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Theorem thincid 47908
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 47900 . . 3 (𝜑𝐶 ∈ Cat)
83, 4, 5, 7, 1catidcl 17633 . 2 (𝜑 → ( 1𝑋) ∈ (𝑋𝐻𝑋))
91, 1, 2, 8, 3, 4, 6thincmo2 47903 1 (𝜑𝐹 = ( 1𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1533  wcel 2098  cfv 6536  (class class class)co 7404  Basecbs 17151  Hom chom 17215  Idccid 17616  ThinCatcthinc 47894
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rmo 3370  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6488  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7360  df-ov 7407  df-cat 17619  df-cid 17620  df-thinc 47895
This theorem is referenced by:  functhinclem4  47919  thincsect  47932
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