Theorem fulltermc 49116

Description: A functor to a terminal category is full iff all hom-sets of the source category are non-empty. (Contributed by Zhi Wang, 17-Oct-2025.)

Hypotheses

Ref	Expression
fulltermc.b	⊢ 𝐵 = (Base‘𝐶)
fulltermc.h	⊢ 𝐻 = (Hom ‘𝐶)
fulltermc.d	⊢ (𝜑 → 𝐷 ∈ TermCat)
fulltermc.f	⊢ (𝜑 → 𝐹(𝐶 Func 𝐷)𝐺)

Assertion

Ref	Expression
fulltermc	⊢ (𝜑 → (𝐹(𝐶 Full 𝐷)𝐺 ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ (𝑥𝐻𝑦) = ∅))

Distinct variable groups:   𝑥,𝐵,𝑦   𝑥,𝐶,𝑦   𝑥,𝐷,𝑦   𝑥,𝐹,𝑦   𝑥,𝐺,𝑦   𝑥,𝐻,𝑦   𝜑,𝑥,𝑦

Proof of Theorem fulltermc

Step	Hyp	Ref	Expression
1		fulltermc.b	. . 3 ⊢ 𝐵 = (Base‘𝐶)
2		eqid 2736	. . 3 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
3		fulltermc.h	. . 3 ⊢ 𝐻 = (Hom ‘𝐶)
4		fulltermc.d	. . . 4 ⊢ (𝜑 → 𝐷 ∈ TermCat)
5	4	termcthind 49098	. . 3 ⊢ (𝜑 → 𝐷 ∈ ThinCat)
6		fulltermc.f	. . 3 ⊢ (𝜑 → 𝐹(𝐶 Func 𝐷)𝐺)
7	1, 2, 3, 5, 6	fullthinc 49072	. 2 ⊢ (𝜑 → (𝐹(𝐶 Full 𝐷)𝐺 ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝑥𝐻𝑦) = ∅ → ((𝐹‘𝑥)(Hom ‘𝐷)(𝐹‘𝑦)) = ∅)))
8	4	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝐷 ∈ TermCat)
9		eqid 2736	. . . . . 6 ⊢ (Base‘𝐷) = (Base‘𝐷)
10	1, 9, 6	funcf1 17907	. . . . . . . 8 ⊢ (𝜑 → 𝐹:𝐵⟶(Base‘𝐷))
11	10	ffvelcdmda 7102	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝐹‘𝑥) ∈ (Base‘𝐷))
12	11	adantrr 717	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝐹‘𝑥) ∈ (Base‘𝐷))
13	10	ffvelcdmda 7102	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → (𝐹‘𝑦) ∈ (Base‘𝐷))
14	13	adantrl 716	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝐹‘𝑦) ∈ (Base‘𝐷))
15	8, 9, 12, 14, 2	termchomn0 49102	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ¬ ((𝐹‘𝑥)(Hom ‘𝐷)(𝐹‘𝑦)) = ∅)
16		biimt 360	. . . . 5 ⊢ (¬ ((𝐹‘𝑥)(Hom ‘𝐷)(𝐹‘𝑦)) = ∅ → (¬ (𝑥𝐻𝑦) = ∅ ↔ (¬ ((𝐹‘𝑥)(Hom ‘𝐷)(𝐹‘𝑦)) = ∅ → ¬ (𝑥𝐻𝑦) = ∅)))
17	15, 16	syl 17	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (¬ (𝑥𝐻𝑦) = ∅ ↔ (¬ ((𝐹‘𝑥)(Hom ‘𝐷)(𝐹‘𝑦)) = ∅ → ¬ (𝑥𝐻𝑦) = ∅)))
18		con34b 316	. . . 4 ⊢ (((𝑥𝐻𝑦) = ∅ → ((𝐹‘𝑥)(Hom ‘𝐷)(𝐹‘𝑦)) = ∅) ↔ (¬ ((𝐹‘𝑥)(Hom ‘𝐷)(𝐹‘𝑦)) = ∅ → ¬ (𝑥𝐻𝑦) = ∅))
19	17, 18	bitr4di 289	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (¬ (𝑥𝐻𝑦) = ∅ ↔ ((𝑥𝐻𝑦) = ∅ → ((𝐹‘𝑥)(Hom ‘𝐷)(𝐹‘𝑦)) = ∅)))
20	19	2ralbidva 3218	. 2 ⊢ (𝜑 → (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ (𝑥𝐻𝑦) = ∅ ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝑥𝐻𝑦) = ∅ → ((𝐹‘𝑥)(Hom ‘𝐷)(𝐹‘𝑦)) = ∅)))
21	7, 20	bitr4d 282	1 ⊢ (𝜑 → (𝐹(𝐶 Full 𝐷)𝐺 ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ (𝑥𝐻𝑦) = ∅))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2108  ∀wral 3060  ∅c0 4332   class class class wbr 5141  ‘cfv 6559  (class class class)co 7429  Basecbs 17243  Hom chom 17304   Func cfunc 17895   Full cful 17945  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-full 17947  df-thinc 49041  df-termc 49093

This theorem is referenced by:  fulltermc2  49117