Theorem discthing 49295

Description: A discrete category, i.e., a category where all morphisms are identity morphisms, is thin. Example 3.26(1) of [Adamek] p. 33. (Contributed by Zhi Wang, 11-Nov-2025.)

Hypotheses

Ref	Expression
indcthing.b	⊢ (𝜑 → 𝐵 = (Base‘𝐶))
indcthing.h	⊢ (𝜑 → 𝐻 = (Hom ‘𝐶))
indcthing.c	⊢ (𝜑 → 𝐶 ∈ Cat)
discthing.i	⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥𝐻𝑦) = if(𝑥 = 𝑦, {𝐼}, ∅))

Assertion

Ref	Expression
discthing	⊢ (𝜑 → 𝐶 ∈ ThinCat)

Distinct variable groups:   𝑦,𝐵   𝑥,𝐶,𝑦   𝜑,𝑥,𝑦

Allowed substitution hints:   𝐵(𝑥)   𝐻(𝑥,𝑦)   𝐼(𝑥,𝑦)

Proof of Theorem discthing

Dummy variable 𝑖 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		indcthing.b	. 2 ⊢ (𝜑 → 𝐵 = (Base‘𝐶))
2		indcthing.h	. 2 ⊢ (𝜑 → 𝐻 = (Hom ‘𝐶))
3		eleq2w2 2731	. . . . 5 ⊢ ({𝐼} = if(𝑥 = 𝑦, {𝐼}, ∅) → (𝑖 ∈ {𝐼} ↔ 𝑖 ∈ if(𝑥 = 𝑦, {𝐼}, ∅)))
4	3	mobidv 2548	. . . 4 ⊢ ({𝐼} = if(𝑥 = 𝑦, {𝐼}, ∅) → (∃*𝑖 𝑖 ∈ {𝐼} ↔ ∃*𝑖 𝑖 ∈ if(𝑥 = 𝑦, {𝐼}, ∅)))
5		eleq2w2 2731	. . . . 5 ⊢ (∅ = if(𝑥 = 𝑦, {𝐼}, ∅) → (𝑖 ∈ ∅ ↔ 𝑖 ∈ if(𝑥 = 𝑦, {𝐼}, ∅)))
6	5	mobidv 2548	. . . 4 ⊢ (∅ = if(𝑥 = 𝑦, {𝐼}, ∅) → (∃*𝑖 𝑖 ∈ ∅ ↔ ∃*𝑖 𝑖 ∈ if(𝑥 = 𝑦, {𝐼}, ∅)))
7		eqid 2735	. . . . 5 ⊢ {𝐼} = {𝐼}
8		mosn 48739	. . . . 5 ⊢ ({𝐼} = {𝐼} → ∃*𝑖 𝑖 ∈ {𝐼})
9	7, 8	mp1i 13	. . . 4 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑥 = 𝑦) → ∃*𝑖 𝑖 ∈ {𝐼})
10		eqid 2735	. . . . 5 ⊢ ∅ = ∅
11		mo0 48740	. . . . 5 ⊢ (∅ = ∅ → ∃*𝑖 𝑖 ∈ ∅)
12	10, 11	mp1i 13	. . . 4 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ ¬ 𝑥 = 𝑦) → ∃*𝑖 𝑖 ∈ ∅)
13	4, 6, 9, 12	ifbothda 4539	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ∃*𝑖 𝑖 ∈ if(𝑥 = 𝑦, {𝐼}, ∅))
14		discthing.i	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥𝐻𝑦) = if(𝑥 = 𝑦, {𝐼}, ∅))
15	14	eleq2d 2820	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑖 ∈ (𝑥𝐻𝑦) ↔ 𝑖 ∈ if(𝑥 = 𝑦, {𝐼}, ∅)))
16	15	mobidv 2548	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (∃*𝑖 𝑖 ∈ (𝑥𝐻𝑦) ↔ ∃*𝑖 𝑖 ∈ if(𝑥 = 𝑦, {𝐼}, ∅)))
17	13, 16	mpbird 257	. 2 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ∃*𝑖 𝑖 ∈ (𝑥𝐻𝑦))
18		indcthing.c	. 2 ⊢ (𝜑 → 𝐶 ∈ Cat)
19	1, 2, 17, 18	isthincd 49270	1 ⊢ (𝜑 → 𝐶 ∈ ThinCat)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2108  ∃*wmo 2537  ∅c0 4308  ifcif 4500  {csn 4601  ‘cfv 6530  (class class class)co 7403  Basecbs 17226  Hom chom 17280  Catccat 17674  ThinCatcthinc 49251

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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-sep 5266  ax-nul 5276

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-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rmo 3359  df-reu 3360  df-rab 3416  df-v 3461  df-sbc 3766  df-dif 3929  df-un 3931  df-ss 3943  df-nul 4309  df-if 4501  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-br 5120  df-iota 6483  df-fv 6538  df-ov 7406  df-thinc 49252

This theorem is referenced by: (None)