Theorem functermceu 49755

Description: There exists a unique functor to a terminal category. (Contributed by Zhi Wang, 17-Oct-2025.)

Hypotheses

Ref	Expression
functermceu.c	⊢ (𝜑 → 𝐶 ∈ Cat)
functermceu.d	⊢ (𝜑 → 𝐷 ∈ TermCat)

Assertion

Ref	Expression
functermceu	⊢ (𝜑 → ∃!𝑓 𝑓 ∈ (𝐶 Func 𝐷))

Distinct variable groups:   𝐶,𝑓   𝐷,𝑓

Allowed substitution hint:   𝜑(𝑓)

Proof of Theorem functermceu

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

Step	Hyp	Ref	Expression
1		opex 5412	. . . 4 ⊢ ⟨((Base‘𝐶) × (Base‘𝐷)), (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ ((𝑥(Hom ‘𝐶)𝑦) × ((((Base‘𝐶) × (Base‘𝐷))‘𝑥)(Hom ‘𝐷)(((Base‘𝐶) × (Base‘𝐷))‘𝑦))))⟩ ∈ V
2	1	a1i 11	. . 3 ⊢ (𝜑 → ⟨((Base‘𝐶) × (Base‘𝐷)), (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ ((𝑥(Hom ‘𝐶)𝑦) × ((((Base‘𝐶) × (Base‘𝐷))‘𝑥)(Hom ‘𝐷)(((Base‘𝐶) × (Base‘𝐷))‘𝑦))))⟩ ∈ V)
3		functermceu.c	. . . 4 ⊢ (𝜑 → 𝐶 ∈ Cat)
4		functermceu.d	. . . 4 ⊢ (𝜑 → 𝐷 ∈ TermCat)
5		eqid 2736	. . . 4 ⊢ (Base‘𝐶) = (Base‘𝐶)
6		eqid 2736	. . . 4 ⊢ (Base‘𝐷) = (Base‘𝐷)
7		eqid 2736	. . . 4 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
8		eqid 2736	. . . 4 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
9		eqid 2736	. . . 4 ⊢ ((Base‘𝐶) × (Base‘𝐷)) = ((Base‘𝐶) × (Base‘𝐷))
10		eqid 2736	. . . 4 ⊢ (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ ((𝑥(Hom ‘𝐶)𝑦) × ((((Base‘𝐶) × (Base‘𝐷))‘𝑥)(Hom ‘𝐷)(((Base‘𝐶) × (Base‘𝐷))‘𝑦)))) = (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ ((𝑥(Hom ‘𝐶)𝑦) × ((((Base‘𝐶) × (Base‘𝐷))‘𝑥)(Hom ‘𝐷)(((Base‘𝐶) × (Base‘𝐷))‘𝑦))))
11	3, 4, 5, 6, 7, 8, 9, 10	functermc2 49754	. . 3 ⊢ (𝜑 → (𝐶 Func 𝐷) = {⟨((Base‘𝐶) × (Base‘𝐷)), (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ ((𝑥(Hom ‘𝐶)𝑦) × ((((Base‘𝐶) × (Base‘𝐷))‘𝑥)(Hom ‘𝐷)(((Base‘𝐶) × (Base‘𝐷))‘𝑦))))⟩})
12		sneq 4590	. . . 4 ⊢ (𝑓 = ⟨((Base‘𝐶) × (Base‘𝐷)), (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ ((𝑥(Hom ‘𝐶)𝑦) × ((((Base‘𝐶) × (Base‘𝐷))‘𝑥)(Hom ‘𝐷)(((Base‘𝐶) × (Base‘𝐷))‘𝑦))))⟩ → {𝑓} = {⟨((Base‘𝐶) × (Base‘𝐷)), (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ ((𝑥(Hom ‘𝐶)𝑦) × ((((Base‘𝐶) × (Base‘𝐷))‘𝑥)(Hom ‘𝐷)(((Base‘𝐶) × (Base‘𝐷))‘𝑦))))⟩})
13	12	eqeq2d 2747	. . 3 ⊢ (𝑓 = ⟨((Base‘𝐶) × (Base‘𝐷)), (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ ((𝑥(Hom ‘𝐶)𝑦) × ((((Base‘𝐶) × (Base‘𝐷))‘𝑥)(Hom ‘𝐷)(((Base‘𝐶) × (Base‘𝐷))‘𝑦))))⟩ → ((𝐶 Func 𝐷) = {𝑓} ↔ (𝐶 Func 𝐷) = {⟨((Base‘𝐶) × (Base‘𝐷)), (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ ((𝑥(Hom ‘𝐶)𝑦) × ((((Base‘𝐶) × (Base‘𝐷))‘𝑥)(Hom ‘𝐷)(((Base‘𝐶) × (Base‘𝐷))‘𝑦))))⟩}))
14	2, 11, 13	spcedv 3552	. 2 ⊢ (𝜑 → ∃𝑓(𝐶 Func 𝐷) = {𝑓})
15		eusn 4687	. 2 ⊢ (∃!𝑓 𝑓 ∈ (𝐶 Func 𝐷) ↔ ∃𝑓(𝐶 Func 𝐷) = {𝑓})
16	14, 15	sylibr 234	1 ⊢ (𝜑 → ∃!𝑓 𝑓 ∈ (𝐶 Func 𝐷))

Syntax hints:   → wi 4   = wceq 1541  ∃wex 1780   ∈ wcel 2113  ∃!weu 2568  Vcvv 3440  {csn 4580  ⟨cop 4586   × cxp 5622  ‘cfv 6492  (class class class)co 7358   ∈ cmpo 7360  Basecbs 17136  Hom chom 17188  Catccat 17587   Func cfunc 17778  TermCatctermc 49717

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-rep 5224  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rmo 3350  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-riota 7315  df-ov 7361  df-oprab 7362  df-mpo 7363  df-1st 7933  df-2nd 7934  df-map 8765  df-ixp 8836  df-cat 17591  df-cid 17592  df-func 17782  df-thinc 49663  df-termc 49718

This theorem is referenced by:  termcterm  49758  termc  49764