Theorem isthincd 46206

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 3114	. . 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 7272	. . . . . . 7 ⊢ (𝜑 → (𝑥𝐻𝑦) = (𝑥(Hom ‘𝐶)𝑦))
7	6	eleq2d 2824	. . . . . 6 ⊢ (𝜑 → (𝑓 ∈ (𝑥𝐻𝑦) ↔ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)))
8	7	mobidv 2549	. . . . 5 ⊢ (𝜑 → (∃*𝑓 𝑓 ∈ (𝑥𝐻𝑦) ↔ ∃*𝑓 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)))
9	4, 8	raleqbidv 3327	. . . 4 ⊢ (𝜑 → (∀𝑦 ∈ 𝐵 ∃*𝑓 𝑓 ∈ (𝑥𝐻𝑦) ↔ ∀𝑦 ∈ (Base‘𝐶)∃*𝑓 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)))
10	4, 9	raleqbidv 3327	. . 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 2738	. . 3 ⊢ (Base‘𝐶) = (Base‘𝐶)
13		eqid 2738	. . 3 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
14	12, 13	isthinc 46190	. 2 ⊢ (𝐶 ∈ ThinCat ↔ (𝐶 ∈ Cat ∧ ∀𝑥 ∈ (Base‘𝐶)∀𝑦 ∈ (Base‘𝐶)∃*𝑓 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)))
15	1, 11, 14	sylanbrc 582	1 ⊢ (𝜑 → 𝐶 ∈ ThinCat)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1539   ∈ wcel 2108  ∃*wmo 2538  ∀wral 3063  ‘cfv 6418  (class class class)co 7255  Basecbs 16840  Hom chom 16899  Catccat 17290  ThinCatcthinc 46188

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-nul 5225

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-sbc 3712  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-iota 6376  df-fv 6426  df-ov 7258  df-thinc 46189

This theorem is referenced by:  isthincd2  46207  oppcthin  46208  subthinc  46209  setcthin  46224