Theorem functermceu 49115

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 5467	. . . 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 49114	. . 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 4634	. . . 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 3597	. 2 ⊢ (𝜑 → ∃𝑓(𝐶 Func 𝐷) = {𝑓})
15		eusn 4728	. 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 1540  ∃wex 1779   ∈ wcel 2108  ∃!weu 2567  Vcvv 3479  {csn 4624  ⟨cop 4630   × cxp 5681  ‘cfv 6559  (class class class)co 7429   ∈ cmpo 7431  Basecbs 17243  Hom chom 17304  Catccat 17703   Func cfunc 17895  TermCatctermc 49092

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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-rep 5277  ax-sep 5294  ax-nul 5304  ax-pow 5363  ax-pr 5430  ax-un 7751

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rmo 3379  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4906  df-iun 4991  df-br 5142  df-opab 5204  df-mpt 5224  df-id 5576  df-xp 5689  df-rel 5690  df-cnv 5691  df-co 5692  df-dm 5693  df-rn 5694  df-res 5695  df-ima 5696  df-iota 6512  df-fun 6561  df-fn 6562  df-f 6563  df-f1 6564  df-fo 6565  df-f1o 6566  df-fv 6567  df-riota 7386  df-ov 7432  df-oprab 7433  df-mpo 7434  df-1st 8010  df-2nd 8011  df-map 8864  df-ixp 8934  df-cat 17707  df-cid 17708  df-func 17899  df-thinc 49041  df-termc 49093

This theorem is referenced by:  termcterm  49118  termc  49122