Theorem functermceu 50168

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 5443	. . . 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 2769	. . . 4 ⊢ (Base‘𝐶) = (Base‘𝐶)
6		eqid 2769	. . . 4 ⊢ (Base‘𝐷) = (Base‘𝐷)
7		eqid 2769	. . . 4 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
8		eqid 2769	. . . 4 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
9		eqid 2769	. . . 4 ⊢ ((Base‘𝐶) × (Base‘𝐷)) = ((Base‘𝐶) × (Base‘𝐷))
10		eqid 2769	. . . 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 50167	. . 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 4601	. . . 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 2780	. . 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 3566	. 2 ⊢ (𝜑 → ∃𝑓(𝐶 Func 𝐷) = {𝑓})
15		eusn 4698	. 2 ⊢ (∃!𝑓 𝑓 ∈ (𝐶 Func 𝐷) ↔ ∃𝑓(𝐶 Func 𝐷) = {𝑓})
16	14, 15	sylibr 237	1 ⊢ (𝜑 → ∃!𝑓 𝑓 ∈ (𝐶 Func 𝐷))

Syntax hints:   → wi 4   = wceq 1567  ∃wex 1806   ∈ wcel 2149  ∃!weu 2602  Vcvv 3463  {csn 4591  ⟨cop 4597   × cxp 5657  ‘cfv 6534  (class class class)co 7408   ∈ cmpo 7410  Basecbs 17265  Hom chom 17317  Catccat 17716   Func cfunc 17907  TermCatctermc 50130

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5239  ax-sep 5258  ax-nul 5268  ax-pow 5334  ax-pr 5402  ax-un 7730

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-rmo 3376  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-iun 4959  df-br 5111  df-opab 5175  df-mpt 5194  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-iota 6490  df-fun 6536  df-fn 6537  df-f 6538  df-f1 6539  df-fo 6540  df-f1o 6541  df-fv 6542  df-riota 7365  df-ov 7411  df-oprab 7412  df-mpo 7413  df-1st 7982  df-2nd 7983  df-map 8822  df-ixp 8892  df-cat 17720  df-cid 17721  df-func 17911  df-thinc 50076  df-termc 50131

This theorem is referenced by:  termcterm  50171  termc  50177