Theorem idfudiag1 50676

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 2761	. . 3 ⊢ (𝜑 → (Hom ‘𝐶) = (Hom ‘𝐶))
4		fveq2 6885	. . . . . . . . . . 11 ⊢ (𝑝 = ⟨𝑦, 𝑧⟩ → ((Hom ‘𝐶)‘𝑝) = ((Hom ‘𝐶)‘⟨𝑦, 𝑧⟩))
5		df-ov 7423	. . . . . . . . . . 11 ⊢ (𝑦(Hom ‘𝐶)𝑧) = ((Hom ‘𝐶)‘⟨𝑦, 𝑧⟩)
6	4, 5	eqtr4di 2813	. . . . . . . . . 10 ⊢ (𝑝 = ⟨𝑦, 𝑧⟩ → ((Hom ‘𝐶)‘𝑝) = (𝑦(Hom ‘𝐶)𝑧))
7	6	reseq2d 5969	. . . . . . . . 9 ⊢ (𝑝 = ⟨𝑦, 𝑧⟩ → ( I ↾ ((Hom ‘𝐶)‘𝑝)) = ( I ↾ (𝑦(Hom ‘𝐶)𝑧)))
8	7	mpompt 7534	. . . . . . . 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 7453	. . . . . . . 8 ⊢ (𝑦(Hom ‘𝐶)𝑧) ∈ V
11		resiexg 7924	. . . . . . . 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 50018	. . . . . 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 2760	. . . . . . . . . 10 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
18	15, 1, 16, 17	idfuval 18051	. . . . . . . . 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 2760	. . . . . . . . . 10 ⊢ (Id‘𝐶) = (Id‘𝐶)
23	19, 16, 16, 1, 20, 21, 1, 17, 22	diag1a 50456	. . . . . . . . 9 ⊢ (𝜑 → 𝐾 = ⟨(𝐵 × {𝑋}), (𝑦 ∈ 𝐵, 𝑧 ∈ 𝐵 ↦ ((𝑦(Hom ‘𝐶)𝑧) × {((Id‘𝐶)‘𝑋)}))⟩)
24	14, 18, 23	3eqtr3d 2803	. . . . . . . 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 6899	. . . . . . . . . . 11 ⊢ 𝐵 ∈ V
26		resiexg 7924	. . . . . . . . . . 11 ⊢ (𝐵 ∈ V → ( I ↾ 𝐵) ∈ V)
27	25, 26	ax-mp 5	. . . . . . . . . 10 ⊢ ( I ↾ 𝐵) ∈ V
28	25, 25	xpex 7767	. . . . . . . . . . 11 ⊢ (𝐵 × 𝐵) ∈ V
29	28	mptex 7229	. . . . . . . . . 10 ⊢ (𝑝 ∈ (𝐵 × 𝐵) ↦ ( I ↾ ((Hom ‘𝐶)‘𝑝))) ∈ V
30	27, 29	opth 5444	. . . . . . . . 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 503	. . . . . . . 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 5396	. . . . . . . . 9 ⊢ {((Id‘𝐶)‘𝑋)} ∈ V
34	10, 33	xpex 7767	. . . . . . . 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 50018	. . . . . 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 2797	. . . . 5 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → ( I ↾ (𝑦(Hom ‘𝐶)𝑧)) = ((𝑦(Hom ‘𝐶)𝑧) × {((Id‘𝐶)‘𝑋)}))
38	16	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → 𝐶 ∈ Cat)
39		simprl 783	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → 𝑦 ∈ 𝐵)
40	1, 17, 22, 38, 39	catidcl 17856	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → ((Id‘𝐶)‘𝑦) ∈ (𝑦(Hom ‘𝐶)𝑦))
41	15, 19, 16, 1, 20, 21, 14	idfudiag1bas 50675	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐵 = {𝑋})
42	41	adantr 486	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → 𝐵 = {𝑋})
43	39, 42	eleqtrd 2862	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → 𝑦 ∈ {𝑋})
44		elsni 4600	. . . . . . . . . 10 ⊢ (𝑦 ∈ {𝑋} → 𝑦 = 𝑋)
45	43, 44	syl 18	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → 𝑦 = 𝑋)
46		simprr 785	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → 𝑧 ∈ 𝐵)
47	46, 42	eleqtrd 2862	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → 𝑧 ∈ {𝑋})
48		elsni 4600	. . . . . . . . . 10 ⊢ (𝑧 ∈ {𝑋} → 𝑧 = 𝑋)
49	47, 48	syl 18	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → 𝑧 = 𝑋)
50	45, 49	eqtr4d 2798	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → 𝑦 = 𝑧)
51	50	oveq2d 7436	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → (𝑦(Hom ‘𝐶)𝑦) = (𝑦(Hom ‘𝐶)𝑧))
52	40, 51	eleqtrd 2862	. . . . . 6 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → ((Id‘𝐶)‘𝑦) ∈ (𝑦(Hom ‘𝐶)𝑧))
53	52	ne0d 4287	. . . . 5 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → (𝑦(Hom ‘𝐶)𝑧) ≠ ∅)
54	37, 53	idfudiag1lem 50674	. . . 4 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → (𝑦(Hom ‘𝐶)𝑧) = {((Id‘𝐶)‘𝑋)})
55		mosn 49966	. . . 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 50587	. 2 ⊢ (𝜑 → 𝐶 ∈ ThinCat)
58		sneq 4593	. . . 4 ⊢ (𝑥 = 𝑋 → {𝑥} = {𝑋})
59	58	eqeq2d 2771	. . 3 ⊢ (𝑥 = 𝑋 → (𝐵 = {𝑥} ↔ 𝐵 = {𝑋}))
60	20, 41, 59	spcedv 3552	. 2 ⊢ (𝜑 → ∃𝑥 𝐵 = {𝑥})
61	1	istermc 50625	. 2 ⊢ (𝐶 ∈ TermCat ↔ (𝐶 ∈ ThinCat ∧ ∃𝑥 𝐵 = {𝑥}))
62	57, 60, 61	sylanbrc 595	1 ⊢ (𝜑 → 𝐶 ∈ TermCat)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1569  ∃wex 1811   ∈ wcel 2144  ∃*wmo 2562  Vcvv 3450  {csn 4583  ⟨cop 4589   ↦ cmpt 5185   I cid 5544   × cxp 5648   ↾ cres 5652  ‘cfv 6538  (class class class)co 7420   ∈ cmpo 7422  1^st c1st 7999  Basecbs 17387  Hom chom 17439  Catccat 17838  Idccid 17839  id_funccidfu 18030  Δ_funccdiag 18386  ThinCatcthinc 50568  TermCatctermc 50623

