Theorem thincid 48354

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 48346	. . 3 ⊢ (𝜑 → 𝐶 ∈ Cat)
8	3, 4, 5, 7, 1	catidcl 17695	. 2 ⊢ (𝜑 → ( 1 ‘𝑋) ∈ (𝑋𝐻𝑋))
9	1, 1, 2, 8, 3, 4, 6	thincmo2 48349	1 ⊢ (𝜑 → 𝐹 = ( 1 ‘𝑋))

Syntax hints:   → wi 4   = wceq 1534   ∈ wcel 2099  ‘cfv 6554  (class class class)co 7424  Basecbs 17213  Hom chom 17277  Idccid 17678  ThinCatcthinc 48340

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-rep 5290  ax-sep 5304  ax-nul 5311  ax-pr 5433

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-rmo 3364  df-reu 3365  df-rab 3420  df-v 3464  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4326  df-if 4534  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4914  df-iun 5003  df-br 5154  df-opab 5216  df-mpt 5237  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6506  df-fun 6556  df-fn 6557  df-f 6558  df-f1 6559  df-fo 6560  df-f1o 6561  df-fv 6562  df-riota 7380  df-ov 7427  df-cat 17681  df-cid 17682  df-thinc 48341

This theorem is referenced by:  functhinclem4  48365  thincsect  48378