Theorem isthincd 45934

Description: The predicate "is a thin category" (deduction form). (Contributed by Zhi Wang, 17-Sep-2024.)

Hypotheses

Ref	Expression
isthincd.b	⊢ (𝜑 → 𝐵 = (Base‘𝐶))
isthincd.h	⊢ (𝜑 → 𝐻 = (Hom ‘𝐶))
isthincd.t	⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ∃*𝑓 𝑓 ∈ (𝑥𝐻𝑦))
isthincd.c	⊢ (𝜑 → 𝐶 ∈ Cat)

Assertion

Ref	Expression
isthincd	⊢ (𝜑 → 𝐶 ∈ ThinCat)

Distinct variable groups:   𝑦,𝐵   𝐶,𝑓,𝑥,𝑦   𝜑,𝑓,𝑥,𝑦

Allowed substitution hints:   𝐵(𝑥,𝑓)   𝐻(𝑥,𝑦,𝑓)

Proof of Theorem isthincd

Step	Hyp	Ref	Expression
1		isthincd.c	. 2 ⊢ (𝜑 → 𝐶 ∈ Cat)
2		isthincd.t	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ∃*𝑓 𝑓 ∈ (𝑥𝐻𝑦))
3	2	ralrimivva 3102	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∃*𝑓 𝑓 ∈ (𝑥𝐻𝑦))
4		isthincd.b	. . . 4 ⊢ (𝜑 → 𝐵 = (Base‘𝐶))
5		isthincd.h	. . . . . . . 8 ⊢ (𝜑 → 𝐻 = (Hom ‘𝐶))
6	5	oveqd 7208	. . . . . . 7 ⊢ (𝜑 → (𝑥𝐻𝑦) = (𝑥(Hom ‘𝐶)𝑦))
7	6	eleq2d 2816	. . . . . 6 ⊢ (𝜑 → (𝑓 ∈ (𝑥𝐻𝑦) ↔ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)))
8	7	mobidv 2548	. . . . 5 ⊢ (𝜑 → (∃*𝑓 𝑓 ∈ (𝑥𝐻𝑦) ↔ ∃*𝑓 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)))
9	4, 8	raleqbidv 3303	. . . 4 ⊢ (𝜑 → (∀𝑦 ∈ 𝐵 ∃*𝑓 𝑓 ∈ (𝑥𝐻𝑦) ↔ ∀𝑦 ∈ (Base‘𝐶)∃*𝑓 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)))
10	4, 9	raleqbidv 3303	. . 3 ⊢ (𝜑 → (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∃*𝑓 𝑓 ∈ (𝑥𝐻𝑦) ↔ ∀𝑥 ∈ (Base‘𝐶)∀𝑦 ∈ (Base‘𝐶)∃*𝑓 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)))
11	3, 10	mpbid 235	. 2 ⊢ (𝜑 → ∀𝑥 ∈ (Base‘𝐶)∀𝑦 ∈ (Base‘𝐶)∃*𝑓 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))
12		eqid 2736	. . 3 ⊢ (Base‘𝐶) = (Base‘𝐶)
13		eqid 2736	. . 3 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
14	12, 13	isthinc 45918	. 2 ⊢ (𝐶 ∈ ThinCat ↔ (𝐶 ∈ Cat ∧ ∀𝑥 ∈ (Base‘𝐶)∀𝑦 ∈ (Base‘𝐶)∃*𝑓 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)))
15	1, 11, 14	sylanbrc 586	1 ⊢ (𝜑 → 𝐶 ∈ ThinCat)

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1543   ∈ wcel 2112  ∃*wmo 2537  ∀wral 3051  ‘cfv 6358  (class class class)co 7191  Basecbs 16666  Hom chom 16760  Catccat 17121  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-nul 5184

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-ral 3056  df-rex 3057  df-rab 3060  df-v 3400  df-sbc 3684  df-dif 3856  df-un 3858  df-in 3860  df-ss 3870  df-nul 4224  df-if 4426  df-sn 4528  df-pr 4530  df-op 4534  df-uni 4806  df-br 5040  df-iota 6316  df-fv 6366  df-ov 7194  df-thinc 45917

This theorem is referenced by:  isthincd2  45935  oppcthin  45936  subthinc  45937  setcthin  45952