Theorem discthing 50090

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 2761	. . . . 5 ⊢ ({𝐼} = if(𝑥 = 𝑦, {𝐼}, ∅) → (𝑖 ∈ {𝐼} ↔ 𝑖 ∈ if(𝑥 = 𝑦, {𝐼}, ∅)))
4	3	mobidv 2579	. . . 4 ⊢ ({𝐼} = if(𝑥 = 𝑦, {𝐼}, ∅) → (∃*𝑖 𝑖 ∈ {𝐼} ↔ ∃*𝑖 𝑖 ∈ if(𝑥 = 𝑦, {𝐼}, ∅)))
5		eleq2w2 2761	. . . . 5 ⊢ (∅ = if(𝑥 = 𝑦, {𝐼}, ∅) → (𝑖 ∈ ∅ ↔ 𝑖 ∈ if(𝑥 = 𝑦, {𝐼}, ∅)))
6	5	mobidv 2579	. . . 4 ⊢ (∅ = if(𝑥 = 𝑦, {𝐼}, ∅) → (∃*𝑖 𝑖 ∈ ∅ ↔ ∃*𝑖 𝑖 ∈ if(𝑥 = 𝑦, {𝐼}, ∅)))
7		eqid 2765	. . . . 5 ⊢ {𝐼} = {𝐼}
8		mosn 49442	. . . . 5 ⊢ ({𝐼} = {𝐼} → ∃*𝑖 𝑖 ∈ {𝐼})
9	7, 8	mp1i 14	. . . 4 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑥 = 𝑦) → ∃*𝑖 𝑖 ∈ {𝐼})
10		eqid 2765	. . . . 5 ⊢ ∅ = ∅
11		mo0 49443	. . . . 5 ⊢ (∅ = ∅ → ∃*𝑖 𝑖 ∈ ∅)
12	10, 11	mp1i 14	. . . 4 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ ¬ 𝑥 = 𝑦) → ∃*𝑖 𝑖 ∈ ∅)
13	4, 6, 9, 12	ifbothda 4522	. . 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 2851	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑖 ∈ (𝑥𝐻𝑦) ↔ 𝑖 ∈ if(𝑥 = 𝑦, {𝐼}, ∅)))
16	15	mobidv 2579	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (∃*𝑖 𝑖 ∈ (𝑥𝐻𝑦) ↔ ∃*𝑖 𝑖 ∈ if(𝑥 = 𝑦, {𝐼}, ∅)))
17	13, 16	mpbird 260	. 2 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ∃*𝑖 𝑖 ∈ (𝑥𝐻𝑦))
18		indcthing.c	. 2 ⊢ (𝜑 → 𝐶 ∈ Cat)
19	1, 2, 17, 18	isthincd 50065	1 ⊢ (𝜑 → 𝐶 ∈ ThinCat)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 400   = wceq 1563   ∈ wcel 2145  ∃*wmo 2567  ∅c0 4288  ifcif 4483  {csn 4585  ‘cfv 6525  (class class class)co 7400  Basecbs 17259  Hom chom 17311  Catccat 17710  ThinCatcthinc 50046

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-sep 5251  ax-nul 5261

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3080  df-rex 3090  df-rmo 3370  df-reu 3371  df-rab 3418  df-v 3459  df-sbc 3748  df-dif 3910  df-un 3912  df-ss 3924  df-nul 4289  df-if 4484  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4869  df-br 5106  df-iota 6481  df-fv 6533  df-ov 7403  df-thinc 50047

This theorem is referenced by: (None)