Theorem funcsn 50018

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 17932	. . . 4 ⊢ (𝐶 Func 𝐷) = (Base‘𝑄)
3	2	a1i 11	. . 3 ⊢ (𝜑 → (𝐶 Func 𝐷) = (Base‘𝑄))
4		eqid 2737	. . . . 5 ⊢ (𝐶 Nat 𝐷) = (𝐶 Nat 𝐷)
5	1, 4	fuchom 17933	. . . 4 ⊢ (𝐶 Nat 𝐷) = (Hom ‘𝑄)
6	5	a1i 11	. . 3 ⊢ (𝜑 → (𝐶 Nat 𝐷) = (Hom ‘𝑄))
7		simprl 771	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑎 ∈ (𝑓(𝐶 Nat 𝐷)𝑔) ∧ 𝑏 ∈ (𝑓(𝐶 Nat 𝐷)𝑔))) → 𝑎 ∈ (𝑓(𝐶 Nat 𝐷)𝑔))
8	4, 7	nat1st2nd 17923	. . . . . . . 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 2737	. . . . . . . 8 ⊢ (Base‘𝐶) = (Base‘𝐶)
10	4, 8, 9	natfn 17926	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑎 ∈ (𝑓(𝐶 Nat 𝐷)𝑔) ∧ 𝑏 ∈ (𝑓(𝐶 Nat 𝐷)𝑔))) → 𝑎 Fn (Base‘𝐶))
11		simprr 773	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑎 ∈ (𝑓(𝐶 Nat 𝐷)𝑔) ∧ 𝑏 ∈ (𝑓(𝐶 Nat 𝐷)𝑔))) → 𝑏 ∈ (𝑓(𝐶 Nat 𝐷)𝑔))
12	4, 11	nat1st2nd 17923	. . . . . . . 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 17926	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑎 ∈ (𝑓(𝐶 Nat 𝐷)𝑔) ∧ 𝑏 ∈ (𝑓(𝐶 Nat 𝐷)𝑔))) → 𝑏 Fn (Base‘𝐶))
14		eqid 2737	. . . . . . . . . 10 ⊢ (Base‘𝐷) = (Base‘𝐷)
15	4, 8	natrcl2 49701	. . . . . . . . . 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 17835	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑎 ∈ (𝑓(𝐶 Nat 𝐷)𝑔) ∧ 𝑏 ∈ (𝑓(𝐶 Nat 𝐷)𝑔))) → (1st ‘𝑓):(Base‘𝐶)⟶(Base‘𝐷))
17	16	ffvelcdmda 7038	. . . . . . . 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 49702	. . . . . . . . . 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 17835	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑎 ∈ (𝑓(𝐶 Nat 𝐷)𝑔) ∧ 𝑏 ∈ (𝑓(𝐶 Nat 𝐷)𝑔))) → (1st ‘𝑔):(Base‘𝐶)⟶(Base‘𝐷))
20	19	ffvelcdmda 7038	. . . . . . . 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 2737	. . . . . . . . 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 17925	. . . . . . . 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 17925	. . . . . . . 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 727	. . . . . . . 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 49903	. . . . . . 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 6988	. . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ∈ (𝑓(𝐶 Nat 𝐷)𝑔) ∧ 𝑏 ∈ (𝑓(𝐶 Nat 𝐷)𝑔))) → 𝑎 = 𝑏)
31	30	ralrimivva 3181	. . . . 5 ⊢ (𝜑 → ∀𝑎 ∈ (𝑓(𝐶 Nat 𝐷)𝑔)∀𝑏 ∈ (𝑓(𝐶 Nat 𝐷)𝑔)𝑎 = 𝑏)
32		moel 3363	. . . . 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 4605	. . . . . . . 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 2840	. . . . . 6 ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))
40	39	func1st2nd 49553	. . . . 5 ⊢ (𝜑 → (1st ‘𝐹)(𝐶 Func 𝐷)(2nd ‘𝐹))
41	40	funcrcl2 49556	. . . 4 ⊢ (𝜑 → 𝐶 ∈ Cat)
42	27	thinccd 49900	. . . 4 ⊢ (𝜑 → 𝐷 ∈ Cat)
43	1, 41, 42	fuccat 17942	. . 3 ⊢ (𝜑 → 𝑄 ∈ Cat)
44	3, 6, 34, 43	isthincd 49913	. 2 ⊢ (𝜑 → 𝑄 ∈ ThinCat)
45		sneq 4578	. . . 4 ⊢ (𝑓 = 𝐹 → {𝑓} = {𝐹})
46	45	eqeq2d 2748	. . 3 ⊢ (𝑓 = 𝐹 → ((𝐶 Func 𝐷) = {𝑓} ↔ (𝐶 Func 𝐷) = {𝐹}))
47	35, 38, 46	spcedv 3541	. 2 ⊢ (𝜑 → ∃𝑓(𝐶 Func 𝐷) = {𝑓})
48	2	istermc 49951	. 2 ⊢ (𝑄 ∈ TermCat ↔ (𝑄 ∈ ThinCat ∧ ∃𝑓(𝐶 Func 𝐷) = {𝑓}))
49	44, 47, 48	sylanbrc 584	1 ⊢ (𝜑 → 𝑄 ∈ TermCat)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542  ∃wex 1781   ∈ wcel 2114  ∃*wmo 2538  ∀wral 3052  {csn 4568  ⟨cop 4574  ‘cfv 6500  (class class class)co 7369  1^st c1st 7942  2^nd c2nd 7943  Basecbs 17181  Hom chom 17233   Func cfunc 17823   Nat cnat 17913   FuncCat cfuc 17914  ThinCatcthinc 49894  TermCatctermc 49949

