Theorem fulltermc 50437

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 2760	. . 3 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
3		fulltermc.h	. . 3 ⊢ 𝐻 = (Hom ‘𝐶)
4		fulltermc.d	. . . 4 ⊢ (𝜑 → 𝐷 ∈ TermCat)
5	4	termcthind 50404	. . 3 ⊢ (𝜑 → 𝐷 ∈ ThinCat)
6		fulltermc.f	. . 3 ⊢ (𝜑 → 𝐹(𝐶 Func 𝐷)𝐺)
7	1, 2, 3, 5, 6	fullthinc 50376	. 2 ⊢ (𝜑 → (𝐹(𝐶 Full 𝐷)𝐺 ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝑥𝐻𝑦) = ∅ → ((𝐹‘𝑥)(Hom ‘𝐷)(𝐹‘𝑦)) = ∅)))
8	4	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝐷 ∈ TermCat)
9		eqid 2760	. . . . . 6 ⊢ (Base‘𝐷) = (Base‘𝐷)
10	1, 9, 6	funcf1 17955	. . . . . . . 8 ⊢ (𝜑 → 𝐹:𝐵⟶(Base‘𝐷))
11	10	ffvelcdmda 7077	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝐹‘𝑥) ∈ (Base‘𝐷))
12	11	adantrr 730	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝐹‘𝑥) ∈ (Base‘𝐷))
13	10	ffvelcdmda 7077	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → (𝐹‘𝑦) ∈ (Base‘𝐷))
14	13	adantrl 729	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝐹‘𝑦) ∈ (Base‘𝐷))
15	8, 9, 12, 14, 2	termchomn0 50410	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ¬ ((𝐹‘𝑥)(Hom ‘𝐷)(𝐹‘𝑦)) = ∅)
16		biimt 363	. . . . 5 ⊢ (¬ ((𝐹‘𝑥)(Hom ‘𝐷)(𝐹‘𝑦)) = ∅ → (¬ (𝑥𝐻𝑦) = ∅ ↔ (¬ ((𝐹‘𝑥)(Hom ‘𝐷)(𝐹‘𝑦)) = ∅ → ¬ (𝑥𝐻𝑦) = ∅)))
17	15, 16	syl 18	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (¬ (𝑥𝐻𝑦) = ∅ ↔ (¬ ((𝐹‘𝑥)(Hom ‘𝐷)(𝐹‘𝑦)) = ∅ → ¬ (𝑥𝐻𝑦) = ∅)))
18		con34b 319	. . . 4 ⊢ (((𝑥𝐻𝑦) = ∅ → ((𝐹‘𝑥)(Hom ‘𝐷)(𝐹‘𝑦)) = ∅) ↔ (¬ ((𝐹‘𝑥)(Hom ‘𝐷)(𝐹‘𝑦)) = ∅ → ¬ (𝑥𝐻𝑦) = ∅))
19	17, 18	bitr4di 292	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (¬ (𝑥𝐻𝑦) = ∅ ↔ ((𝑥𝐻𝑦) = ∅ → ((𝐹‘𝑥)(Hom ‘𝐷)(𝐹‘𝑦)) = ∅)))
20	19	2ralbidva 3224	. 2 ⊢ (𝜑 → (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ (𝑥𝐻𝑦) = ∅ ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝑥𝐻𝑦) = ∅ → ((𝐹‘𝑥)(Hom ‘𝐷)(𝐹‘𝑦)) = ∅)))
21	7, 20	bitr4d 285	1 ⊢ (𝜑 → (𝐹(𝐶 Full 𝐷)𝐺 ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ (𝑥𝐻𝑦) = ∅))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ∀wral 3076  ∅c0 4279   class class class wbr 5103  ‘cfv 6533  (class class class)co 7413  Basecbs 17301  Hom chom 17353   Func cfunc 17943   Full cful 17993  TermCatctermc 50398

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-rep 5232  ax-sep 5251  ax-nul 5263  ax-pow 5330  ax-pr 5398  ax-un 7736

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  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 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-riota 7370  df-ov 7416  df-oprab 7417  df-mpo 7418  df-1st 7986  df-2nd 7987  df-map 8828  df-ixp 8905  df-cat 17756  df-cid 17757  df-func 17947  df-full 17995  df-thinc 50344  df-termc 50399

This theorem is used by:  fulltermc2  50438