Theorem isthincd 49911

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 3180	. . 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 7384	. . . . . . 7 ⊢ (𝜑 → (𝑥𝐻𝑦) = (𝑥(Hom ‘𝐶)𝑦))
7	6	eleq2d 2822	. . . . . 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 3311	. . . 4 ⊢ (𝜑 → (∀𝑦 ∈ 𝐵 ∃*𝑓 𝑓 ∈ (𝑥𝐻𝑦) ↔ ∀𝑦 ∈ (Base‘𝐶)∃*𝑓 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)))
10	4, 9	raleqbidv 3311	. . 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 2736	. . 3 ⊢ (Base‘𝐶) = (Base‘𝐶)
13		eqid 2736	. . 3 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
14	12, 13	isthinc 49894	. 2 ⊢ (𝐶 ∈ ThinCat ↔ (𝐶 ∈ Cat ∧ ∀𝑥 ∈ (Base‘𝐶)∀𝑦 ∈ (Base‘𝐶)∃*𝑓 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)))
15	1, 11, 14	sylanbrc 584	1 ⊢ (𝜑 → 𝐶 ∈ ThinCat)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ∃*wmo 2537  ∀wral 3051  ‘cfv 6498  (class class class)co 7367  Basecbs 17179  Hom chom 17231  Catccat 17630  ThinCatcthinc 49892

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2708  ax-nul 5241

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-mo 2539  df-clab 2715  df-cleq 2728  df-clel 2811  df-ne 2933  df-ral 3052  df-rex 3062  df-rab 3390  df-v 3431  df-sbc 3729  df-dif 3892  df-un 3894  df-ss 3906  df-nul 4274  df-if 4467  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-br 5086  df-iota 6454  df-fv 6506  df-ov 7370  df-thinc 49893

This theorem is referenced by:  isthincd2  49912  oppcthin  49913  subthinc  49918  thincciso2  49930  indcthing  49935  discthing  49936  setcthin  49940  idfudiag1  50000  arweuthinc  50004  funcsn  50016  0fucterm  50018