Theorem thincid 50094

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 50085	. . 3 ⊢ (𝜑 → 𝐶 ∈ Cat)
8	3, 4, 5, 7, 1	catidcl 17737	. 2 ⊢ (𝜑 → ( 1 ‘𝑋) ∈ (𝑋𝐻𝑋))
9	1, 1, 2, 8, 3, 4, 6	thincmo2 50088	1 ⊢ (𝜑 → 𝐹 = ( 1 ‘𝑋))

Syntax hints:   → wi 4   = wceq 1567   ∈ wcel 2149  ‘cfv 6537  (class class class)co 7411  Basecbs 17268  Hom chom 17320  Idccid 17720  ThinCatcthinc 50079

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5242  ax-sep 5261  ax-nul 5271  ax-pr 5405

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-rmo 3376  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-iun 4962  df-br 5114  df-opab 5178  df-mpt 5197  df-id 5557  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7368  df-ov 7414  df-cat 17723  df-cid 17724  df-thinc 50080

This theorem is referenced by:  functhinclem4  50109  thincsect  50129  termcid  50148