Theorem fullthinc2 45944

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 17367	. . . . . . 7 ⊢ (𝐶 Full 𝐷) ⊆ (𝐶 Func 𝐷)
9	8	ssbri 5084	. . . . . 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 45943	. . . 4 ⊢ (𝜑 → (𝐹(𝐶 Full 𝐷)𝐺 ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝑥𝐻𝑦) = ∅ → ((𝐹‘𝑥)𝐽(𝐹‘𝑦)) = ∅)))
12	3, 11	mpbid 235	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝑥𝐻𝑦) = ∅ → ((𝐹‘𝑥)𝐽(𝐹‘𝑦)) = ∅))
13		oveq12 7200	. . . . . . 7 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → (𝑥𝐻𝑦) = (𝑋𝐻𝑌))
14	13	eqeq1d 2738	. . . . . 6 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → ((𝑥𝐻𝑦) = ∅ ↔ (𝑋𝐻𝑌) = ∅))
15		simpl 486	. . . . . . . . 9 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → 𝑥 = 𝑋)
16	15	fveq2d 6699	. . . . . . . 8 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → (𝐹‘𝑥) = (𝐹‘𝑋))
17		simpr 488	. . . . . . . . 9 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → 𝑦 = 𝑌)
18	17	fveq2d 6699	. . . . . . . 8 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → (𝐹‘𝑦) = (𝐹‘𝑌))
19	16, 18	oveq12d 7209	. . . . . . 7 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → ((𝐹‘𝑥)𝐽(𝐹‘𝑦)) = ((𝐹‘𝑋)𝐽(𝐹‘𝑌)))
20	19	eqeq1d 2738	. . . . . 6 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → (((𝐹‘𝑥)𝐽(𝐹‘𝑦)) = ∅ ↔ ((𝐹‘𝑋)𝐽(𝐹‘𝑌)) = ∅))
21	14, 20	imbi12d 348	. . . . 5 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → (((𝑥𝐻𝑦) = ∅ → ((𝐹‘𝑥)𝐽(𝐹‘𝑦)) = ∅) ↔ ((𝑋𝐻𝑌) = ∅ → ((𝐹‘𝑋)𝐽(𝐹‘𝑌)) = ∅)))
22	21	rspc2gv 3536	. . . 4 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝑥𝐻𝑦) = ∅ → ((𝐹‘𝑥)𝐽(𝐹‘𝑦)) = ∅) → ((𝑋𝐻𝑌) = ∅ → ((𝐹‘𝑋)𝐽(𝐹‘𝑌)) = ∅)))
23	22	imp 410	. . 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 838	. 2 ⊢ (𝜑 → ((𝑋𝐻𝑌) = ∅ → ((𝐹‘𝑋)𝐽(𝐹‘𝑌)) = ∅))
25	4, 6, 5, 10, 1, 2	funcf2 17328	. . 3 ⊢ (𝜑 → (𝑋𝐺𝑌):(𝑋𝐻𝑌)⟶((𝐹‘𝑋)𝐽(𝐹‘𝑌)))
26	25	f002 45797	. 2 ⊢ (𝜑 → (((𝐹‘𝑋)𝐽(𝐹‘𝑌)) = ∅ → (𝑋𝐻𝑌) = ∅))
27	24, 26	impbid 215	1 ⊢ (𝜑 → ((𝑋𝐻𝑌) = ∅ ↔ ((𝐹‘𝑋)𝐽(𝐹‘𝑌)) = ∅))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1543   ∈ wcel 2112  ∀wral 3051  ∅c0 4223   class class class wbr 5039  ‘cfv 6358  (class class class)co 7191  Basecbs 16666  Hom chom 16760   Func cfunc 17314   Full cful 17363  ThinCatcthinc 45916

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2018  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2160  ax-12 2177  ax-ext 2708  ax-rep 5164  ax-sep 5177  ax-nul 5184  ax-pow 5243  ax-pr 5307  ax-un 7501

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2073  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2728  df-clel 2809  df-nfc 2879  df-ne 2933  df-ral 3056  df-rex 3057  df-reu 3058  df-rab 3060  df-v 3400  df-sbc 3684  df-csb 3799  df-dif 3856  df-un 3858  df-in 3860  df-ss 3870  df-nul 4224  df-if 4426  df-pw 4501  df-sn 4528  df-pr 4530  df-op 4534  df-uni 4806  df-iun 4892  df-br 5040  df-opab 5102  df-mpt 5121  df-id 5440  df-xp 5542  df-rel 5543  df-cnv 5544  df-co 5545  df-dm 5546  df-rn 5547  df-res 5548  df-ima 5549  df-iota 6316  df-fun 6360  df-fn 6361  df-f 6362  df-f1 6363  df-fo 6364  df-f1o 6365  df-fv 6366  df-ov 7194  df-oprab 7195  df-mpo 7196  df-1st 7739  df-2nd 7740  df-map 8488  df-ixp 8557  df-func 17318  df-full 17365  df-thinc 45917

This theorem is referenced by: (None)