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751  ax-cnex 11256  ax-resscn 11257  ax-1cn 11258  ax-icn 11259  ax-addcl 11260  ax-addrcl 11261  ax-mulcl 11262  ax-mulrcl 11263  ax-mulcom 11264  ax-addass 11265  ax-mulass 11266  ax-distr 11267  ax-i2m1 11268  ax-1ne0 11269  ax-1rid 11270  ax-rnegex 11271  ax-rrecex 11272  ax-cnre 11273  ax-pre-lttri 11274  ax-pre-lttrn 11275  ax-pre-ltadd 11276  ax-pre-mulgt0 11277

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-tp 4588  df-op 4590  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-riota 7377  df-ov 7423  df-oprab 7424  df-mpo 7425  df-om 7878  df-1st 8001  df-2nd 8002  df-frecs 8299  df-wrecs 8330  df-recs 8379  df-rdg 8418  df-1o 8476  df-er 8717  df-map 8849  df-ixp 8926  df-en 8974  df-dom 8975  df-sdom 8976  df-fin 8977  df-pnf 11345  df-mnf 11346  df-xr 11347  df-ltxr 11348  df-le 11349  df-sub 11543  df-neg 11544  df-nn 12336  df-2 12405  df-3 12406  df-4 12407  df-5 12408  df-6 12409  df-7 12410  df-8 12411  df-9 12412  df-n0 12607  df-z 12694  df-dec 12815  df-uz 12966  df-fz 13640  df-struct 17325  df-slot 17360  df-ndx 17372  df-base 17388  df-hom 17452  df-cco 17453  df-cat 17842  df-cid 17843  df-func 18033  df-idfu 18034  df-nat 18121  df-fuc 18122  df-xpc 18346  df-1stf 18347  df-curf 18388  df-diag 18390  df-thinc 50569  df-termc 50624

This theorem is used by:  euendfunc  50677