Theorem functermc 49113

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 484	. . . 4 ⊢ ((𝜑 ∧ 𝐾(𝐷 Func 𝐸)𝐿) → 𝐾(𝐷 Func 𝐸)𝐿)
4	1, 2, 3	funcf1 17907	. . 3 ⊢ ((𝜑 ∧ 𝐾(𝐷 Func 𝐸)𝐿) → 𝐾:𝐵⟶𝐶)
5		functermc.e	. . . . . 6 ⊢ (𝜑 → 𝐸 ∈ TermCat)
6	5, 2	termcbas 49100	. . . . 5 ⊢ (𝜑 → ∃𝑧 𝐶 = {𝑧})
7		feq3 6716	. . . . . . 7 ⊢ (𝐶 = {𝑧} → (𝐾:𝐵⟶𝐶 ↔ 𝐾:𝐵⟶{𝑧}))
8		vex 3483	. . . . . . . . 9 ⊢ 𝑧 ∈ V
9	8	fconst2 7223	. . . . . . . 8 ⊢ (𝐾:𝐵⟶{𝑧} ↔ 𝐾 = (𝐵 × {𝑧}))
10		functermc.f	. . . . . . . . . 10 ⊢ 𝐹 = (𝐵 × 𝐶)
11		xpeq2 5704	. . . . . . . . . 10 ⊢ (𝐶 = {𝑧} → (𝐵 × 𝐶) = (𝐵 × {𝑧}))
12	10, 11	eqtrid 2788	. . . . . . . . 9 ⊢ (𝐶 = {𝑧} → 𝐹 = (𝐵 × {𝑧}))
13	12	eqeq2d 2747	. . . . . . . 8 ⊢ (𝐶 = {𝑧} → (𝐾 = 𝐹 ↔ 𝐾 = (𝐵 × {𝑧})))
14	9, 13	bitr4id 290	. . . . . . 7 ⊢ (𝐶 = {𝑧} → (𝐾:𝐵⟶{𝑧} ↔ 𝐾 = 𝐹))
15	7, 14	bitrd 279	. . . . . 6 ⊢ (𝐶 = {𝑧} → (𝐾:𝐵⟶𝐶 ↔ 𝐾 = 𝐹))
16	15	exlimiv 1930	. . . . 5 ⊢ (∃𝑧 𝐶 = {𝑧} → (𝐾:𝐵⟶𝐶 ↔ 𝐾 = 𝐹))
17	6, 16	syl 17	. . . 4 ⊢ (𝜑 → (𝐾:𝐵⟶𝐶 ↔ 𝐾 = 𝐹))
18	17	biimpa 476	. . 3 ⊢ ((𝜑 ∧ 𝐾:𝐵⟶𝐶) → 𝐾 = 𝐹)
19	4, 18	syldan 591	. 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 49098	. . 3 ⊢ (𝜑 → 𝐸 ∈ ThinCat)
24	8	fconst 6792	. . . . . 6 ⊢ (𝐵 × {𝑧}):𝐵⟶{𝑧}
25	12	feq1d 6718	. . . . . . 7 ⊢ (𝐶 = {𝑧} → (𝐹:𝐵⟶𝐶 ↔ (𝐵 × {𝑧}):𝐵⟶𝐶))
26		feq3 6716	. . . . . . 7 ⊢ (𝐶 = {𝑧} → ((𝐵 × {𝑧}):𝐵⟶𝐶 ↔ (𝐵 × {𝑧}):𝐵⟶{𝑧}))
27	25, 26	bitrd 279	. . . . . 6 ⊢ (𝐶 = {𝑧} → (𝐹:𝐵⟶𝐶 ↔ (𝐵 × {𝑧}):𝐵⟶{𝑧}))
28	24, 27	mpbiri 258	. . . . 5 ⊢ (𝐶 = {𝑧} → 𝐹:𝐵⟶𝐶)
29	28	exlimiv 1930	. . . 4 ⊢ (∃𝑧 𝐶 = {𝑧} → 𝐹:𝐵⟶𝐶)
30	6, 29	syl 17	. . 3 ⊢ (𝜑 → 𝐹:𝐵⟶𝐶)
31		functermc.g	. . 3 ⊢ 𝐺 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ((𝑥𝐻𝑦) × ((𝐹‘𝑥)𝐽(𝐹‘𝑦))))
32	5	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → 𝐸 ∈ TermCat)
33	30	ffvelcdmda 7102	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐵) → (𝐹‘𝑧) ∈ 𝐶)
34	33	adantrr 717	. . . . . 6 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → (𝐹‘𝑧) ∈ 𝐶)
35	30	ffvelcdmda 7102	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐵) → (𝐹‘𝑤) ∈ 𝐶)
36	35	adantrl 716	. . . . . 6 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → (𝐹‘𝑤) ∈ 𝐶)
37	32, 2, 34, 36, 21	termchomn0 49102	. . . . 5 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → ¬ ((𝐹‘𝑧)𝐽(𝐹‘𝑤)) = ∅)
38	37	pm2.21d 121	. . . 4 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → (((𝐹‘𝑧)𝐽(𝐹‘𝑤)) = ∅ → (𝑧𝐻𝑤) = ∅))
39	38	ralrimivva 3201	. . 3 ⊢ (𝜑 → ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐵 (((𝐹‘𝑧)𝐽(𝐹‘𝑤)) = ∅ → (𝑧𝐻𝑤) = ∅))
40	1, 2, 20, 21, 22, 23, 30, 31, 39	functhinc 49070	. 2 ⊢ (𝜑 → (𝐹(𝐷 Func 𝐸)𝐿 ↔ 𝐿 = 𝐺))
41	19, 40	functermclem 49112	1 ⊢ (𝜑 → (𝐾(𝐷 Func 𝐸)𝐿 ↔ (𝐾 = 𝐹 ∧ 𝐿 = 𝐺)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540  ∃wex 1779   ∈ wcel 2108  ∅c0 4332  {csn 4624   class class class wbr 5141   × cxp 5681  ⟶wf 6555  ‘cfv 6559  (class class class)co 7429   ∈ cmpo 7431  Basecbs 17243  Hom chom 17304  Catccat 17703   Func cfunc 17895  TermCatctermc 49092

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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-rep 5277  ax-sep 5294  ax-nul 5304  ax-pow 5363  ax-pr 5430  ax-un 7751

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rmo 3379  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4906  df-iun 4991  df-br 5142  df-opab 5204  df-mpt 5224  df-id 5576  df-xp 5689  df-rel 5690  df-cnv 5691  df-co 5692  df-dm 5693  df-rn 5694  df-res 5695  df-ima 5696  df-iota 6512  df-fun 6561  df-fn 6562  df-f 6563  df-f1 6564  df-fo 6565  df-f1o 6566  df-fv 6567  df-riota 7386  df-ov 7432  df-oprab 7433  df-mpo 7434  df-1st 8010  df-2nd 8011  df-map 8864  df-ixp 8934  df-cat 17707  df-cid 17708  df-func 17899  df-thinc 49041  df-termc 49093

This theorem is referenced by:  functermc2  49114