐶 Func 𝐷)(2nd ‘𝐹))
41	40	funcrcl2 49068	. . . 4 ⊢ (𝜑 → 𝐶 ∈ Cat)
42	27	thinccd 49412	. . . 4 ⊢ (𝜑 → 𝐷 ∈ Cat)
43	1, 41, 42	fuccat 17935	. . 3 ⊢ (𝜑 → 𝑄 ∈ Cat)
44	3, 6, 34, 43	isthincd 49425	. 2 ⊢ (𝜑 → 𝑄 ∈ ThinCat)
45		sneq 4599	. . . 4 ⊢ (𝑓 = 𝐹 → {𝑓} = {𝐹})
46	45	eqeq2d 2740	. . 3 ⊢ (𝑓 = 𝐹 → ((𝐶 Func 𝐷) = {𝑓} ↔ (𝐶 Func 𝐷) = {𝐹}))
47	35, 38, 46	spcedv 3564	. 2 ⊢ (𝜑 → ∃𝑓(𝐶 Func 𝐷) = {𝑓})
48	2	istermc 49463	. 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 2531  ∀wral 3044  {csn 4589  ⟨cop 4595  ‘cfv 6511  (class class class)co 7387  1^st c1st 7966  2^nd c2nd 7967  Basecbs 17179  Hom chom 17231   Func cfunc 17816   Nat cnat 17906   FuncCat cfuc 17907  ThinCatcthinc 49406  TermCatctermc 49461

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 2701  ax-rep 5234  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711  ax-cnex 11124  ax-resscn 11125  ax-1cn 11126  ax-icn 11127  ax-addcl 11128  ax-addrcl 11129  ax-mulcl 11130  ax-mulrcl 11131  ax-mulcom 11132  ax-addass 11133  ax-mulass 11134  ax-distr 11135  ax-i2m1 11136  ax-1ne0 11137  ax-1rid 11138  ax-rnegex 11139  ax-rrecex 11140  ax-cnre 11141  ax-pre-lttri 11142  ax-pre-lttrn 11143  ax-pre-ltadd 11144  ax-pre-mulgt0 11145

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 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-rmo 3354  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3934  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-tp 4594  df-op 4596  df-uni 4872  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-tr 5215  df-id 5533  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5591  df-we 5593  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6274  df-ord 6335  df-on 6336  df-lim 6337  df-suc 6338  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-riota 7344  df-ov 7390  df-oprab 7391  df-mpo 7392  df-om 7843  df-1st 7968  df-2nd 7969  df-frecs 8260  df-wrecs 8291  df-recs 8340  df-rdg 8378  df-1o 8434  df-er 8671  df-map 8801  df-ixp 8871  df-en 8919  df-dom 8920  df-sdom 8921  df-fin 8922  df-pnf 11210  df-mnf 11211  df-xr 11212  df-ltxr 11213  df-le 11214  df-sub 11407  df-neg 11408  df-nn 12187  df-2 12249  df-3 12250  df-4 12251  df-5 12252  df-6 12253  df-7 12254  df-8 12255  df-9 12256  df-n0 12443  df-z 12530  df-dec 12650  df-uz 12794  df-fz 13469  df-struct 17117  df-slot 17152  df-ndx 17164  df-base 17180  df-hom 17244  df-cco 17245  df-cat 17629  df-cid 17630  df-func 17820  df-nat 17908  df-fuc 17909  df-thinc 49407  df-termc 49462

This theorem is referenced by:  fucterm  49531