Theorem idfudiag1 49511

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 2730	. . 3 ⊢ (𝜑 → (Hom ‘𝐶) = (Hom ‘𝐶))
4		fveq2 6858	. . . . . . . . . . 11 ⊢ (𝑝 = ⟨𝑦, 𝑧⟩ → ((Hom ‘𝐶)‘𝑝) = ((Hom ‘𝐶)‘⟨𝑦, 𝑧⟩))
5		df-ov 7390	. . . . . . . . . . 11 ⊢ (𝑦(Hom ‘𝐶)𝑧) = ((Hom ‘𝐶)‘⟨𝑦, 𝑧⟩)
6	4, 5	eqtr4di 2782	. . . . . . . . . 10 ⊢ (𝑝 = ⟨𝑦, 𝑧⟩ → ((Hom ‘𝐶)‘𝑝) = (𝑦(Hom ‘𝐶)𝑧))
7	6	reseq2d 5950	. . . . . . . . 9 ⊢ (𝑝 = ⟨𝑦, 𝑧⟩ → ( I ↾ ((Hom ‘𝐶)‘𝑝)) = ( I ↾ (𝑦(Hom ‘𝐶)𝑧)))
8	7	mpompt 7503	. . . . . . . 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 7420	. . . . . . . 8 ⊢ (𝑦(Hom ‘𝐶)𝑧) ∈ V
11		resiexg 7888	. . . . . . . 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 48850	. . . . . 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 2729	. . . . . . . . . 10 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
18	15, 1, 16, 17	idfuval 17838	. . . . . . . . 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 2729	. . . . . . . . . 10 ⊢ (Id‘𝐶) = (Id‘𝐶)
23	19, 16, 16, 1, 20, 21, 1, 17, 22	diag1a 49291	. . . . . . . . 9 ⊢ (𝜑 → 𝐾 = ⟨(𝐵 × {𝑋}), (𝑦 ∈ 𝐵, 𝑧 ∈ 𝐵 ↦ ((𝑦(Hom ‘𝐶)𝑧) × {((Id‘𝐶)‘𝑋)}))⟩)
24	14, 18, 23	3eqtr3d 2772	. . . . . . . 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 6872	. . . . . . . . . . 11 ⊢ 𝐵 ∈ V
26		resiexg 7888	. . . . . . . . . . 11 ⊢ (𝐵 ∈ V → ( I ↾ 𝐵) ∈ V)
27	25, 26	ax-mp 5	. . . . . . . . . 10 ⊢ ( I ↾ 𝐵) ∈ V
28	25, 25	xpex 7729	. . . . . . . . . . 11 ⊢ (𝐵 × 𝐵) ∈ V
29	28	mptex 7197	. . . . . . . . . 10 ⊢ (𝑝 ∈ (𝐵 × 𝐵) ↦ ( I ↾ ((Hom ‘𝐶)‘𝑝))) ∈ V
30	27, 29	opth 5436	. . . . . . . . 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 496	. . . . . . . 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 5391	. . . . . . . . 9 ⊢ {((Id‘𝐶)‘𝑋)} ∈ V
34	10, 33	xpex 7729	. . . . . . . 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 48850	. . . . . 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 2766	. . . . 5 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → ( I ↾ (𝑦(Hom ‘𝐶)𝑧)) = ((𝑦(Hom ‘𝐶)𝑧) × {((Id‘𝐶)‘𝑋)}))
38	16	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → 𝐶 ∈ Cat)
39		simprl 770	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → 𝑦 ∈ 𝐵)
40	1, 17, 22, 38, 39	catidcl 17643	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → ((Id‘𝐶)‘𝑦) ∈ (𝑦(Hom ‘𝐶)𝑦))
41	15, 19, 16, 1, 20, 21, 14	idfudiag1bas 49510	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐵 = {𝑋})
42	41	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → 𝐵 = {𝑋})
43	39, 42	eleqtrd 2830	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → 𝑦 ∈ {𝑋})
44		elsni 4606	. . . . . . . . . 10 ⊢ (𝑦 ∈ {𝑋} → 𝑦 = 𝑋)
45	43, 44	syl 17	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → 𝑦 = 𝑋)
46		simprr 772	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → 𝑧 ∈ 𝐵)
47	46, 42	eleqtrd 2830	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → 𝑧 ∈ {𝑋})
48		elsni 4606	. . . . . . . . . 10 ⊢ (𝑧 ∈ {𝑋} → 𝑧 = 𝑋)
49	47, 48	syl 17	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → 𝑧 = 𝑋)
50	45, 49	eqtr4d 2767	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → 𝑦 = 𝑧)
51	50	oveq2d 7403	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → (𝑦(Hom ‘𝐶)𝑦) = (𝑦(Hom ‘𝐶)𝑧))
52	40, 51	eleqtrd 2830	. . . . . 6 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → ((Id‘𝐶)‘𝑦) ∈ (𝑦(Hom ‘𝐶)𝑧))
53	52	ne0d 4305	. . . . 5 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → (𝑦(Hom ‘𝐶)𝑧) ≠ ∅)
54	37, 53	idfudiag1lem 49509	. . . 4 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → (𝑦(Hom ‘𝐶)𝑧) = {((Id‘𝐶)‘𝑋)})
55		mosn 48798	. . . 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 49422	. 2 ⊢ (𝜑 → 𝐶 ∈ ThinCat)
58		sneq 4599	. . . 4 ⊢ (𝑥 = 𝑋 → {𝑥} = {𝑋})
59	58	eqeq2d 2740	. . 3 ⊢ (𝑥 = 𝑋 → (𝐵 = {𝑥} ↔ 𝐵 = {𝑋}))
60	20, 41, 59	spcedv 3564	. 2 ⊢ (𝜑 → ∃𝑥 𝐵 = {𝑥})
61	1	istermc 49460	. 2 ⊢ (𝐶 ∈ TermCat ↔ (𝐶 ∈ ThinCat ∧ ∃𝑥 𝐵 = {𝑥}))
62	57, 60, 61	sylanbrc 583	1 ⊢ (𝜑 → 𝐶 ∈ TermCat)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540  ∃wex 1779   ∈ wcel 2109  ∃*wmo 2531  Vcvv 3447  {csn 4589  ⟨cop 4595   ↦ cmpt 5188   I cid 5532   × cxp 5636   ↾ cres 5640  ‘cfv 6511  (class class class)co 7387   ∈ cmpo 7389  1^st c1st 7966  Basecbs 17179  Hom chom 17231  Catccat 17625  Idccid 17626  id_funccidfu 17817  Δ_funccdiag 18173  ThinCatcthinc 49403  TermCatctermc 49458

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-idfu 17821  df-nat 17908  df-fuc 17909  df-xpc 18133  df-1stf 18134  df-curf 18175  df-diag 18177  df-thinc 49404  df-termc 49459

This theorem is referenced by:  euendfunc  49512