Theorem fullthinc2 46216

Description: A full functor to a thin category maps empty hom-sets to empty hom-sets. (Contributed by Zhi Wang, 1-Oct-2024.)

Hypotheses

Ref	Expression
fullthinc.b	⊢ 𝐵 = (Base‘𝐶)
fullthinc.j	⊢ 𝐽 = (Hom ‘𝐷)
fullthinc.h	⊢ 𝐻 = (Hom ‘𝐶)
fullthinc.d	⊢ (𝜑 → 𝐷 ∈ ThinCat)
fullthinc2.f	⊢ (𝜑 → 𝐹(𝐶 Full 𝐷)𝐺)
fullthinc2.x	⊢ (𝜑 → 𝑋 ∈ 𝐵)
fullthinc2.y	⊢ (𝜑 → 𝑌 ∈ 𝐵)

Assertion

Ref	Expression
fullthinc2	⊢ (𝜑 → ((𝑋𝐻𝑌) = ∅ ↔ ((𝐹‘𝑋)𝐽(𝐹‘𝑌)) = ∅))

Proof of Theorem fullthinc2

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

Step	Hyp	Ref	Expression
1		fullthinc2.x	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
2		fullthinc2.y	. . 3 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
3		fullthinc2.f	. . . 4 ⊢ (𝜑 → 𝐹(𝐶 Full 𝐷)𝐺)
4		fullthinc.b	. . . . 5 ⊢ 𝐵 = (Base‘𝐶)
5		fullthinc.j	. . . . 5 ⊢ 𝐽 = (Hom ‘𝐷)
6		fullthinc.h	. . . . 5 ⊢ 𝐻 = (Hom ‘𝐶)
7		fullthinc.d	. . . . 5 ⊢ (𝜑 → 𝐷 ∈ ThinCat)
8		fullfunc 17538	. . . . . . 7 ⊢ (𝐶 Full 𝐷) ⊆ (𝐶 Func 𝐷)
9	8	ssbri 5115	. . . . . 6 ⊢ (𝐹(𝐶 Full 𝐷)𝐺 → 𝐹(𝐶 Func 𝐷)𝐺)
10	3, 9	syl 17	. . . . 5 ⊢ (𝜑 → 𝐹(𝐶 Func 𝐷)𝐺)
11	4, 5, 6, 7, 10	fullthinc 46215	. . . 4 ⊢ (𝜑 → (𝐹(𝐶 Full 𝐷)𝐺 ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝑥𝐻𝑦) = ∅ → ((𝐹‘𝑥)𝐽(𝐹‘𝑦)) = ∅)))
12	3, 11	mpbid 231	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝑥𝐻𝑦) = ∅ → ((𝐹‘𝑥)𝐽(𝐹‘𝑦)) = ∅))
13		oveq12 7264	. . . . . . 7 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → (𝑥𝐻𝑦) = (𝑋𝐻𝑌))
14	13	eqeq1d 2740	. . . . . 6 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → ((𝑥𝐻𝑦) = ∅ ↔ (𝑋𝐻𝑌) = ∅))
15		simpl 482	. . . . . . . . 9 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → 𝑥 = 𝑋)
16	15	fveq2d 6760	. . . . . . . 8 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → (𝐹‘𝑥) = (𝐹‘𝑋))
17		simpr 484	. . . . . . . . 9 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → 𝑦 = 𝑌)
18	17	fveq2d 6760	. . . . . . . 8 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → (𝐹‘𝑦) = (𝐹‘𝑌))
19	16, 18	oveq12d 7273	. . . . . . 7 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → ((𝐹‘𝑥)𝐽(𝐹‘𝑦)) = ((𝐹‘𝑋)𝐽(𝐹‘𝑌)))
20	19	eqeq1d 2740	. . . . . 6 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → (((𝐹‘𝑥)𝐽(𝐹‘𝑦)) = ∅ ↔ ((𝐹‘𝑋)𝐽(𝐹‘𝑌)) = ∅))
21	14, 20	imbi12d 344	. . . . 5 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → (((𝑥𝐻𝑦) = ∅ → ((𝐹‘𝑥)𝐽(𝐹‘𝑦)) = ∅) ↔ ((𝑋𝐻𝑌) = ∅ → ((𝐹‘𝑋)𝐽(𝐹‘𝑌)) = ∅)))
22	21	rspc2gv 3561	. . . 4 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝑥𝐻𝑦) = ∅ → ((𝐹‘𝑥)𝐽(𝐹‘𝑦)) = ∅) → ((𝑋𝐻𝑌) = ∅ → ((𝐹‘𝑋)𝐽(𝐹‘𝑌)) = ∅)))
23	22	imp 406	. . 3 ⊢ (((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝑥𝐻𝑦) = ∅ → ((𝐹‘𝑥)𝐽(𝐹‘𝑦)) = ∅)) → ((𝑋𝐻𝑌) = ∅ → ((𝐹‘𝑋)𝐽(𝐹‘𝑌)) = ∅))
24	1, 2, 12, 23	syl21anc 834	. 2 ⊢ (𝜑 → ((𝑋𝐻𝑌) = ∅ → ((𝐹‘𝑋)𝐽(𝐹‘𝑌)) = ∅))
25	4, 6, 5, 10, 1, 2	funcf2 17499	. . 3 ⊢ (𝜑 → (𝑋𝐺𝑌):(𝑋𝐻𝑌)⟶((𝐹‘𝑋)𝐽(𝐹‘𝑌)))
26	25	f002 46069	. 2 ⊢ (𝜑 → (((𝐹‘𝑋)𝐽(𝐹‘𝑌)) = ∅ → (𝑋𝐻𝑌) = ∅))
27	24, 26	impbid 211	1 ⊢ (𝜑 → ((𝑋𝐻𝑌) = ∅ ↔ ((𝐹‘𝑋)𝐽(𝐹‘𝑌)) = ∅))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1539   ∈ wcel 2108  ∀wral 3063  ∅c0 4253   class class class wbr 5070  ‘cfv 6418  (class class class)co 7255  Basecbs 16840  Hom chom 16899   Func cfunc 17485   Full cful 17534  ThinCatcthinc 46188

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-ov 7258  df-oprab 7259  df-mpo 7260  df-1st 7804  df-2nd 7805  df-map 8575  df-ixp 8644  df-func 17489  df-full 17536  df-thinc 46189

This theorem is referenced by: (None)