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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 (𝜑𝐶 ∈ ThinCat)
thincid.b 𝐵 = (Base‘𝐶)
thincid.h 𝐻 = (Hom ‘𝐶)
thincid.x (𝜑𝑋𝐵)
thincid.i 1 = (Id‘𝐶)
thincid.f (𝜑𝐹 ∈ (𝑋𝐻𝑋))
Assertion
Ref Expression
thincid (𝜑𝐹 = ( 1𝑋))

Proof of Theorem thincid
StepHypRef Expression
1 thincid.x . 2 (𝜑𝑋𝐵)
2 thincid.f . 2 (𝜑𝐹 ∈ (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 (𝜑𝐶 ∈ ThinCat)
76thinccd 49073 . . 3 (𝜑𝐶 ∈ Cat)
83, 4, 5, 7, 1catidcl 17725 . 2 (𝜑 → ( 1𝑋) ∈ (𝑋𝐻𝑋))
91, 1, 2, 8, 3, 4, 6thincmo2 49076 1 (𝜑𝐹 = ( 1𝑋))
Colors of variables: wff setvar class
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
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