Theorem functermc2 49478

Description: Functor to a terminal category. (Contributed by Zhi Wang, 17-Oct-2025.)

Hypotheses

Ref	Expression
functermc.d	⊢ (𝜑 → 𝐷 ∈ Cat)
functermc.e	⊢ (𝜑 → 𝐸 ∈ TermCat)
functermc.b	⊢ 𝐵 = (Base‘𝐷)
functermc.c	⊢ 𝐶 = (Base‘𝐸)
functermc.h	⊢ 𝐻 = (Hom ‘𝐷)
functermc.j	⊢ 𝐽 = (Hom ‘𝐸)
functermc.f	⊢ 𝐹 = (𝐵 × 𝐶)
functermc.g	⊢ 𝐺 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ((𝑥𝐻𝑦) × ((𝐹‘𝑥)𝐽(𝐹‘𝑦))))

Assertion

Ref	Expression
functermc2	⊢ (𝜑 → (𝐷 Func 𝐸) = {⟨𝐹, 𝐺⟩})

Distinct variable groups:   𝑥,𝐵,𝑦   𝑥,𝐹,𝑦   𝑥,𝐻,𝑦   𝑥,𝐽,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐶(𝑥,𝑦)   𝐷(𝑥,𝑦)   𝐸(𝑥,𝑦)   𝐺(𝑥,𝑦)

Proof of Theorem functermc2

Dummy variables 𝑤 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		relfunc 17830	. 2 ⊢ Rel (𝐷 Func 𝐸)
2		functermc.f	. . . 4 ⊢ 𝐹 = (𝐵 × 𝐶)
3		functermc.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝐷)
4	3	fvexi 6874	. . . . 5 ⊢ 𝐵 ∈ V
5		functermc.c	. . . . . 6 ⊢ 𝐶 = (Base‘𝐸)
6	5	fvexi 6874	. . . . 5 ⊢ 𝐶 ∈ V
7	4, 6	xpex 7731	. . . 4 ⊢ (𝐵 × 𝐶) ∈ V
8	2, 7	eqeltri 2825	. . 3 ⊢ 𝐹 ∈ V
9		functermc.g	. . . 4 ⊢ 𝐺 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ((𝑥𝐻𝑦) × ((𝐹‘𝑥)𝐽(𝐹‘𝑦))))
10	4, 4	mpoex 8060	. . . 4 ⊢ (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ((𝑥𝐻𝑦) × ((𝐹‘𝑥)𝐽(𝐹‘𝑦)))) ∈ V
11	9, 10	eqeltri 2825	. . 3 ⊢ 𝐺 ∈ V
12	8, 11	relsnop 5770	. 2 ⊢ Rel {⟨𝐹, 𝐺⟩}
13		functermc.d	. . . 4 ⊢ (𝜑 → 𝐷 ∈ Cat)
14		functermc.e	. . . 4 ⊢ (𝜑 → 𝐸 ∈ TermCat)
15		functermc.h	. . . 4 ⊢ 𝐻 = (Hom ‘𝐷)
16		functermc.j	. . . 4 ⊢ 𝐽 = (Hom ‘𝐸)
17	13, 14, 3, 5, 15, 16, 2, 9	functermc 49477	. . 3 ⊢ (𝜑 → (𝑧(𝐷 Func 𝐸)𝑤 ↔ (𝑧 = 𝐹 ∧ 𝑤 = 𝐺)))
18		brsnop 5484	. . . 4 ⊢ ((𝐹 ∈ V ∧ 𝐺 ∈ V) → (𝑧{⟨𝐹, 𝐺⟩}𝑤 ↔ (𝑧 = 𝐹 ∧ 𝑤 = 𝐺)))
19	8, 11, 18	mp2an 692	. . 3 ⊢ (𝑧{⟨𝐹, 𝐺⟩}𝑤 ↔ (𝑧 = 𝐹 ∧ 𝑤 = 𝐺))
20	17, 19	bitr4di 289	. 2 ⊢ (𝜑 → (𝑧(𝐷 Func 𝐸)𝑤 ↔ 𝑧{⟨𝐹, 𝐺⟩}𝑤))
21	1, 12, 20	eqbrrdiv 5759	1 ⊢ (𝜑 → (𝐷 Func 𝐸) = {⟨𝐹, 𝐺⟩})

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  Vcvv 3450  {csn 4591  ⟨cop 4597   class class class wbr 5109   × 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:  functermceu  49479  fucterm  49511