Theorem functermc 50082

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 488	. . . 4 ⊢ ((𝜑 ∧ 𝐾(𝐷 Func 𝐸)𝐿) → 𝐾(𝐷 Func 𝐸)𝐿)
4	1, 2, 3	funcf1 17880	. . 3 ⊢ ((𝜑 ∧ 𝐾(𝐷 Func 𝐸)𝐿) → 𝐾:𝐵⟶𝐶)
5		functermc.e	. . . . . 6 ⊢ (𝜑 → 𝐸 ∈ TermCat)
6	5, 2	termcbas 50054	. . . . 5 ⊢ (𝜑 → ∃𝑧 𝐶 = {𝑧})
7		feq3 6665	. . . . . . 7 ⊢ (𝐶 = {𝑧} → (𝐾:𝐵⟶𝐶 ↔ 𝐾:𝐵⟶{𝑧}))
8		vex 3457	. . . . . . . . 9 ⊢ 𝑧 ∈ V
9	8	fconst2 7183	. . . . . . . 8 ⊢ (𝐾:𝐵⟶{𝑧} ↔ 𝐾 = (𝐵 × {𝑧}))
10		functermc.f	. . . . . . . . . 10 ⊢ 𝐹 = (𝐵 × 𝐶)
11		xpeq2 5666	. . . . . . . . . 10 ⊢ (𝐶 = {𝑧} → (𝐵 × 𝐶) = (𝐵 × {𝑧}))
12	10, 11	eqtrid 2808	. . . . . . . . 9 ⊢ (𝐶 = {𝑧} → 𝐹 = (𝐵 × {𝑧}))
13	12	eqeq2d 2772	. . . . . . . 8 ⊢ (𝐶 = {𝑧} → (𝐾 = 𝐹 ↔ 𝐾 = (𝐵 × {𝑧})))
14	9, 13	bitr4id 292	. . . . . . 7 ⊢ (𝐶 = {𝑧} → (𝐾:𝐵⟶{𝑧} ↔ 𝐾 = 𝐹))
15	7, 14	bitrd 281	. . . . . 6 ⊢ (𝐶 = {𝑧} → (𝐾:𝐵⟶𝐶 ↔ 𝐾 = 𝐹))
16	15	exlimiv 1949	. . . . 5 ⊢ (∃𝑧 𝐶 = {𝑧} → (𝐾:𝐵⟶𝐶 ↔ 𝐾 = 𝐹))
17	6, 16	syl 17	. . . 4 ⊢ (𝜑 → (𝐾:𝐵⟶𝐶 ↔ 𝐾 = 𝐹))
18	17	biimpa 480	. . 3 ⊢ ((𝜑 ∧ 𝐾:𝐵⟶𝐶) → 𝐾 = 𝐹)
19	4, 18	syldan 600	. 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 50052	. . 3 ⊢ (𝜑 → 𝐸 ∈ ThinCat)
24	8	fconst 6744	. . . . . 6 ⊢ (𝐵 × {𝑧}):𝐵⟶{𝑧}
25	12	feq1d 6667	. . . . . . 7 ⊢ (𝐶 = {𝑧} → (𝐹:𝐵⟶𝐶 ↔ (𝐵 × {𝑧}):𝐵⟶𝐶))
26		feq3 6665	. . . . . . 7 ⊢ (𝐶 = {𝑧} → ((𝐵 × {𝑧}):𝐵⟶𝐶 ↔ (𝐵 × {𝑧}):𝐵⟶{𝑧}))
27	25, 26	bitrd 281	. . . . . 6 ⊢ (𝐶 = {𝑧} → (𝐹:𝐵⟶𝐶 ↔ (𝐵 × {𝑧}):𝐵⟶{𝑧}))
28	24, 27	mpbiri 260	. . . . 5 ⊢ (𝐶 = {𝑧} → 𝐹:𝐵⟶𝐶)
29	28	exlimiv 1949	. . . 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 484	. . . . . 6 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → 𝐸 ∈ TermCat)
33	30	ffvelcdmda 7059	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐵) → (𝐹‘𝑧) ∈ 𝐶)
34	33	adantrr 727	. . . . . 6 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → (𝐹‘𝑧) ∈ 𝐶)
35	30	ffvelcdmda 7059	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐵) → (𝐹‘𝑤) ∈ 𝐶)
36	35	adantrl 726	. . . . . 6 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → (𝐹‘𝑤) ∈ 𝐶)
37	32, 2, 34, 36, 21	termchomn0 50058	. . . . 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 3204	. . 3 ⊢ (𝜑 → ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐵 (((𝐹‘𝑧)𝐽(𝐹‘𝑤)) = ∅ → (𝑧𝐻𝑤) = ∅))
40	1, 2, 20, 21, 22, 23, 30, 31, 39	functhinc 50022	. 2 ⊢ (𝜑 → (𝐹(𝐷 Func 𝐸)𝐿 ↔ 𝐿 = 𝐺))
41	19, 40	functermclem 50081	1 ⊢ (𝜑 → (𝐾(𝐷 Func 𝐸)𝐿 ↔ (𝐾 = 𝐹 ∧ 𝐿 = 𝐺)))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399   = wceq 1559  ∃wex 1798   ∈ wcel 2141  ∅c0 4285  {csn 4581   class class class wbr 5099   × cxp 5643  ⟶wf 6511  ‘cfv 6515  (class class class)co 7390   ∈ cmpo 7392  Basecbs 17226  Hom chom 17278  Catccat 17677   Func cfunc 17868  TermCatctermc 50046

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5226  ax-sep 5245  ax-nul 5255  ax-pow 5321  ax-pr 5389  ax-un 7712

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3455  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-iun 4950  df-br 5100  df-opab 5162  df-mpt 5181  df-id 5540  df-xp 5651  df-rel 5652  df-cnv 5653  df-co 5654  df-dm 5655  df-rn 5656  df-res 5657  df-ima 5658  df-iota 6471  df-fun 6517  df-fn 6518  df-f 6519  df-f1 6520  df-fo 6521  df-f1o 6522  df-fv 6523  df-riota 7347  df-ov 7393  df-oprab 7394  df-mpo 7395  df-1st 7964  df-2nd 7965  df-map 8803  df-ixp 8874  df-cat 17681  df-cid 17682  df-func 17872  df-thinc 49992  df-termc 50047

This theorem is referenced by:  functermc2  50083