Theorem idfudiag1 50084

Description: If the identity functor of a category is the same as a constant functor to the category, then the category is terminal. (Contributed by Zhi Wang, 19-Oct-2025.)

Hypotheses

Ref	Expression
idfudiag1.i	⊢ 𝐼 = (idfunc‘𝐶)
idfudiag1.l	⊢ 𝐿 = (𝐶Δfunc𝐶)
idfudiag1.c	⊢ (𝜑 → 𝐶 ∈ Cat)
idfudiag1.b	⊢ 𝐵 = (Base‘𝐶)
idfudiag1.x	⊢ (𝜑 → 𝑋 ∈ 𝐵)
idfudiag1.k	⊢ 𝐾 = ((1st ‘𝐿)‘𝑋)
idfudiag1.e	⊢ (𝜑 → 𝐼 = 𝐾)

Assertion

Ref	Expression
idfudiag1	⊢ (𝜑 → 𝐶 ∈ TermCat)

Proof of Theorem idfudiag1

Dummy variables 𝑓 𝑝 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		idfudiag1.b	. . . 4 ⊢ 𝐵 = (Base‘𝐶)
2	1	a1i 11	. . 3 ⊢ (𝜑 → 𝐵 = (Base‘𝐶))
3		eqidd 2753	. . 3 ⊢ (𝜑 → (Hom ‘𝐶) = (Hom ‘𝐶))
4		fveq2 6852	. . . . . . . . . . 11 ⊢ (𝑝 = ⟨𝑦, 𝑧⟩ → ((Hom ‘𝐶)‘𝑝) = ((Hom ‘𝐶)‘⟨𝑦, 𝑧⟩))
5		df-ov 7384	. . . . . . . . . . 11 ⊢ (𝑦(Hom ‘𝐶)𝑧) = ((Hom ‘𝐶)‘⟨𝑦, 𝑧⟩)
6	4, 5	eqtr4di 2805	. . . . . . . . . 10 ⊢ (𝑝 = ⟨𝑦, 𝑧⟩ → ((Hom ‘𝐶)‘𝑝) = (𝑦(Hom ‘𝐶)𝑧))
7	6	reseq2d 5954	. . . . . . . . 9 ⊢ (𝑝 = ⟨𝑦, 𝑧⟩ → ( I ↾ ((Hom ‘𝐶)‘𝑝)) = ( I ↾ (𝑦(Hom ‘𝐶)𝑧)))
8	7	mpompt 7495	. . . . . . . 8 ⊢ (𝑝 ∈ (𝐵 × 𝐵) ↦ ( I ↾ ((Hom ‘𝐶)‘𝑝))) = (𝑦 ∈ 𝐵, 𝑧 ∈ 𝐵 ↦ ( I ↾ (𝑦(Hom ‘𝐶)𝑧)))
9	8	a1i 11	. . . . . . 7 ⊢ (𝜑 → (𝑝 ∈ (𝐵 × 𝐵) ↦ ( I ↾ ((Hom ‘𝐶)‘𝑝))) = (𝑦 ∈ 𝐵, 𝑧 ∈ 𝐵 ↦ ( I ↾ (𝑦(Hom ‘𝐶)𝑧))))
10		ovex 7414	. . . . . . . 8 ⊢ (𝑦(Hom ‘𝐶)𝑧) ∈ V
11		resiexg 7878	. . . . . . . 8 ⊢ ((𝑦(Hom ‘𝐶)𝑧) ∈ V → ( I ↾ (𝑦(Hom ‘𝐶)𝑧)) ∈ V)
12	10, 11	mp1i 13	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → ( I ↾ (𝑦(Hom ‘𝐶)𝑧)) ∈ V)
13	9, 12	ovmpt4d 49424	. . . . . 6 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → (𝑦(𝑝 ∈ (𝐵 × 𝐵) ↦ ( I ↾ ((Hom ‘𝐶)‘𝑝)))𝑧) = ( I ↾ (𝑦(Hom ‘𝐶)𝑧)))
14		idfudiag1.e	. . . . . . . . 9 ⊢ (𝜑 → 𝐼 = 𝐾)
15		idfudiag1.i	. . . . . . . . . 10 ⊢ 𝐼 = (idfunc‘𝐶)
16		idfudiag1.c	. . . . . . . . . 10 ⊢ (𝜑 → 𝐶 ∈ Cat)
17		eqid 2752	. . . . . . . . . 10 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
18	15, 1, 16, 17	idfuval 17881	. . . . . . . . 9 ⊢ (𝜑 → 𝐼 = ⟨( I ↾ 𝐵), (𝑝 ∈ (𝐵 × 𝐵) ↦ ( I ↾ ((Hom ‘𝐶)‘𝑝)))⟩)
19		idfudiag1.l	. . . . . . . . . 10 ⊢ 𝐿 = (𝐶Δfunc𝐶)