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5213  ax-sep 5232  ax-nul 5242  ax-pow 5308  ax-pr 5376  ax-un 7691  ax-cnex 11096  ax-resscn 11097  ax-1cn 11098  ax-icn 11099  ax-addcl 11100  ax-addrcl 11101  ax-mulcl 11102  ax-mulrcl 11103  ax-mulcom 11104  ax-addass 11105  ax-mulass 11106  ax-distr 11107  ax-i2m1 11108  ax-1ne0 11109  ax-1rid 11110  ax-rnegex 11111  ax-rrecex 11112  ax-cnre 11113  ax-pre-lttri 11114  ax-pre-lttrn 11115  ax-pre-ltadd 11116  ax-pre-mulgt0 11117

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-rmo 3343  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-tp 4573  df-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5527  df-eprel 5532  df-po 5540  df-so 5541  df-fr 5585  df-we 5587  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-pred 6267  df-ord 6328  df-on 6329  df-lim 6330  df-suc 6331  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-riota 7326  df-ov 7372  df-oprab 7373  df-mpo 7374  df-om 7820  df-1st 7944  df-2nd 7945  df-frecs 8233  df-wrecs 8264  df-recs 8313  df-rdg 8351  df-1o 8407  df-er 8645  df-map 8777  df-ixp 8848  df-en 8896  df-dom 8897  df-sdom 8898  df-fin 8899  df-pnf 11183  df-mnf 11184  df-xr 11185  df-ltxr 11186  df-le 11187  df-sub 11381  df-neg 11382  df-nn 12177  df-2 12246  df-3 12247  df-4 12248  df-5 12249  df-6 12250  df-7 12251  df-8 12252  df-9 12253  df-n0 12440  df-z 12527  df-dec 12647  df-uz 12791  df-fz 13464  df-struct 17119  df-slot 17154  df-ndx 17166  df-base 17182  df-hom 17246  df-cco 17247  df-cat 17636  df-cid 17637  df-func 17827  df-nat 17915  df-fuc 17916  df-thinc 49895  df-termc 49950

This theorem is referenced by:  fucterm  50019