Theorem functermc2 50586

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 18030	. 2 ⊢ Rel (𝐷 Func 𝐸)
2		functermc.f	. . . 4 ⊢ 𝐹 = (𝐵 × 𝐶)
3		functermc.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝐷)
4	3	fvexi 6897	. . . . 5 ⊢ 𝐵 ∈ V
5		functermc.c	. . . . . 6 ⊢ 𝐶 = (Base‘𝐸)
6	5	fvexi 6897	. . . . 5 ⊢ 𝐶 ∈ V
7	4, 6	xpex 7765	. . . 4 ⊢ (𝐵 × 𝐶) ∈ V
8	2, 7	eqeltri 2857	. . 3 ⊢ 𝐹 ∈ V
9		functermc.g	. . . 4 ⊢ 𝐺 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ((𝑥𝐻𝑦) × ((𝐹‘𝑥)𝐽(𝐹‘𝑦))))
10	4, 4	mpoex 8090	. . . 4 ⊢ (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ((𝑥𝐻𝑦) × ((𝐹‘𝑥)𝐽(𝐹‘𝑦)))) ∈ V
11	9, 10	eqeltri 2857	. . 3 ⊢ 𝐺 ∈ V
12	8, 11	relsnop 5783	. 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 50585	. . 3 ⊢ (𝜑 → (𝑧(𝐷 Func 𝐸)𝑤 ↔ (𝑧 = 𝐹 ∧ 𝑤 = 𝐺)))
18		brsnop 5496	. . . 4 ⊢ ((𝐹 ∈ V ∧ 𝐺 ∈ V) → (𝑧{⟨𝐹, 𝐺⟩}𝑤 ↔ (𝑧 = 𝐹 ∧ 𝑤 = 𝐺)))
19	8, 11, 18	mp2an 705	. . 3 ⊢ (𝑧{⟨𝐹, 𝐺⟩}𝑤 ↔ (𝑧 = 𝐹 ∧ 𝑤 = 𝐺))
20	17, 19	bitr4di 292	. 2 ⊢ (𝜑 → (𝑧(𝐷 Func 𝐸)𝑤 ↔ 𝑧{⟨𝐹, 𝐺⟩}𝑤))
21	1, 12, 20	eqbrrdiv 5770	1 ⊢ (𝜑 → (𝐷 Func 𝐸) = {⟨𝐹, 𝐺⟩})

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145  Vcvv 3451  {csn 4584  ⟨cop 4590   class class class wbr 5103   × cxp 5649  ‘cfv 6537  (class class class)co 7418   ∈ cmpo 7420  Basecbs 17380  Hom chom 17432  Catccat 17831   Func cfunc 18022  TermCatctermc 50549

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7749

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5546  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 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7375  df-ov 7421  df-oprab 7422  df-mpo 7423  df-1st 7999  df-2nd 8000  df-map 8842  df-ixp 8919  df-cat 17835  df-cid 17836  df-func 18026  df-thinc 50495  df-termc 50550

This theorem is used by:  functermceu  50587  fucterm  50619