Theorem functermceu 49479

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 5426	. . . 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 2730	. . . 4 ⊢ (Base‘𝐶) = (Base‘𝐶)
6		eqid 2730	. . . 4 ⊢ (Base‘𝐷) = (Base‘𝐷)
7		eqid 2730	. . . 4 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
8		eqid 2730	. . . 4 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
9		eqid 2730	. . . 4 ⊢ ((Base‘𝐶) × (Base‘𝐷)) = ((Base‘𝐶) × (Base‘𝐷))
10		eqid 2730	. . . 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 49478	. . 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 2741	. . 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 3567	. 2 ⊢ (𝜑 → ∃𝑓(𝐶 Func 𝐷) = {𝑓})
15		eusn 4696	. 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 2109  ∃!weu 2562  Vcvv 3450  {csn 4591  ⟨cop 4597   × cxp 5638  ‘cfv 6513  (class class class)co 7389   ∈ cmpo 7391  Basecbs 17185  Hom chom 17237  Catccat 17631   Func cfunc 17822  TermCatctermc 49441

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 2702  ax-rep 5236  ax-sep 5253  ax-nul 5263  ax-pow 5322  ax-pr 5389  ax-un 7713

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rmo 3356  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3756  df-csb 3865  df-dif 3919  df-un 3921  df-in 3923  df-ss 3933  df-nul 4299  df-if 4491  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-iun 4959  df-br 5110  df-opab 5172  df-mpt 5191  df-id 5535  df-xp 5646  df-rel 5647  df-cnv 5648  df-co 5649  df-dm 5650  df-rn 5651  df-res 5652  df-ima 5653  df-iota 6466  df-fun 6515  df-fn 6516  df-f 6517  df-f1 6518  df-fo 6519  df-f1o 6520  df-fv 6521  df-riota 7346  df-ov 7392  df-oprab 7393  df-mpo 7394  df-1st 7970  df-2nd 7971  df-map 8803  df-ixp 8873  df-cat 17635  df-cid 17636  df-func 17826  df-thinc 49387  df-termc 49442

This theorem is referenced by:  termcterm  49482  termc  49488