Theorem termcid 49475

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 49467	. 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 49472	. . . 4 ⊢ (𝜑 → 𝑋 = 𝑌)
10	9	oveq2d 7403	. . 3 ⊢ (𝜑 → (𝑋𝐻𝑋) = (𝑋𝐻𝑌))
11	7, 10	eleqtrrd 2831	. 2 ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐻𝑋))
12	2, 3, 4, 5, 6, 11	thincid 49421	1 ⊢ (𝜑 → 𝐹 = ( 1 ‘𝑋))

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2109  ‘cfv 6511  (class class class)co 7387  Basecbs 17179  Hom chom 17231  Idccid 17626  TermCatctermc 49461

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5234  ax-sep 5251  ax-nul 5261  ax-pr 5387

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rmo 3354  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-id 5533  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-riota 7344  df-ov 7390  df-cat 17629  df-cid 17630  df-thinc 49407  df-termc 49462

This theorem is referenced by:  termcid2  49476  termchom  49477