Theorem idfudiag1 50248

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 2762	. . 3 ⊢ (𝜑 → (Hom ‘𝐶) = (Hom ‘𝐶))
4		fveq2 6881	. . . . . . . . . . 11 ⊢ (𝑝 = ⟨𝑦, 𝑧⟩ → ((Hom ‘𝐶)‘𝑝) = ((Hom ‘𝐶)‘⟨𝑦, 𝑧⟩))
5		df-ov 7413	. . . . . . . . . . 11 ⊢ (𝑦(Hom ‘𝐶)𝑧) = ((Hom ‘𝐶)‘⟨𝑦, 𝑧⟩)
6	4, 5	eqtr4di 2814	. . . . . . . . . 10 ⊢ (𝑝 = ⟨𝑦, 𝑧⟩ → ((Hom ‘𝐶)‘𝑝) = (𝑦(Hom ‘𝐶)𝑧))
7	6	reseq2d 5978	. . . . . . . . 9 ⊢ (𝑝 = ⟨𝑦, 𝑧⟩ → ( I ↾ ((Hom ‘𝐶)‘𝑝)) = ( I ↾ (𝑦(Hom ‘𝐶)𝑧)))
8	7	mpompt 7524	. . . . . . . 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 7443	. . . . . . . 8 ⊢ (𝑦(Hom ‘𝐶)𝑧) ∈ V
11		resiexg 7908	. . . . . . . 8 ⊢ ((𝑦(Hom ‘𝐶)𝑧) ∈ V → ( I ↾ (𝑦(Hom ‘𝐶)𝑧)) ∈ V)
12	10, 11	mp1i 14	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → ( I ↾ (𝑦(Hom ‘𝐶)𝑧)) ∈ V)
13	9, 12	ovmpt4d 49588	. . . . . 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 2761	. . . . . . . . . 10 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
18	15, 1, 16, 17	idfuval 17932	. . . . . . . . 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 2761	. . . . . . . . . 10 ⊢ (Id‘𝐶) = (Id‘𝐶)
23	19, 16, 16, 1, 20, 21, 1, 17, 22	diag1a 50028	. . . . . . . . 9 ⊢ (𝜑 → 𝐾 = ⟨(𝐵 × {𝑋}), (𝑦 ∈ 𝐵, 𝑧 ∈ 𝐵 ↦ ((𝑦(Hom ‘𝐶)𝑧) × {((Id‘𝐶)‘𝑋)}))⟩)
24	14, 18, 23	3eqtr3d 2804	. . . . . . . 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 6895	. . . . . . . . . . 11 ⊢ 𝐵 ∈ V
26		resiexg 7908	. . . . . . . . . . 11 ⊢ (𝐵 ∈ V → ( I ↾ 𝐵) ∈ V)
27	25, 26	ax-mp 5	. . . . . . . . . 10 ⊢ ( I ↾ 𝐵) ∈ V
28	25, 25	xpex 7751	. . . . . . . . . . 11 ⊢ (𝐵 × 𝐵) ∈ V
29	28	mptex 7221	. . . . . . . . . 10 ⊢ (𝑝 ∈ (𝐵 × 𝐵) ↦ ( I ↾ ((Hom ‘𝐶)‘𝑝))) ∈ V
30	27, 29	opth 5458	. . . . . . . . 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 502	. . . . . . . 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 18	. . . . . . 7 ⊢ (𝜑 → (𝑝 ∈ (𝐵 × 𝐵) ↦ ( I ↾ ((Hom ‘𝐶)‘𝑝))) = (𝑦 ∈ 𝐵, 𝑧 ∈ 𝐵 ↦ ((𝑦(Hom ‘𝐶)𝑧) × {((Id‘𝐶)‘𝑋)})))
33		snex 5410	. . . . . . . . 9 ⊢ {((Id‘𝐶)‘𝑋)} ∈ V
34	10, 33	xpex 7751	. . . . . . . 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 49588	. . . . . 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 2798	. . . . 5 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → ( I ↾ (𝑦(Hom ‘𝐶)𝑧)) = ((𝑦(Hom ‘𝐶)𝑧) × {((Id‘𝐶)‘𝑋)}))
38	16	adantr 485	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → 𝐶 ∈ Cat)
39		simprl 782	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → 𝑦 ∈ 𝐵)
40	1, 17, 22, 38, 39	catidcl 17737	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → ((Id‘𝐶)‘𝑦) ∈ (𝑦(Hom ‘𝐶)𝑦))
41	15, 19, 16, 1, 20, 21, 14	idfudiag1bas 50247	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐵 = {𝑋})
42	41	adantr 485	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → 𝐵 = {𝑋})
43	39, 42	eleqtrd 2863	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → 𝑦 ∈ {𝑋})
44		elsni 4605	. . . . . . . . . 10 ⊢ (𝑦 ∈ {𝑋} → 𝑦 = 𝑋)
45	43, 44	syl 18	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → 𝑦 = 𝑋)
46		simprr 784	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → 𝑧 ∈ 𝐵)
47	46, 42	eleqtrd 2863	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → 𝑧 ∈ {𝑋})
48		elsni 4605	. . . . . . . . . 10 ⊢ (𝑧 ∈ {𝑋} → 𝑧 = 𝑋)
49	47, 48	syl 18	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → 𝑧 = 𝑋)
50	45, 49	eqtr4d 2799	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → 𝑦 = 𝑧)
51	50	oveq2d 7426	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → (𝑦(Hom ‘𝐶)𝑦) = (𝑦(Hom ‘𝐶)𝑧))
52	40, 51	eleqtrd 2863	. . . . . 6 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → ((Id‘𝐶)‘𝑦) ∈ (𝑦(Hom ‘𝐶)𝑧))
53	52	ne0d 4294	. . . . 5 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → (𝑦(Hom ‘𝐶)𝑧) ≠ ∅)
54	37, 53	idfudiag1lem 50246	. . . 4 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → (𝑦(Hom ‘𝐶)𝑧) = {((Id‘𝐶)‘𝑋)})
55		mosn 49536	. . . 4 ⊢ ((𝑦(Hom ‘𝐶)𝑧) = {((Id‘𝐶)‘𝑋)} → ∃*𝑓 𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))
56	54, 55	syl 18	. . 3 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → ∃*𝑓 𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))
57	2, 3, 56, 16	isthincd 50159	. 2 ⊢ (𝜑 → 𝐶 ∈ ThinCat)
58		sneq 4598	. . . 4 ⊢ (𝑥 = 𝑋 → {𝑥} = {𝑋})
59	58	eqeq2d 2772	. . 3 ⊢ (𝑥 = 𝑋 → (𝐵 = {𝑥} ↔ 𝐵 = {𝑋}))
60	20, 41, 59	spcedv 3556	. 2 ⊢ (𝜑 → ∃𝑥 𝐵 = {𝑥})
61	1	istermc 50197	. 2 ⊢ (𝐶 ∈ TermCat ↔ (𝐶 ∈ ThinCat ∧ ∃𝑥 𝐵 = {𝑥}))
62	57, 60, 61	sylanbrc 594	1 ⊢ (𝜑 → 𝐶 ∈ TermCat)

