Theorem fulltermc 50753

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 50720	. . 3 ⊢ (𝜑 → 𝐷 ∈ ThinCat)
6		fulltermc.f	. . 3 ⊢ (𝜑 → 𝐹(𝐶 Func 𝐷)𝐺)
7	1, 2, 3, 5, 6	fullthinc 50692	. 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 18048	. . . . . . . 8 ⊢ (𝜑 → 𝐹:𝐵⟶(Base‘𝐷))
11	10	ffvelcdmda 7087	. . . . . . 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 7087	. . . . . . 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 50726	. . . . 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 1569   ∈ wcel 2144  ∀wral 3076  ∅c0 4277   class class class wbr 5102  ‘cfv 6541  (class class class)co 7422  Basecbs 17394  Hom chom 17446   Func cfunc 18036   Full cful 18086  TermCatctermc 50714

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7753

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  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 3739  df-csb 3848  df-ss 3897  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-iota 6496  df-fun 6543  df-fn 6544  df-f 6545  df-f1 6546  df-fo 6547  df-f1o 6548  df-fv 6549  df-riota 7379  df-ov 7425  df-oprab 7426  df-mpo 7427  df-1st 8003  df-2nd 8004  df-map 8854  df-ixp 8931  df-cat 17849  df-cid 17850  df-func 18040  df-full 18088  df-thinc 50660  df-termc 50715

This theorem is used by:  fulltermc2  50754