20		idfudiag1.x	. . . . . . . . . 10 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
21		idfudiag1.k	. . . . . . . . . 10 ⊢ 𝐾 = ((1st ‘𝐿)‘𝑋)
22		eqid 2752	. . . . . . . . . 10 ⊢ (Id‘𝐶) = (Id‘𝐶)
23	19, 16, 16, 1, 20, 21, 1, 17, 22	diag1a 49864	. . . . . . . . 9 ⊢ (𝜑 → 𝐾 = ⟨(𝐵 × {𝑋}), (𝑦 ∈ 𝐵, 𝑧 ∈ 𝐵 ↦ ((𝑦(Hom ‘𝐶)𝑧) × {((Id‘𝐶)‘𝑋)}))⟩)
24	14, 18, 23	3eqtr3d 2795	. . . . . . . 8 ⊢ (𝜑 → ⟨( I ↾ 𝐵), (𝑝 ∈ (𝐵 × 𝐵) ↦ ( I ↾ ((Hom ‘𝐶)‘𝑝)))⟩ = ⟨(𝐵 × {𝑋}), (𝑦 ∈ 𝐵, 𝑧 ∈ 𝐵 ↦ ((𝑦(Hom ‘𝐶)𝑧) × {((Id‘𝐶)‘𝑋)}))⟩)
25	1	fvexi 6866	. . . . . . . . . . 11 ⊢ 𝐵 ∈ V
26		resiexg 7878	. . . . . . . . . . 11 ⊢ (𝐵 ∈ V → ( I ↾ 𝐵) ∈ V)
27	25, 26	ax-mp 5	. . . . . . . . . 10 ⊢ ( I ↾ 𝐵) ∈ V
28	25, 25	xpex 7721	. . . . . . . . . . 11 ⊢ (𝐵 × 𝐵) ∈ V
29	28	mptex 7192	. . . . . . . . . 10 ⊢ (𝑝 ∈ (𝐵 × 𝐵) ↦ ( I ↾ ((Hom ‘𝐶)‘𝑝))) ∈ V
30	27, 29	opth 5434	. . . . . . . . 9 ⊢ (⟨( I ↾ 𝐵), (𝑝 ∈ (𝐵 × 𝐵) ↦ ( I ↾ ((Hom ‘𝐶)‘𝑝)))⟩ = ⟨(𝐵 × {𝑋}), (𝑦 ∈ 𝐵, 𝑧 ∈ 𝐵 ↦ ((𝑦(Hom ‘𝐶)𝑧) × {((Id‘𝐶)‘𝑋)}))⟩ ↔ (( I ↾ 𝐵) = (𝐵 × {𝑋}) ∧ (𝑝 ∈ (𝐵 × 𝐵) ↦ ( I ↾ ((Hom ‘𝐶)‘𝑝))) = (𝑦 ∈ 𝐵, 𝑧 ∈ 𝐵 ↦ ((𝑦(Hom ‘𝐶)𝑧) × {((Id‘𝐶)‘𝑋)}))))
31	30	simprbi 500	. . . . . . . 8 ⊢ (⟨( I ↾ 𝐵), (𝑝 ∈ (𝐵 × 𝐵) ↦ ( I ↾ ((Hom ‘𝐶)‘𝑝)))⟩ = ⟨(𝐵 × {𝑋}), (𝑦 ∈ 𝐵, 𝑧 ∈ 𝐵 ↦ ((𝑦(Hom ‘𝐶)𝑧) × {((Id‘𝐶)‘𝑋)}))⟩ → (𝑝 ∈ (𝐵 × 𝐵) ↦ ( I ↾ ((Hom ‘𝐶)‘𝑝))) = (𝑦 ∈ 𝐵, 𝑧 ∈ 𝐵 ↦ ((𝑦(Hom ‘𝐶)𝑧) × {((Id‘𝐶)‘𝑋)})))
32	24, 31	syl 17	. . . . . . 7 ⊢ (𝜑 → (𝑝 ∈ (𝐵 × 𝐵) ↦ ( I ↾ ((Hom ‘𝐶)‘𝑝))) = (𝑦 ∈ 𝐵, 𝑧 ∈ 𝐵 ↦ ((𝑦(Hom ‘𝐶)𝑧) × {((Id‘𝐶)‘𝑋)})))
33		snex 5386	. . . . . . . . 9 ⊢ {((Id‘𝐶)‘𝑋)} ∈ V
34	10, 33	xpex 7721	. . . . . . . 8 ⊢ ((𝑦(Hom ‘𝐶)𝑧) × {((Id‘𝐶)‘𝑋)}) ∈ V
35	34	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → ((𝑦(Hom ‘𝐶)𝑧) × {((Id‘𝐶)‘𝑋)}) ∈ V)
36	32, 35	ovmpt4d 49424	. . . . . 6 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → (𝑦(𝑝 ∈ (𝐵 × 𝐵) ↦ ( I ↾ ((Hom ‘𝐶)‘𝑝)))𝑧) = ((𝑦(Hom ‘𝐶)𝑧) × {((Id‘𝐶)‘𝑋)}))
37	13, 36	eqtr3d 2789	. . . . 5 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → ( I ↾ (𝑦(Hom ‘𝐶)𝑧)) = ((𝑦(Hom ‘𝐶)𝑧) × {((Id‘𝐶)‘𝑋)}))
38	16	adantr 483	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → 𝐶 ∈ Cat)
