Theorem isthinc 46190

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

Hypotheses

Ref	Expression
isthinc.b	⊢ 𝐵 = (Base‘𝐶)
isthinc.h	⊢ 𝐻 = (Hom ‘𝐶)

Assertion

Ref	Expression
isthinc	⊢ (𝐶 ∈ ThinCat ↔ (𝐶 ∈ Cat ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∃*𝑓 𝑓 ∈ (𝑥𝐻𝑦)))

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

Proof of Theorem isthinc

Dummy variables 𝑏 𝑐 ℎ are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fvexd 6771	. . 3 ⊢ (𝑐 = 𝐶 → (Base‘𝑐) ∈ V)
2		fveq2 6756	. . . 4 ⊢ (𝑐 = 𝐶 → (Base‘𝑐) = (Base‘𝐶))
3		isthinc.b	. . . 4 ⊢ 𝐵 = (Base‘𝐶)
4	2, 3	eqtr4di 2797	. . 3 ⊢ (𝑐 = 𝐶 → (Base‘𝑐) = 𝐵)
5		fvexd 6771	. . . 4 ⊢ ((𝑐 = 𝐶 ∧ 𝑏 = 𝐵) → (Hom ‘𝑐) ∈ V)
6		fveq2 6756	. . . . . 6 ⊢ (𝑐 = 𝐶 → (Hom ‘𝑐) = (Hom ‘𝐶))
7		isthinc.h	. . . . . 6 ⊢ 𝐻 = (Hom ‘𝐶)
8	6, 7	eqtr4di 2797	. . . . 5 ⊢ (𝑐 = 𝐶 → (Hom ‘𝑐) = 𝐻)
9	8	adantr 480	. . . 4 ⊢ ((𝑐 = 𝐶 ∧ 𝑏 = 𝐵) → (Hom ‘𝑐) = 𝐻)
10		raleq 3333	. . . . . . 7 ⊢ (𝑏 = 𝐵 → (∀𝑦 ∈ 𝑏 ∃*𝑓 𝑓 ∈ (𝑥ℎ𝑦) ↔ ∀𝑦 ∈ 𝐵 ∃*𝑓 𝑓 ∈ (𝑥ℎ𝑦)))
11	10	raleqbi1dv 3331	. . . . . 6 ⊢ (𝑏 = 𝐵 → (∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 ∃*𝑓 𝑓 ∈ (𝑥ℎ𝑦) ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∃*𝑓 𝑓 ∈ (𝑥ℎ𝑦)))
12	11	ad2antlr 723	. . . . 5 ⊢ (((𝑐 = 𝐶 ∧ 𝑏 = 𝐵) ∧ ℎ = 𝐻) → (∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 ∃*𝑓 𝑓 ∈ (𝑥ℎ𝑦) ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∃*𝑓 𝑓 ∈ (𝑥ℎ𝑦)))
13		oveq 7261	. . . . . . . . 9 ⊢ (ℎ = 𝐻 → (𝑥ℎ𝑦) = (𝑥𝐻𝑦))
14	13	eleq2d 2824	. . . . . . . 8 ⊢ (ℎ = 𝐻 → (𝑓 ∈ (𝑥ℎ𝑦) ↔ 𝑓 ∈ (𝑥𝐻𝑦)))
15	14	mobidv 2549	. . . . . . 7 ⊢ (ℎ = 𝐻 → (∃*𝑓 𝑓 ∈ (𝑥ℎ𝑦) ↔ ∃*𝑓 𝑓 ∈ (𝑥𝐻𝑦)))
16	15	2ralbidv 3122	. . . . . 6 ⊢ (ℎ = 𝐻 → (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∃*𝑓 𝑓 ∈ (𝑥ℎ𝑦) ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∃*𝑓 𝑓 ∈ (𝑥𝐻𝑦)))
17	16	adantl 481	. . . . 5 ⊢ (((𝑐 = 𝐶 ∧ 𝑏 = 𝐵) ∧ ℎ = 𝐻) → (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∃*𝑓 𝑓 ∈ (𝑥ℎ𝑦) ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∃*𝑓 𝑓 ∈ (𝑥𝐻𝑦)))
18	12, 17	bitrd 278	. . . 4 ⊢ (((𝑐 = 𝐶 ∧ 𝑏 = 𝐵) ∧ ℎ = 𝐻) → (∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 ∃*𝑓 𝑓 ∈ (𝑥ℎ𝑦) ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∃*𝑓 𝑓 ∈ (𝑥𝐻𝑦)))
19	5, 9, 18	sbcied2 3758	. . 3 ⊢ ((𝑐 = 𝐶 ∧ 𝑏 = 𝐵) → ([(Hom ‘𝑐) / ℎ]∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 ∃*𝑓 𝑓 ∈ (𝑥ℎ𝑦) ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∃*𝑓 𝑓 ∈ (𝑥𝐻𝑦)))
20	1, 4, 19	sbcied2 3758	. 2 ⊢ (𝑐 = 𝐶 → ([(Base‘𝑐) / 𝑏][(Hom ‘𝑐) / ℎ]∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 ∃*𝑓 𝑓 ∈ (𝑥ℎ𝑦) ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∃*𝑓 𝑓 ∈ (𝑥𝐻𝑦)))
21		df-thinc 46189	. 2 ⊢ ThinCat = {𝑐 ∈ Cat ∣ [(Base‘𝑐) / 𝑏][(Hom ‘𝑐) / ℎ]∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 ∃*𝑓 𝑓 ∈ (𝑥ℎ𝑦)}
22	20, 21	elrab2 3620	1 ⊢ (𝐶 ∈ ThinCat ↔ (𝐶 ∈ Cat ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∃*𝑓 𝑓 ∈ (𝑥𝐻𝑦)))

Syntax hints:   ↔ wb 205   ∧ wa 395   = wceq 1539   ∈ wcel 2108  ∃*wmo 2538  ∀wral 3063  Vcvv 3422  [wsbc 3711  ‘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:  isthinc2  46191  isthinc3  46192  thincc  46193  thincmo2  46197  thincmoALT  46199  isthincd  46206  0thincg  46219