Theorem isthinc 49412

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 6876	. . 3 ⊢ (𝑐 = 𝐶 → (Base‘𝑐) ∈ V)
2		fveq2 6861	. . . 4 ⊢ (𝑐 = 𝐶 → (Base‘𝑐) = (Base‘𝐶))
3		isthinc.b	. . . 4 ⊢ 𝐵 = (Base‘𝐶)
4	2, 3	eqtr4di 2783	. . 3 ⊢ (𝑐 = 𝐶 → (Base‘𝑐) = 𝐵)
5		fvexd 6876	. . . 4 ⊢ ((𝑐 = 𝐶 ∧ 𝑏 = 𝐵) → (Hom ‘𝑐) ∈ V)
6		fveq2 6861	. . . . . 6 ⊢ (𝑐 = 𝐶 → (Hom ‘𝑐) = (Hom ‘𝐶))
7		isthinc.h	. . . . . 6 ⊢ 𝐻 = (Hom ‘𝐶)
8	6, 7	eqtr4di 2783	. . . . 5 ⊢ (𝑐 = 𝐶 → (Hom ‘𝑐) = 𝐻)
9	8	adantr 480	. . . 4 ⊢ ((𝑐 = 𝐶 ∧ 𝑏 = 𝐵) → (Hom ‘𝑐) = 𝐻)
10		raleq 3298	. . . . . . 7 ⊢ (𝑏 = 𝐵 → (∀𝑦 ∈ 𝑏 ∃*𝑓 𝑓 ∈ (𝑥ℎ𝑦) ↔ ∀𝑦 ∈ 𝐵 ∃*𝑓 𝑓 ∈ (𝑥ℎ𝑦)))
11	10	raleqbi1dv 3313	. . . . . 6 ⊢ (𝑏 = 𝐵 → (∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 ∃*𝑓 𝑓 ∈ (𝑥ℎ𝑦) ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∃*𝑓 𝑓 ∈ (𝑥ℎ𝑦)))
12	11	ad2antlr 727	. . . . 5 ⊢ (((𝑐 = 𝐶 ∧ 𝑏 = 𝐵) ∧ ℎ = 𝐻) → (∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 ∃*𝑓 𝑓 ∈ (𝑥ℎ𝑦) ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∃*𝑓 𝑓 ∈ (𝑥ℎ𝑦)))
13		oveq 7396	. . . . . . . . 9 ⊢ (ℎ = 𝐻 → (𝑥ℎ𝑦) = (𝑥𝐻𝑦))
14	13	eleq2d 2815	. . . . . . . 8 ⊢ (ℎ = 𝐻 → (𝑓 ∈ (𝑥ℎ𝑦) ↔ 𝑓 ∈ (𝑥𝐻𝑦)))
15	14	mobidv 2543	. . . . . . 7 ⊢ (ℎ = 𝐻 → (∃*𝑓 𝑓 ∈ (𝑥ℎ𝑦) ↔ ∃*𝑓 𝑓 ∈ (𝑥𝐻𝑦)))
16	15	2ralbidv 3202	. . . . . 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 279	. . . 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 3801	. . 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 3801	. 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 49411	. 2 ⊢ ThinCat = {𝑐 ∈ Cat ∣ [(Base‘𝑐) / 𝑏][(Hom ‘𝑐) / ℎ]∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 ∃*𝑓 𝑓 ∈ (𝑥ℎ𝑦)}
22	20, 21	elrab2 3665	1 ⊢ (𝐶 ∈ ThinCat ↔ (𝐶 ∈ Cat ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∃*𝑓 𝑓 ∈ (𝑥𝐻𝑦)))

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∃*wmo 2532  ∀wral 3045  Vcvv 3450  [wsbc 3756  ‘cfv 6514  (class class class)co 7390  Basecbs 17186  Hom chom 17238  Catccat 17632  ThinCatcthinc 49410

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2702  ax-nul 5264

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-mo 2534  df-clab 2709  df-cleq 2722  df-clel 2804  df-ne 2927  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-sbc 3757  df-dif 3920  df-un 3922  df-ss 3934  df-nul 4300  df-if 4492  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-br 5111  df-iota 6467  df-fv 6522  df-ov 7393  df-thinc 49411

This theorem is referenced by:  isthinc2  49413  isthinc3  49414  thincc  49415  thincmo2  49419  thincmoALT  49422  isthincd  49429  thincpropd  49435  0thincg  49451