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1568  ∃wex 1807   ∈ wcel 2141  ∃*wmo 2563  Vcvv 3453  {csn 4588  ⟨cop 4594   ↦ cmpt 5191   I cid 5555   × cxp 5659   ↾ cres 5663  ‘cfv 6536  (class class class)co 7410   ∈ cmpo 7412  1^st c1st 7983  Basecbs 17268  Hom chom 17320  Catccat 17719  Idccid 17720  id_funccidfu 17911  Δ_funccdiag 18267  ThinCatcthinc 50140  TermCatctermc 50195

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5237  ax-sep 5256  ax-nul 5268  ax-pow 5336  ax-pr 5404  ax-un 7732  ax-cnex 11155  ax-resscn 11156  ax-1cn 11157  ax-icn 11158  ax-addcl 11159  ax-addrcl 11160  ax-mulcl 11161  ax-mulrcl 11162  ax-mulcom 11163  ax-addass 11164  ax-mulass 11165  ax-distr 11166  ax-i2m1 11167  ax-1ne0 11168  ax-1rid 11169  ax-rnegex 11170  ax-rrecex 11171  ax-cnre 11172  ax-pre-lttri 11173  ax-pre-lttrn 11174  ax-pre-ltadd 11175  ax-pre-mulgt0 11176

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1571  df-fal 1581  df-ex 1808  df-nf 1812  df-sb 2095  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3063  df-ral 3078  df-rex 3088  df-rmo 3367  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3744  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4487  df-pw 4563  df-sn 4589  df-pr 4591  df-tp 4593  df-op 4595  df-uni 4872  df-iun 4957  df-br 5109  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5556  df-eprel 5561  df-po 5569  df-so 5570  df-fr 5614  df-we 5616  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-pred 6302  df-ord 6363  df-on 6364  df-lim 6365  df-suc 6366  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7367  df-ov 7413  df-oprab 7414  df-mpo 7415  df-om 7862  df-1st 7985  df-2nd 7986  df-frecs 8277  df-wrecs 8308  df-recs 8357  df-rdg 8396  df-1o 8452  df-er 8693  df-map 8825  df-ixp 8895  df-en 8943  df-dom 8944  df-sdom 8945  df-fin 8946  df-pnf 11244  df-mnf 11245  df-xr 11246  df-ltxr 11247  df-le 11248  df-sub 11442  df-neg 11443  df-nn 12233  df-2 12302  df-3 12303  df-4 12304  df-5 12305  df-6 12306  df-7 12307  df-8 12308  df-9 12309  df-n0 12504  df-z 12591  df-dec 12711  df-uz 12862  df-fz 13535  df-struct 17206  df-slot 17241  df-ndx 17253  df-base 17269  df-hom 17333  df-cco 17334  df-cat 17723  df-cid 17724  df-func 17914  df-idfu 17915  df-nat 18002  df-fuc 18003  df-xpc 18227  df-1stf 18228  df-curf 18269  df-diag 18271  df-thinc 50141  df-termc 50196

This theorem is referenced by:  euendfunc  50249