Theorem isthinc 50661

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 6903	. . 3 ⊢ (𝑐 = 𝐶 → (Base‘𝑐) ∈ V)
2		fveq2 6888	. . . 4 ⊢ (𝑐 = 𝐶 → (Base‘𝑐) = (Base‘𝐶))
3		isthinc.b	. . . 4 ⊢ 𝐵 = (Base‘𝐶)
4	2, 3	eqtr4di 2813	. . 3 ⊢ (𝑐 = 𝐶 → (Base‘𝑐) = 𝐵)
5		fvexd 6903	. . . 4 ⊢ ((𝑐 = 𝐶 ∧ 𝑏 = 𝐵) → (Hom ‘𝑐) ∈ V)
6		fveq2 6888	. . . . . 6 ⊢ (𝑐 = 𝐶 → (Hom ‘𝑐) = (Hom ‘𝐶))
7		isthinc.h	. . . . . 6 ⊢ 𝐻 = (Hom ‘𝐶)
8	6, 7	eqtr4di 2813	. . . . 5 ⊢ (𝑐 = 𝐶 → (Hom ‘𝑐) = 𝐻)
9	8	adantr 486	. . . 4 ⊢ ((𝑐 = 𝐶 ∧ 𝑏 = 𝐵) → (Hom ‘𝑐) = 𝐻)
10		raleq 3316	. . . . . . 7 ⊢ (𝑏 = 𝐵 → (∀𝑦 ∈ 𝑏 ∃*𝑓 𝑓 ∈ (𝑥ℎ𝑦) ↔ ∀𝑦 ∈ 𝐵 ∃*𝑓 𝑓 ∈ (𝑥ℎ𝑦)))
11	10	raleqbi1dv 3329	. . . . . 6 ⊢ (𝑏 = 𝐵 → (∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 ∃*𝑓 𝑓 ∈ (𝑥ℎ𝑦) ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∃*𝑓 𝑓 ∈ (𝑥ℎ𝑦)))
12	11	ad2antlr 740	. . . . 5 ⊢ (((𝑐 = 𝐶 ∧ 𝑏 = 𝐵) ∧ ℎ = 𝐻) → (∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 ∃*𝑓 𝑓 ∈ (𝑥ℎ𝑦) ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∃*𝑓 𝑓 ∈ (𝑥ℎ𝑦)))
13		oveq 7428	. . . . . . . . 9 ⊢ (ℎ = 𝐻 → (𝑥ℎ𝑦) = (𝑥𝐻𝑦))
14	13	eleq2d 2846	. . . . . . . 8 ⊢ (ℎ = 𝐻 → (𝑓 ∈ (𝑥ℎ𝑦) ↔ 𝑓 ∈ (𝑥𝐻𝑦)))
15	14	mobidv 2574	. . . . . . 7 ⊢ (ℎ = 𝐻 → (∃*𝑓 𝑓 ∈ (𝑥ℎ𝑦) ↔ ∃*𝑓 𝑓 ∈ (𝑥𝐻𝑦)))
16	15	2ralbidv 3226	. . . . . 6 ⊢ (ℎ = 𝐻 → (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∃*𝑓 𝑓 ∈ (𝑥ℎ𝑦) ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∃*𝑓 𝑓 ∈ (𝑥𝐻𝑦)))
17	16	adantl 487	. . . . 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 282	. . . 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 3782	. . 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 3782	. 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 50660	. 2 ⊢ ThinCat = {𝑐 ∈ Cat ∣ [(Base‘𝑐) / 𝑏][(Hom ‘𝑐) / ℎ]∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 ∃*𝑓 𝑓 ∈ (𝑥ℎ𝑦)}
22	20, 21	elrab2 3648	1 ⊢ (𝐶 ∈ ThinCat ↔ (𝐶 ∈ Cat ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∃*𝑓 𝑓 ∈ (𝑥𝐻𝑦)))

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401   = wceq 1569   ∈ wcel 2144  ∃*wmo 2562  ∀wral 3076  Vcvv 3450  [wsbc 3738  ‘cfv 6541  (class class class)co 7422  Basecbs 17394  Hom chom 17446  Catccat 17845  ThinCatcthinc 50659

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732  ax-nul 5259

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-mo 2564  df-clab 2739  df-cleq 2752  df-clel 2835  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-sbc 3739  df-ss 3897  df-dif 4046  df-un 4150  df-nul 4278  df-if 4482  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-br 5103  df-iota 6496  df-fv 6549  df-ov 7425  df-thinc 50660

This theorem is used by:  isthinc2  50662  isthinc3  50663  thinccat  50664  thincmo2  50668  thincmoALT  50671  isthincd  50678  thincpropd  50684  0thincg  50700