Theorem termcid 50116

Description: The morphism of a terminal category is an identity morphism. (Contributed by Zhi Wang, 16-Oct-2025.)

Hypotheses

Ref	Expression
termcbas.c	⊢ (𝜑 → 𝐶 ∈ TermCat)
termcbas.b	⊢ 𝐵 = (Base‘𝐶)
termcbasmo.x	⊢ (𝜑 → 𝑋 ∈ 𝐵)
termcbasmo.y	⊢ (𝜑 → 𝑌 ∈ 𝐵)
termcid.h	⊢ 𝐻 = (Hom ‘𝐶)
termcid.f	⊢ (𝜑 → 𝐹 ∈ (𝑋𝐻𝑌))
termcid.i	⊢ 1 = (Id‘𝐶)

Assertion

Ref	Expression
termcid	⊢ (𝜑 → 𝐹 = ( 1 ‘𝑋))

Proof of Theorem termcid

Step	Hyp	Ref	Expression
1		termcbas.c	. . 3 ⊢ (𝜑 → 𝐶 ∈ TermCat)
2	1	termcthind 50108	. 2 ⊢ (𝜑 → 𝐶 ∈ ThinCat)
3		termcbas.b	. 2 ⊢ 𝐵 = (Base‘𝐶)
4		termcid.h	. 2 ⊢ 𝐻 = (Hom ‘𝐶)
5		termcbasmo.x	. 2 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
6		termcid.i	. 2 ⊢ 1 = (Id‘𝐶)
7		termcid.f	. . 3 ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐻𝑌))
8		termcbasmo.y	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
9	1, 3, 5, 8	termcbasmo 50113	. . . 4 ⊢ (𝜑 → 𝑋 = 𝑌)
10	9	oveq2d 7416	. . 3 ⊢ (𝜑 → (𝑋𝐻𝑋) = (𝑋𝐻𝑌))
11	7, 10	eleqtrrd 2868	. 2 ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐻𝑋))
12	2, 3, 4, 5, 6, 11	thincid 50062	1 ⊢ (𝜑 → 𝐹 = ( 1 ‘𝑋))

Syntax hints:   → wi 4   = wceq 1563   ∈ wcel 2145  ‘cfv 6525  (class class class)co 7400  Basecbs 17257  Hom chom 17309  Idccid 17709  TermCatctermc 50102

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-rep 5231  ax-sep 5250  ax-nul 5260  ax-pr 5394

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3080  df-rex 3090  df-rmo 3370  df-reu 3371  df-rab 3418  df-v 3459  df-sbc 3748  df-csb 3856  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-iun 4953  df-br 5105  df-opab 5167  df-mpt 5186  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-iota 6481  df-fun 6527  df-fn 6528  df-f 6529  df-f1 6530  df-fo 6531  df-f1o 6532  df-fv 6533  df-riota 7357  df-ov 7403  df-cat 17712  df-cid 17713  df-thinc 50048  df-termc 50103

This theorem is referenced by:  termcid2  50117  termchom  50118