Theorem thincid 49543

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 49534	. . 3 ⊢ (𝜑 → 𝐶 ∈ Cat)
8	3, 4, 5, 7, 1	catidcl 17588	. 2 ⊢ (𝜑 → ( 1 ‘𝑋) ∈ (𝑋𝐻𝑋))
9	1, 1, 2, 8, 3, 4, 6	thincmo2 49537	1 ⊢ (𝜑 → 𝐹 = ( 1 ‘𝑋))

Syntax hints:   → wi 4   = wceq 1541   ∈ wcel 2111  ‘cfv 6481  (class class class)co 7346  Basecbs 17120  Hom chom 17172  Idccid 17571  ThinCatcthinc 49528

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5215  ax-sep 5232  ax-nul 5242  ax-pr 5368

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rmo 3346  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4281  df-if 4473  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-iun 4941  df-br 5090  df-opab 5152  df-mpt 5171  df-id 5509  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-riota 7303  df-ov 7349  df-cat 17574  df-cid 17575  df-thinc 49529

This theorem is referenced by:  functhinclem4  49558  thincsect  49578  termcid  49597