Theorem fulltermc 49473

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 2729	. . 3 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
3		fulltermc.h	. . 3 ⊢ 𝐻 = (Hom ‘𝐶)
4		fulltermc.d	. . . 4 ⊢ (𝜑 → 𝐷 ∈ TermCat)
5	4	termcthind 49440	. . 3 ⊢ (𝜑 → 𝐷 ∈ ThinCat)
6		fulltermc.f	. . 3 ⊢ (𝜑 → 𝐹(𝐶 Func 𝐷)𝐺)
7	1, 2, 3, 5, 6	fullthinc 49412	. 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 2729	. . . . . 6 ⊢ (Base‘𝐷) = (Base‘𝐷)
10	1, 9, 6	funcf1 17804	. . . . . . . 8 ⊢ (𝜑 → 𝐹:𝐵⟶(Base‘𝐷))
11	10	ffvelcdmda 7038	. . . . . . 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 7038	. . . . . . 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 49446	. . . . 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 3197	. 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 2109  ∀wral 3044  ∅c0 4292   class class class wbr 5102  ‘cfv 6499  (class class class)co 7369  Basecbs 17155  Hom chom 17207   Func cfunc 17792   Full cful 17842  TermCatctermc 49434

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5229  ax-sep 5246  ax-nul 5256  ax-pow 5315  ax-pr 5382  ax-un 7691

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rmo 3351  df-reu 3352  df-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-riota 7326  df-ov 7372  df-oprab 7373  df-mpo 7374  df-1st 7947  df-2nd 7948  df-map 8778  df-ixp 8848  df-cat 17605  df-cid 17606  df-func 17796  df-full 17844  df-thinc 49380  df-termc 49435

This theorem is referenced by:  fulltermc2  49474