Theorem fulltermc2 49474

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 7376	. . . 4 ⊢ (𝑥 = 𝑋 → (𝑥𝐻𝑦) = (𝑋𝐻𝑦))
2	1	eqeq1d 2731	. . 3 ⊢ (𝑥 = 𝑋 → ((𝑥𝐻𝑦) = ∅ ↔ (𝑋𝐻𝑦) = ∅))
3	2	notbid 318	. 2 ⊢ (𝑥 = 𝑋 → (¬ (𝑥𝐻𝑦) = ∅ ↔ ¬ (𝑋𝐻𝑦) = ∅))
4		oveq2 7377	. . . 4 ⊢ (𝑦 = 𝑌 → (𝑋𝐻𝑦) = (𝑋𝐻𝑌))
5	4	eqeq1d 2731	. . 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 17846	. . . . . 6 ⊢ (𝐶 Full 𝐷) ⊆ (𝐶 Func 𝐷)
12	11	ssbri 5147	. . . . 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 49473	. . 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 3600	1 ⊢ (𝜑 → ¬ (𝑋𝐻𝑌) = ∅)

Syntax hints:  ¬ wn 3   → wi 4   = 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: (None)