Theorem thincid 50053

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 50044	. . 3 ⊢ (𝜑 → 𝐶 ∈ Cat)
8	3, 4, 5, 7, 1	catidcl 17714	. 2 ⊢ (𝜑 → ( 1 ‘𝑋) ∈ (𝑋𝐻𝑋))
9	1, 1, 2, 8, 3, 4, 6	thincmo2 50047	1 ⊢ (𝜑 → 𝐹 = ( 1 ‘𝑋))

Syntax hints:   → wi 4   = wceq 1560   ∈ wcel 2142  ‘cfv 6521  (class class class)co 7396  Basecbs 17245  Hom chom 17297  Idccid 17697  ThinCatcthinc 50038

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-rep 5227  ax-sep 5246  ax-nul 5256  ax-pr 5390

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-nf 1804  df-sb 2091  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3077  df-rex 3087  df-rmo 3367  df-reu 3368  df-rab 3415  df-v 3456  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4481  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4951  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5542  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-iota 6477  df-fun 6523  df-fn 6524  df-f 6525  df-f1 6526  df-fo 6527  df-f1o 6528  df-fv 6529  df-riota 7353  df-ov 7399  df-cat 17700  df-cid 17701  df-thinc 50039

This theorem is referenced by:  functhinclem4  50068  thincsect  50088  termcid  50107