Theorem isthincd 48229

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 3190	. . 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 7436	. . . . . . 7 ⊢ (𝜑 → (𝑥𝐻𝑦) = (𝑥(Hom ‘𝐶)𝑦))
7	6	eleq2d 2811	. . . . . 6 ⊢ (𝜑 → (𝑓 ∈ (𝑥𝐻𝑦) ↔ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)))
8	7	mobidv 2537	. . . . 5 ⊢ (𝜑 → (∃*𝑓 𝑓 ∈ (𝑥𝐻𝑦) ↔ ∃*𝑓 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)))
9	4, 8	raleqbidv 3329	. . . 4 ⊢ (𝜑 → (∀𝑦 ∈ 𝐵 ∃*𝑓 𝑓 ∈ (𝑥𝐻𝑦) ↔ ∀𝑦 ∈ (Base‘𝐶)∃*𝑓 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)))
10	4, 9	raleqbidv 3329	. . 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 231	. 2 ⊢ (𝜑 → ∀𝑥 ∈ (Base‘𝐶)∀𝑦 ∈ (Base‘𝐶)∃*𝑓 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))
12		eqid 2725	. . 3 ⊢ (Base‘𝐶) = (Base‘𝐶)
13		eqid 2725	. . 3 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
14	12, 13	isthinc 48213	. 2 ⊢ (𝐶 ∈ ThinCat ↔ (𝐶 ∈ Cat ∧ ∀𝑥 ∈ (Base‘𝐶)∀𝑦 ∈ (Base‘𝐶)∃*𝑓 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)))
15	1, 11, 14	sylanbrc 581	1 ⊢ (𝜑 → 𝐶 ∈ ThinCat)

Syntax hints:   → wi 4   ∧ wa 394   = wceq 1533   ∈ wcel 2098  ∃*wmo 2526  ∀wral 3050  ‘cfv 6549  (class class class)co 7419  Basecbs 17183  Hom chom 17247  Catccat 17647  ThinCatcthinc 48211

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2696  ax-nul 5307

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-mo 2528  df-clab 2703  df-cleq 2717  df-clel 2802  df-ne 2930  df-ral 3051  df-rex 3060  df-rab 3419  df-v 3463  df-sbc 3774  df-dif 3947  df-un 3949  df-ss 3961  df-nul 4323  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-br 5150  df-iota 6501  df-fv 6557  df-ov 7422  df-thinc 48212

This theorem is referenced by:  isthincd2  48230  oppcthin  48231  subthinc  48232  setcthin  48247