Theorem functermc 50751

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
functermc	⊢ (𝜑 → (𝐾(𝐷 Func 𝐸)𝐿 ↔ (𝐾 = 𝐹 ∧ 𝐿 = 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)   𝐾(𝑥, 𝑦)   𝐿(𝑥, 𝑦)

Proof of Theorem functermc

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

Step	Hyp	Ref	Expression
1		functermc.b	. . . 4 ⊢ 𝐵 = (Base‘𝐷)
2		functermc.c	. . . 4 ⊢ 𝐶 = (Base‘𝐸)
3		simpr 490	. . . 4 ⊢ ((𝜑 ∧ 𝐾(𝐷 Func 𝐸)𝐿) → 𝐾(𝐷 Func 𝐸)𝐿)
4	1, 2, 3	funcf1 18050	. . 3 ⊢ ((𝜑 ∧ 𝐾(𝐷 Func 𝐸)𝐿) → 𝐾:𝐵⟶𝐶)
5		functermc.e	. . . . . 6 ⊢ (𝜑 → 𝐸 ∈ TermCat)
6	5, 2	termcbas 50723	. . . . 5 ⊢ (𝜑 → ∃𝑧 𝐶 = {𝑧})
7		feq3 6694	. . . . . . 7 ⊢ (𝐶 = {𝑧} → (𝐾:𝐵⟶𝐶 ↔ 𝐾:𝐵⟶{𝑧}))
8		vex 3454	. . . . . . . . 9 ⊢ 𝑧 ∈ V
9	8	fconst2 7215	. . . . . . . 8 ⊢ (𝐾:𝐵⟶{𝑧} ↔ 𝐾 = (𝐵 × {𝑧}))
10		functermc.f	. . . . . . . . . 10 ⊢ 𝐹 = (𝐵 × 𝐶)
11		xpeq2 5672	. . . . . . . . . 10 ⊢ (𝐶 = {𝑧} → (𝐵 × 𝐶) = (𝐵 × {𝑧}))
12	10, 11	eqtrid 2807	. . . . . . . . 9 ⊢ (𝐶 = {𝑧} → 𝐹 = (𝐵 × {𝑧}))
13	12	eqeq2d 2771	. . . . . . . 8 ⊢ (𝐶 = {𝑧} → (𝐾 = 𝐹 ↔ 𝐾 = (𝐵 × {𝑧})))
14	9, 13	bitr4id 293	. . . . . . 7 ⊢ (𝐶 = {𝑧} → (𝐾:𝐵⟶{𝑧} ↔ 𝐾 = 𝐹))
15	7, 14	bitrd 282	. . . . . 6 ⊢ (𝐶 = {𝑧} → (𝐾:𝐵⟶𝐶 ↔ 𝐾 = 𝐹))
16	15	exlimiv 1962	. . . . 5 ⊢ (∃𝑧 𝐶 = {𝑧} → (𝐾:𝐵⟶𝐶 ↔ 𝐾 = 𝐹))
17	6, 16	syl 18	. . . 4 ⊢ (𝜑 → (𝐾:𝐵⟶𝐶 ↔ 𝐾 = 𝐹))
18	17	biimpa 482	. . 3 ⊢ ((𝜑 ∧ 𝐾:𝐵⟶𝐶) → 𝐾 = 𝐹)
19	4, 18	syldan 603	. 2 ⊢ ((𝜑 ∧ 𝐾(𝐷 Func 𝐸)𝐿) → 𝐾 = 𝐹)
20		functermc.h	. . 3 ⊢ 𝐻 = (Hom ‘𝐷)
21		functermc.j	. . 3 ⊢ 𝐽 = (Hom ‘𝐸)
22		functermc.d	. . 3 ⊢ (𝜑 → 𝐷 ∈ Cat)
23	5	termcthind 50721	. . 3 ⊢ (𝜑 → 𝐸 ∈ ThinCat)
24	8	fconst 6773	. . . . . 6 ⊢ (𝐵 × {𝑧}):𝐵⟶{𝑧}
25	12	feq1d 6696	. . . . . . 7 ⊢ (𝐶 = {𝑧} → (𝐹:𝐵⟶𝐶 ↔ (𝐵 × {𝑧}):𝐵⟶𝐶))
26		feq3 6694	. . . . . . 7 ⊢ (𝐶 = {𝑧} → ((𝐵 × {𝑧}):𝐵⟶𝐶 ↔ (𝐵 × {𝑧}):𝐵⟶{𝑧}))
27	25, 26	bitrd 282	. . . . . 6 ⊢ (𝐶 = {𝑧} → (𝐹:𝐵⟶𝐶 ↔ (𝐵 × {𝑧}):𝐵⟶{𝑧}))
28	24, 27	mpbiri 261	. . . . 5 ⊢ (𝐶 = {𝑧} → 𝐹:𝐵⟶𝐶)
29	28	exlimiv 1962	. . . 4 ⊢ (∃𝑧 𝐶 = {𝑧} → 𝐹:𝐵⟶𝐶)
30	6, 29	syl 18	. . 3 ⊢ (𝜑 → 𝐹:𝐵⟶𝐶)
31		functermc.g	. . 3 ⊢ 𝐺 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ((𝑥𝐻𝑦) × ((𝐹‘𝑥)𝐽(𝐹‘𝑦))))
32	5	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → 𝐸 ∈ TermCat)
33	30	ffvelcdmda 7089	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐵) → (𝐹‘𝑧) ∈ 𝐶)
34	33	adantrr 730	. . . . . 6 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → (𝐹‘𝑧) ∈ 𝐶)
35	30	ffvelcdmda 7089	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐵) → (𝐹‘𝑤) ∈ 𝐶)
36	35	adantrl 729	. . . . . 6 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → (𝐹‘𝑤) ∈ 𝐶)
37	32, 2, 34, 36, 21	termchomn0 50727	. . . . 5 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → ¬ ((𝐹‘𝑧)𝐽(𝐹‘𝑤)) = ∅)
38	37	pm2.21d 122	. . . 4 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → (((𝐹‘𝑧)𝐽(𝐹‘𝑤)) = ∅ → (𝑧𝐻𝑤) = ∅))
39	38	ralrimivva 3205	. . 3 ⊢ (𝜑 → ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐵 (((𝐹‘𝑧)𝐽(𝐹‘𝑤)) = ∅ → (𝑧𝐻𝑤) = ∅))
40	1, 2, 20, 21, 22, 23, 30, 31, 39	functhinc 50691	. 2 ⊢ (𝜑 → (𝐹(𝐷 Func 𝐸)𝐿 ↔ 𝐿 = 𝐺))
41	19, 40	functermclem 50750	1 ⊢ (𝜑 → (𝐾(𝐷 Func 𝐸)𝐿 ↔ (𝐾 = 𝐹 ∧ 𝐿 = 𝐺)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1569  ∃wex 1811   ∈ wcel 2144  ∅c0 4277  {csn 4584   class class class wbr 5102   × cxp 5649  ⟶wf 6539  ‘cfv 6543  (class class class)co 7424   ∈ cmpo 7426  Basecbs 17396  Hom chom 17448  Catccat 17847   Func cfunc 18038  TermCatctermc 50715

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5327  ax-pr 5391  ax-un 7755

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3848  df-ss 3897  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  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 6497  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7381  df-ov 7427  df-oprab 7428  df-mpo 7429  df-1st 8005  df-2nd 8006  df-map 8856  df-ixp 8933  df-cat 17851  df-cid 17852  df-func 18042  df-thinc 50661  df-termc 50716

This theorem is used by:  functermc2  50752