39		simprl 778	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → 𝑦 ∈ 𝐵)
40	1, 17, 22, 38, 39	catidcl 17686	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → ((Id‘𝐶)‘𝑦) ∈ (𝑦(Hom ‘𝐶)𝑦))
41	15, 19, 16, 1, 20, 21, 14	idfudiag1bas 50083	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐵 = {𝑋})
42	41	adantr 483	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → 𝐵 = {𝑋})
43	39, 42	eleqtrd 2854	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → 𝑦 ∈ {𝑋})
44		elsni 4589	. . . . . . . . . 10 ⊢ (𝑦 ∈ {𝑋} → 𝑦 = 𝑋)
45	43, 44	syl 17	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → 𝑦 = 𝑋)
46		simprr 780	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → 𝑧 ∈ 𝐵)
47	46, 42	eleqtrd 2854	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → 𝑧 ∈ {𝑋})
48		elsni 4589	. . . . . . . . . 10 ⊢ (𝑧 ∈ {𝑋} → 𝑧 = 𝑋)
49	47, 48	syl 17	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → 𝑧 = 𝑋)
50	45, 49	eqtr4d 2790	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → 𝑦 = 𝑧)
51	50	oveq2d 7397	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → (𝑦(Hom ‘𝐶)𝑦) = (𝑦(Hom ‘𝐶)𝑧))
52	40, 51	eleqtrd 2854	. . . . . 6 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → ((Id‘𝐶)‘𝑦) ∈ (𝑦(Hom ‘𝐶)𝑧))
53	52	ne0d 4285	. . . . 5 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → (𝑦(Hom ‘𝐶)𝑧) ≠ ∅)
54	37, 53	idfudiag1lem 50082	. . . 4 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → (𝑦(Hom ‘𝐶)𝑧) = {((Id‘𝐶)‘𝑋)})
55		mosn 49372	. . . 4 ⊢ ((𝑦(Hom ‘𝐶)𝑧) = {((Id‘𝐶)‘𝑋)} → ∃*𝑓 𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))
56	54, 55	syl 17	. . 3 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → ∃*𝑓 𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))
57	2, 3, 56, 16	isthincd 49995	. 2 ⊢ (𝜑 → 𝐶 ∈ ThinCat)
58		sneq 4582	. . . 4 ⊢ (𝑥 = 𝑋 → {𝑥} = {𝑋})
59	58	eqeq2d 2763	. . 3 ⊢ (𝑥 = 𝑋 → (𝐵 = {𝑥} ↔ 𝐵 = {𝑋}))
60	20, 41, 59	spcedv 3548	. 2 ⊢ (𝜑 → ∃𝑥 𝐵 = {𝑥})
61	1	istermc 50033	. 2 ⊢ (𝐶 ∈ TermCat ↔ (𝐶 ∈ ThinCat ∧ ∃𝑥 𝐵 = {𝑥}))
62	57, 60, 61	sylanbrc 591	1 ⊢ (𝜑 → 𝐶 ∈ TermCat)

