Theorem discthing 49958

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 2736	. . . . 5 ⊢ ({𝐼} = if(𝑥 = 𝑦, {𝐼}, ∅) → (𝑖 ∈ {𝐼} ↔ 𝑖 ∈ if(𝑥 = 𝑦, {𝐼}, ∅)))
4	3	mobidv 2553	. . . 4 ⊢ ({𝐼} = if(𝑥 = 𝑦, {𝐼}, ∅) → (∃*𝑖 𝑖 ∈ {𝐼} ↔ ∃*𝑖 𝑖 ∈ if(𝑥 = 𝑦, {𝐼}, ∅)))
5		eleq2w2 2736	. . . . 5 ⊢ (∅ = if(𝑥 = 𝑦, {𝐼}, ∅) → (𝑖 ∈ ∅ ↔ 𝑖 ∈ if(𝑥 = 𝑦, {𝐼}, ∅)))
6	5	mobidv 2553	. . . 4 ⊢ (∅ = if(𝑥 = 𝑦, {𝐼}, ∅) → (∃*𝑖 𝑖 ∈ ∅ ↔ ∃*𝑖 𝑖 ∈ if(𝑥 = 𝑦, {𝐼}, ∅)))
7		eqid 2740	. . . . 5 ⊢ {𝐼} = {𝐼}
8		mosn 49310	. . . . 5 ⊢ ({𝐼} = {𝐼} → ∃*𝑖 𝑖 ∈ {𝐼})
9	7, 8	mp1i 13	. . . 4 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑥 = 𝑦) → ∃*𝑖 𝑖 ∈ {𝐼})
10		eqid 2740	. . . . 5 ⊢ ∅ = ∅
11		mo0 49311	. . . . 5 ⊢ (∅ = ∅ → ∃*𝑖 𝑖 ∈ ∅)
12	10, 11	mp1i 13	. . . 4 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ ¬ 𝑥 = 𝑦) → ∃*𝑖 𝑖 ∈ ∅)
13	4, 6, 9, 12	ifbothda 4500	. . 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 2826	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑖 ∈ (𝑥𝐻𝑦) ↔ 𝑖 ∈ if(𝑥 = 𝑦, {𝐼}, ∅)))
16	15	mobidv 2553	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (∃*𝑖 𝑖 ∈ (𝑥𝐻𝑦) ↔ ∃*𝑖 𝑖 ∈ if(𝑥 = 𝑦, {𝐼}, ∅)))
17	13, 16	mpbird 258	. 2 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ∃*𝑖 𝑖 ∈ (𝑥𝐻𝑦))
18		indcthing.c	. 2 ⊢ (𝜑 → 𝐶 ∈ Cat)
19	1, 2, 17, 18	isthincd 49933	1 ⊢ (𝜑 → 𝐶 ∈ ThinCat)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 396   = wceq 1547   ∈ wcel 2119  ∃*wmo 2541  ∅c0 4268  ifcif 4461  {csn 4562  ‘cfv 6492  (class class class)co 7363  Basecbs 17177  Hom chom 17229  Catccat 17628  ThinCatcthinc 49914

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2712  ax-sep 5225  ax-nul 5235

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2719  df-cleq 2732  df-clel 2815  df-nfc 2889  df-ne 2936  df-ral 3055  df-rex 3065  df-rmo 3345  df-reu 3346  df-rab 3393  df-v 3434  df-sbc 3731  df-dif 3893  df-un 3895  df-ss 3907  df-nul 4269  df-if 4462  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-br 5080  df-iota 6448  df-fv 6500  df-ov 7366  df-thinc 49915

This theorem is referenced by: (None)