Theorem functermc2 49514

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 17788	. 2 ⊢ Rel (𝐷 Func 𝐸)
2		functermc.f	. . . 4 ⊢ 𝐹 = (𝐵 × 𝐶)
3		functermc.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝐷)
4	3	fvexi 6840	. . . . 5 ⊢ 𝐵 ∈ V
5		functermc.c	. . . . . 6 ⊢ 𝐶 = (Base‘𝐸)
6	5	fvexi 6840	. . . . 5 ⊢ 𝐶 ∈ V
7	4, 6	xpex 7693	. . . 4 ⊢ (𝐵 × 𝐶) ∈ V
8	2, 7	eqeltri 2824	. . 3 ⊢ 𝐹 ∈ V
9		functermc.g	. . . 4 ⊢ 𝐺 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ((𝑥𝐻𝑦) × ((𝐹‘𝑥)𝐽(𝐹‘𝑦))))
10	4, 4	mpoex 8021	. . . 4 ⊢ (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ((𝑥𝐻𝑦) × ((𝐹‘𝑥)𝐽(𝐹‘𝑦)))) ∈ V
11	9, 10	eqeltri 2824	. . 3 ⊢ 𝐺 ∈ V
12	8, 11	relsnop 5752	. 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 49513	. . 3 ⊢ (𝜑 → (𝑧(𝐷 Func 𝐸)𝑤 ↔ (𝑧 = 𝐹 ∧ 𝑤 = 𝐺)))
18		brsnop 5469	. . . 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 5741	1 ⊢ (𝜑 → (𝐷 Func 𝐸) = {⟨𝐹, 𝐺⟩})

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  Vcvv 3438  {csn 4579  ⟨cop 4585   class class class wbr 5095   × cxp 5621  ‘cfv 6486  (class class class)co 7353   ∈ cmpo 7355  Basecbs 17139  Hom chom 17191  Catccat 17589   Func cfunc 17780  TermCatctermc 49477

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 2701  ax-rep 5221  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7675

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 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rmo 3345  df-reu 3346  df-rab 3397  df-v 3440  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4862  df-iun 4946  df-br 5096  df-opab 5158  df-mpt 5177  df-id 5518  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-riota 7310  df-ov 7356  df-oprab 7357  df-mpo 7358  df-1st 7931  df-2nd 7932  df-map 8762  df-ixp 8832  df-cat 17593  df-cid 17594  df-func 17784  df-thinc 49423  df-termc 49478

This theorem is referenced by:  functermceu  49515  fucterm  49547