Syntax hints:   → wi 4   ∧ wa 398   = wceq 1550  ∃wex 1789   ∈ wcel 2132  ∃*wmo 2554  Vcvv 3444  {csn 4572  ⟨cop 4578   ↦ cmpt 5171   I cid 5530   × cxp 5634   ↾ cres 5638  ‘cfv 6506  (class class class)co 7381   ∈ cmpo 7383  1^st c1st 7953  Basecbs 17217  Hom chom 17269  Catccat 17668  Idccid 17669  id_funccidfu 17860  Δ_funccdiag 18216  ThinCatcthinc 49976  TermCatctermc 50031

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1805  ax-4 1819  ax-5 1920  ax-6 1977  ax-7 2018  ax-8 2134  ax-9 2142  ax-10 2165  ax-11 2181  ax-12 2202  ax-ext 2724  ax-rep 5217  ax-sep 5236  ax-nul 5246  ax-pow 5312  ax-pr 5380  ax-un 7703  ax-cnex 11115  ax-resscn 11116  ax-1cn 11117  ax-icn 11118  ax-addcl 11119  ax-addrcl 11120  ax-mulcl 11121  ax-mulrcl 11122  ax-mulcom 11123  ax-addass 11124  ax-mulass 11125  ax-distr 11126  ax-i2m1 11127  ax-1ne0 11128  ax-1rid 11129  ax-rnegex 11130  ax-rrecex 11131  ax-cnre 11132  ax-pre-lttri 11133  ax-pre-lttrn 11134  ax-pre-ltadd 11135  ax-pre-mulgt0 11136

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 857  df-3or 1096  df-3an 1097  df-tru 1553  df-fal 1563  df-ex 1790  df-nf 1794  df-sb 2081  df-mo 2556  df-eu 2586  df-clab 2731  df-cleq 2744  df-clel 2827  df-nfc 2901  df-ne 2948  df-nel 3052  df-ral 3067  df-rex 3077  df-rmo 3357  df-reu 3358  df-rab 3405  df-v 3446  df-sbc 3736  df-csb 3844  df-dif 3898  df-un 3900  df-in 3902  df-ss 3912  df-pss 3915  df-nul 4277  df-if 4471  df-pw 4547  df-sn 4573  df-pr 4575  df-tp 4577  df-op 4579  df-uni 4856  df-iun 4941  df-br 5091  df-opab 5153  df-mpt 5172  df-tr 5198  df-id 5531  df-eprel 5536  df-po 5544  df-so 5545  df-fr 5589  df-we 5591  df-xp 5642  df-rel 5643  df-cnv 5644  df-co 5645  df-dm 5646  df-rn 5647  df-res 5648  df-ima 5649  df-pred 6273  df-ord 6334  df-on 6335  df-lim 6336  df-suc 6337  df-iota 6462  df-fun 6508  df-fn 6509  df-f 6510  df-f1 6511  df-fo 6512  df-f1o 6513  df-fv 6514  df-riota 7338  df-ov 7384  df-oprab 7385  df-mpo 7386  df-om 7832  df-1st 7955  df-2nd 7956  df-frecs 8246  df-wrecs 8277  df-recs 8326  df-rdg 8365  df-1o 8421  df-er 8662  df-map 8794  df-ixp 8865  df-en 8913  df-dom 8914  df-sdom 8915  df-fin 8916  df-pnf 11204  df-mnf 11205  df-xr 11206  df-ltxr 11207  df-le 11208  df-sub 11402  df-neg 11403  df-nn 12197  df-2 12266  df-3 12267  df-4 12268  df-5 12269  df-6 12270  df-7 12271  df-8 12272  df-9 12273  df-n0 12468  df-z 12555  df-dec 12675  df-uz 12826  df-fz 13499  df-struct 17155  df-slot 17190  df-ndx 17202  df-base 17218  df-hom 17282  df-cco 17283  df-cat 17672  df-cid 17673  df-func 17863  df-idfu 17864  df-nat 17951  df-fuc 17952  df-xpc 18176  df-1stf 18177  df-curf 18218  df-diag 18220  df-thinc 49977  df-termc 50032

This theorem is referenced by:  euendfunc  50085