Theorem functermc2 49179

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 17860	. 2 ⊢ Rel (𝐷 Func 𝐸)
2		functermc.f	. . . 4 ⊢ 𝐹 = (𝐵 × 𝐶)
3		functermc.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝐷)
4	3	fvexi 6886	. . . . 5 ⊢ 𝐵 ∈ V
5		functermc.c	. . . . . 6 ⊢ 𝐶 = (Base‘𝐸)
6	5	fvexi 6886	. . . . 5 ⊢ 𝐶 ∈ V
7	4, 6	xpex 7741	. . . 4 ⊢ (𝐵 × 𝐶) ∈ V
8	2, 7	eqeltri 2829	. . 3 ⊢ 𝐹 ∈ V
9		functermc.g	. . . 4 ⊢ 𝐺 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ((𝑥𝐻𝑦) × ((𝐹‘𝑥)𝐽(𝐹‘𝑦))))
10	4, 4	mpoex 8072	. . . 4 ⊢ (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ((𝑥𝐻𝑦) × ((𝐹‘𝑥)𝐽(𝐹‘𝑦)))) ∈ V
11	9, 10	eqeltri 2829	. . 3 ⊢ 𝐺 ∈ V
12	8, 11	relsnop 5781	. 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 49178	. . 3 ⊢ (𝜑 → (𝑧(𝐷 Func 𝐸)𝑤 ↔ (𝑧 = 𝐹 ∧ 𝑤 = 𝐺)))
18		brsnop 5494	. . . 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 5770	1 ⊢ (𝜑 → (𝐷 Func 𝐸) = {⟨𝐹, 𝐺⟩})

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1539   ∈ wcel 2107  Vcvv 3457  {csn 4599  ⟨cop 4605   class class class wbr 5116   × cxp 5649  ‘cfv 6527  (class class class)co 7399   ∈ cmpo 7401  Basecbs 17213  Hom chom 17267  Catccat 17661   Func cfunc 17852  TermCatctermc 49143

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2706  ax-rep 5246  ax-sep 5263  ax-nul 5273  ax-pow 5332  ax-pr 5399  ax-un 7723

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2808  df-nfc 2884  df-ne 2932  df-ral 3051  df-rex 3060  df-rmo 3357  df-reu 3358  df-rab 3414  df-v 3459  df-sbc 3764  df-csb 3873  df-dif 3927  df-un 3929  df-in 3931  df-ss 3941  df-nul 4307  df-if 4499  df-pw 4575  df-sn 4600  df-pr 4602  df-op 4606  df-uni 4881  df-iun 4966  df-br 5117  df-opab 5179  df-mpt 5199  df-id 5545  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-iota 6480  df-fun 6529  df-fn 6530  df-f 6531  df-f1 6532  df-fo 6533  df-f1o 6534  df-fv 6535  df-riota 7356  df-ov 7402  df-oprab 7403  df-mpo 7404  df-1st 7982  df-2nd 7983  df-map 8836  df-ixp 8906  df-cat 17665  df-cid 17666  df-func 17856  df-thinc 49091  df-termc 49144

This theorem is referenced by:  functermceu  49180