Theorem isthincd 49467

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 3175	. . 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 7363	. . . . . . 7 ⊢ (𝜑 → (𝑥𝐻𝑦) = (𝑥(Hom ‘𝐶)𝑦))
7	6	eleq2d 2817	. . . . . 6 ⊢ (𝜑 → (𝑓 ∈ (𝑥𝐻𝑦) ↔ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)))
8	7	mobidv 2544	. . . . 5 ⊢ (𝜑 → (∃*𝑓 𝑓 ∈ (𝑥𝐻𝑦) ↔ ∃*𝑓 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)))
9	4, 8	raleqbidv 3312	. . . 4 ⊢ (𝜑 → (∀𝑦 ∈ 𝐵 ∃*𝑓 𝑓 ∈ (𝑥𝐻𝑦) ↔ ∀𝑦 ∈ (Base‘𝐶)∃*𝑓 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)))
10	4, 9	raleqbidv 3312	. . 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 232	. 2 ⊢ (𝜑 → ∀𝑥 ∈ (Base‘𝐶)∀𝑦 ∈ (Base‘𝐶)∃*𝑓 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))
12		eqid 2731	. . 3 ⊢ (Base‘𝐶) = (Base‘𝐶)
13		eqid 2731	. . 3 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
14	12, 13	isthinc 49450	. 2 ⊢ (𝐶 ∈ ThinCat ↔ (𝐶 ∈ Cat ∧ ∀𝑥 ∈ (Base‘𝐶)∀𝑦 ∈ (Base‘𝐶)∃*𝑓 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)))
15	1, 11, 14	sylanbrc 583	1 ⊢ (𝜑 → 𝐶 ∈ ThinCat)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2111  ∃*wmo 2533  ∀wral 3047  ‘cfv 6481  (class class class)co 7346  Basecbs 17117  Hom chom 17169  Catccat 17567  ThinCatcthinc 49448

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703  ax-nul 5244

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-mo 2535  df-clab 2710  df-cleq 2723  df-clel 2806  df-ne 2929  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-sbc 3742  df-dif 3905  df-un 3907  df-ss 3919  df-nul 4284  df-if 4476  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-br 5092  df-iota 6437  df-fv 6489  df-ov 7349  df-thinc 49449

This theorem is referenced by:  isthincd2  49468  oppcthin  49469  subthinc  49474  thincciso2  49486  indcthing  49491  discthing  49492  setcthin  49496  idfudiag1  49556  arweuthinc  49560  funcsn  49572  0fucterm  49574