Theorem functermc 49206

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 17883	. . 3 ⊢ ((𝜑 ∧ 𝐾(𝐷 Func 𝐸)𝐿) → 𝐾:𝐵⟶𝐶)
5		functermc.e	. . . . . 6 ⊢ (𝜑 → 𝐸 ∈ TermCat)
6	5, 2	termcbas 49179	. . . . 5 ⊢ (𝜑 → ∃𝑧 𝐶 = {𝑧})
7		feq3 6698	. . . . . . 7 ⊢ (𝐶 = {𝑧} → (𝐾:𝐵⟶𝐶 ↔ 𝐾:𝐵⟶{𝑧}))
8		vex 3467	. . . . . . . . 9 ⊢ 𝑧 ∈ V
9	8	fconst2 7207	. . . . . . . 8 ⊢ (𝐾:𝐵⟶{𝑧} ↔ 𝐾 = (𝐵 × {𝑧}))
10		functermc.f	. . . . . . . . . 10 ⊢ 𝐹 = (𝐵 × 𝐶)
11		xpeq2 5686	. . . . . . . . . 10 ⊢ (𝐶 = {𝑧} → (𝐵 × 𝐶) = (𝐵 × {𝑧}))
12	10, 11	eqtrid 2781	. . . . . . . . 9 ⊢ (𝐶 = {𝑧} → 𝐹 = (𝐵 × {𝑧}))
13	12	eqeq2d 2745	. . . . . . . 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 1929	. . . . 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 49177	. . 3 ⊢ (𝜑 → 𝐸 ∈ ThinCat)
24	8	fconst 6774	. . . . . 6 ⊢ (𝐵 × {𝑧}):𝐵⟶{𝑧}
25	12	feq1d 6700	. . . . . . 7 ⊢ (𝐶 = {𝑧} → (𝐹:𝐵⟶𝐶 ↔ (𝐵 × {𝑧}):𝐵⟶𝐶))
26		feq3 6698	. . . . . . 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 1929	. . . 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 7084	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐵) → (𝐹‘𝑧) ∈ 𝐶)
34	33	adantrr 717	. . . . . 6 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → (𝐹‘𝑧) ∈ 𝐶)
35	30	ffvelcdmda 7084	. . . . . . 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 49182	. . . . 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 3189	. . 3 ⊢ (𝜑 → ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐵 (((𝐹‘𝑧)𝐽(𝐹‘𝑤)) = ∅ → (𝑧𝐻𝑤) = ∅))
40	1, 2, 20, 21, 22, 23, 30, 31, 39	functhinc 49149	. 2 ⊢ (𝜑 → (𝐹(𝐷 Func 𝐸)𝐿 ↔ 𝐿 = 𝐺))
41	19, 40	functermclem 49205	1 ⊢ (𝜑 → (𝐾(𝐷 Func 𝐸)𝐿 ↔ (𝐾 = 𝐹 ∧ 𝐿 = 𝐺)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1539  ∃wex 1778   ∈ wcel 2107  ∅c0 4313  {csn 4606   class class class wbr 5123   × cxp 5663  ⟶wf 6537  ‘cfv 6541  (class class class)co 7413   ∈ cmpo 7415  Basecbs 17230  Hom chom 17285  Catccat 17679   Func cfunc 17871  TermCatctermc 49171

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 5259  ax-sep 5276  ax-nul 5286  ax-pow 5345  ax-pr 5412  ax-un 7737

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 3363  df-reu 3364  df-rab 3420  df-v 3465  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4888  df-iun 4973  df-br 5124  df-opab 5186  df-mpt 5206  df-id 5558  df-xp 5671  df-rel 5672  df-cnv 5673  df-co 5674  df-dm 5675  df-rn 5676  df-res 5677  df-ima 5678  df-iota 6494  df-fun 6543  df-fn 6544  df-f 6545  df-f1 6546  df-fo 6547  df-f1o 6548  df-fv 6549  df-riota 7370  df-ov 7416  df-oprab 7417  df-mpo 7418  df-1st 7996  df-2nd 7997  df-map 8850  df-ixp 8920  df-cat 17683  df-cid 17684  df-func 17875  df-thinc 49119  df-termc 49172

This theorem is referenced by:  functermc2  49207