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Theorem thincid 46948
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 46940 . . 3 (𝜑𝐶 ∈ Cat)
83, 4, 5, 7, 1catidcl 17522 . 2 (𝜑 → ( 1𝑋) ∈ (𝑋𝐻𝑋))
91, 1, 2, 8, 3, 4, 6thincmo2 46943 1 (𝜑𝐹 = ( 1𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1542  wcel 2107  cfv 6494  (class class class)co 7352  Basecbs 17043  Hom chom 17104  Idccid 17505  ThinCatcthinc 46934
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2709  ax-rep 5241  ax-sep 5255  ax-nul 5262  ax-pr 5383
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2888  df-ne 2943  df-ral 3064  df-rex 3073  df-rmo 3352  df-reu 3353  df-rab 3407  df-v 3446  df-sbc 3739  df-csb 3855  df-dif 3912  df-un 3914  df-in 3916  df-ss 3926  df-nul 4282  df-if 4486  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4865  df-iun 4955  df-br 5105  df-opab 5167  df-mpt 5188  df-id 5530  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-iota 6446  df-fun 6496  df-fn 6497  df-f 6498  df-f1 6499  df-fo 6500  df-f1o 6501  df-fv 6502  df-riota 7308  df-ov 7355  df-cat 17508  df-cid 17509  df-thinc 46935
This theorem is referenced by:  functhinclem4  46959  thincsect  46972
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