Theorem fullthinc2 50212

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 17966	. . . . . . 7 ⊢ (𝐶 Full 𝐷) ⊆ (𝐶 Func 𝐷)
9	8	ssbri 5157	. . . . . 6 ⊢ (𝐹(𝐶 Full 𝐷)𝐺 → 𝐹(𝐶 Func 𝐷)𝐺)
10	3, 9	syl 18	. . . . 5 ⊢ (𝜑 → 𝐹(𝐶 Func 𝐷)𝐺)
11	4, 5, 6, 7, 10	fullthinc 50211	. . . 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 7421	. . . . . . 7 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → (𝑥𝐻𝑦) = (𝑋𝐻𝑌))
14	13	eqeq1d 2765	. . . . . 6 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → ((𝑥𝐻𝑦) = ∅ ↔ (𝑋𝐻𝑌) = ∅))
15		simpl 487	. . . . . . . . 9 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → 𝑥 = 𝑋)
16	15	fveq2d 6887	. . . . . . . 8 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → (𝐹‘𝑥) = (𝐹‘𝑋))
17		simpr 489	. . . . . . . . 9 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → 𝑦 = 𝑌)
18	17	fveq2d 6887	. . . . . . . 8 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → (𝐹‘𝑦) = (𝐹‘𝑌))
19	16, 18	oveq12d 7430	. . . . . . 7 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → ((𝐹‘𝑥)𝐽(𝐹‘𝑦)) = ((𝐹‘𝑋)𝐽(𝐹‘𝑌)))
20	19	eqeq1d 2765	. . . . . 6 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → (((𝐹‘𝑥)𝐽(𝐹‘𝑦)) = ∅ ↔ ((𝐹‘𝑋)𝐽(𝐹‘𝑌)) = ∅))
21	14, 20	imbi12d 347	. . . . 5 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → (((𝑥𝐻𝑦) = ∅ → ((𝐹‘𝑥)𝐽(𝐹‘𝑦)) = ∅) ↔ ((𝑋𝐻𝑌) = ∅ → ((𝐹‘𝑋)𝐽(𝐹‘𝑌)) = ∅)))
22	21	rspc2gv 3592	. . . 4 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝑥𝐻𝑦) = ∅ → ((𝐹‘𝑥)𝐽(𝐹‘𝑦)) = ∅) → ((𝑋𝐻𝑌) = ∅ → ((𝐹‘𝑋)𝐽(𝐹‘𝑌)) = ∅)))
23	22	imp 411	. . 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 850	. 2 ⊢ (𝜑 → ((𝑋𝐻𝑌) = ∅ → ((𝐹‘𝑋)𝐽(𝐹‘𝑌)) = ∅))
25	4, 6, 5, 10, 1, 2	funcf2 17926	. . 3 ⊢ (𝜑 → (𝑋𝐺𝑌):(𝑋𝐻𝑌)⟶((𝐹‘𝑋)𝐽(𝐹‘𝑌)))
26	25	f002 49615	. 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 400   = wceq 1570   ∈ wcel 2143  ∀wral 3079  ∅c0 4287   class class class wbr 5110  ‘cfv 6538  (class class class)co 7412  Basecbs 17270  Hom chom 17322   Func cfunc 17912   Full cful 17962  ThinCatcthinc 50178

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5239  ax-sep 5258  ax-nul 5270  ax-pow 5338  ax-pr 5406  ax-un 7734

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-sbc 3746  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-pw 4565  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-iun 4959  df-br 5111  df-opab 5175  df-mpt 5194  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-ov 7415  df-oprab 7416  df-mpo 7417  df-1st 7987  df-2nd 7988  df-map 8827  df-ixp 8897  df-func 17916  df-full 17964  df-thinc 50179

This theorem is referenced by: (None)