Theorem thincid 49081

Description: In a thin category, a morphism from an object to itself is an identity morphism. (Contributed by Zhi Wang, 24-Sep-2024.)

Hypotheses

Ref	Expression
thincid.c	⊢ (𝜑 → 𝐶 ∈ ThinCat)
thincid.b	⊢ 𝐵 = (Base‘𝐶)
thincid.h	⊢ 𝐻 = (Hom ‘𝐶)
thincid.x	⊢ (𝜑 → 𝑋 ∈ 𝐵)
thincid.i	⊢ 1 = (Id‘𝐶)
thincid.f	⊢ (𝜑 → 𝐹 ∈ (𝑋𝐻𝑋))

Assertion

Ref	Expression
thincid	⊢ (𝜑 → 𝐹 = ( 1 ‘𝑋))

Proof of Theorem thincid

Step	Hyp	Ref	Expression
1		thincid.x	. 2 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
2		thincid.f	. 2 ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐻𝑋))
3		thincid.b	. . 3 ⊢ 𝐵 = (Base‘𝐶)
4		thincid.h	. . 3 ⊢ 𝐻 = (Hom ‘𝐶)
5		thincid.i	. . 3 ⊢ 1 = (Id‘𝐶)
6		thincid.c	. . . 4 ⊢ (𝜑 → 𝐶 ∈ ThinCat)
7	6	thinccd 49073	. . 3 ⊢ (𝜑 → 𝐶 ∈ Cat)
8	3, 4, 5, 7, 1	catidcl 17725	. 2 ⊢ (𝜑 → ( 1 ‘𝑋) ∈ (𝑋𝐻𝑋))
9	1, 1, 2, 8, 3, 4, 6	thincmo2 49076	1 ⊢ (𝜑 → 𝐹 = ( 1 ‘𝑋))

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2108  ‘cfv 6561  (class class class)co 7431  Basecbs 17247  Hom chom 17308  Idccid 17708  ThinCatcthinc 49067

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 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pr 5432

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-cat 17711  df-cid 17712  df-thinc 49068

This theorem is referenced by:  functhinclem4  49096  thincsect  49114  termcid  49131