Theorem fulltermc2 49987

Description: Given a full functor to a terminal category, the source category must not have empty hom-sets. (Contributed by Zhi Wang, 17-Oct-2025.) (Proof shortened by Zhi Wang, 6-Nov-2025.)

Hypotheses

Ref	Expression
fulltermc.b	⊢ 𝐵 = (Base‘𝐶)
fulltermc.h	⊢ 𝐻 = (Hom ‘𝐶)
fulltermc.d	⊢ (𝜑 → 𝐷 ∈ TermCat)
fulltermc2.f	⊢ (𝜑 → 𝐹(𝐶 Full 𝐷)𝐺)
fulltermc2.x	⊢ (𝜑 → 𝑋 ∈ 𝐵)
fulltermc2.y	⊢ (𝜑 → 𝑌 ∈ 𝐵)

Assertion

Ref	Expression
fulltermc2	⊢ (𝜑 → ¬ (𝑋𝐻𝑌) = ∅)

Proof of Theorem fulltermc2

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		oveq1 7374	. . . 4 ⊢ (𝑥 = 𝑋 → (𝑥𝐻𝑦) = (𝑋𝐻𝑦))
2	1	eqeq1d 2738	. . 3 ⊢ (𝑥 = 𝑋 → ((𝑥𝐻𝑦) = ∅ ↔ (𝑋𝐻𝑦) = ∅))
3	2	notbid 318	. 2 ⊢ (𝑥 = 𝑋 → (¬ (𝑥𝐻𝑦) = ∅ ↔ ¬ (𝑋𝐻𝑦) = ∅))
4		oveq2 7375	. . . 4 ⊢ (𝑦 = 𝑌 → (𝑋𝐻𝑦) = (𝑋𝐻𝑌))
5	4	eqeq1d 2738	. . 3 ⊢ (𝑦 = 𝑌 → ((𝑋𝐻𝑦) = ∅ ↔ (𝑋𝐻𝑌) = ∅))
6	5	notbid 318	. 2 ⊢ (𝑦 = 𝑌 → (¬ (𝑋𝐻𝑦) = ∅ ↔ ¬ (𝑋𝐻𝑌) = ∅))
7		fulltermc2.f	. . 3 ⊢ (𝜑 → 𝐹(𝐶 Full 𝐷)𝐺)
8		fulltermc.b	. . . 4 ⊢ 𝐵 = (Base‘𝐶)
9		fulltermc.h	. . . 4 ⊢ 𝐻 = (Hom ‘𝐶)
10		fulltermc.d	. . . 4 ⊢ (𝜑 → 𝐷 ∈ TermCat)
11		fullfunc 17875	. . . . . 6 ⊢ (𝐶 Full 𝐷) ⊆ (𝐶 Func 𝐷)
12	11	ssbri 5130	. . . . 5 ⊢ (𝐹(𝐶 Full 𝐷)𝐺 → 𝐹(𝐶 Func 𝐷)𝐺)
13	7, 12	syl 17	. . . 4 ⊢ (𝜑 → 𝐹(𝐶 Func 𝐷)𝐺)
14	8, 9, 10, 13	fulltermc 49986	. . 3 ⊢ (𝜑 → (𝐹(𝐶 Full 𝐷)𝐺 ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ (𝑥𝐻𝑦) = ∅))
15	7, 14	mpbid 232	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ (𝑥𝐻𝑦) = ∅)
16		fulltermc2.x	. 2 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
17		fulltermc2.y	. 2 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
18	3, 6, 15, 16, 17	rspc2dv 3579	1 ⊢ (𝜑 → ¬ (𝑋𝐻𝑌) = ∅)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1542   ∈ wcel 2114  ∀wral 3051  ∅c0 4273   class class class wbr 5085  ‘cfv 6498  (class class class)co 7367  Basecbs 17179  Hom chom 17231   Func cfunc 17821   Full cful 17871  TermCatctermc 49947

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-rmo 3342  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  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 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-riota 7324  df-ov 7370  df-oprab 7371  df-mpo 7372  df-1st 7942  df-2nd 7943  df-map 8775  df-ixp 8846  df-cat 17634  df-cid 17635  df-func 17825  df-full 17873  df-thinc 49893  df-termc 49948

This theorem is referenced by: (None)