Theorem termchom2 49235

Description: The hom-set of a terminal category is a singleton of the identity morphism. (Contributed by Zhi Wang, 21-Oct-2025.)

Hypotheses

Ref	Expression
termchom.c	⊢ (𝜑 → 𝐶 ∈ TermCat)
termchom.b	⊢ 𝐵 = (Base‘𝐶)
termchom.x	⊢ (𝜑 → 𝑋 ∈ 𝐵)
termchom.y	⊢ (𝜑 → 𝑌 ∈ 𝐵)
termchom.h	⊢ 𝐻 = (Hom ‘𝐶)
termchom.i	⊢ 1 = (Id‘𝐶)
termchom2.z	⊢ (𝜑 → 𝑍 ∈ 𝐵)

Assertion

Ref	Expression
termchom2	⊢ (𝜑 → (𝑋𝐻𝑌) = {( 1 ‘𝑍)})

Proof of Theorem termchom2

Step	Hyp	Ref	Expression
1		termchom.c	. . 3 ⊢ (𝜑 → 𝐶 ∈ TermCat)
2		termchom.b	. . 3 ⊢ 𝐵 = (Base‘𝐶)
3		termchom.x	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
4		termchom.y	. . 3 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
5		termchom.h	. . 3 ⊢ 𝐻 = (Hom ‘𝐶)
6		termchom.i	. . 3 ⊢ 1 = (Id‘𝐶)
7	1, 2, 3, 4, 5, 6	termchom 49234	. 2 ⊢ (𝜑 → (𝑋𝐻𝑌) = {( 1 ‘𝑋)})
8		termchom2.z	. . . . 5 ⊢ (𝜑 → 𝑍 ∈ 𝐵)
9	1, 2, 3, 8	termcbasmo 49229	. . . 4 ⊢ (𝜑 → 𝑋 = 𝑍)
10	9	fveq2d 6877	. . 3 ⊢ (𝜑 → ( 1 ‘𝑋) = ( 1 ‘𝑍))
11	10	sneqd 4611	. 2 ⊢ (𝜑 → {( 1 ‘𝑋)} = {( 1 ‘𝑍)})
12	7, 11	eqtrd 2769	1 ⊢ (𝜑 → (𝑋𝐻𝑌) = {( 1 ‘𝑍)})

Syntax hints:   → wi 4   = wceq 1539   ∈ wcel 2107  {csn 4599  ‘cfv 6528  (class class class)co 7400  Basecbs 17215  Hom chom 17269  Idccid 17664  TermCatctermc 49219

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2706  ax-rep 5247  ax-sep 5264  ax-nul 5274  ax-pr 5400

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2808  df-nfc 2884  df-ne 2932  df-ral 3051  df-rex 3060  df-rmo 3357  df-reu 3358  df-rab 3414  df-v 3459  df-sbc 3764  df-csb 3873  df-dif 3927  df-un 3929  df-in 3931  df-ss 3941  df-nul 4307  df-if 4499  df-sn 4600  df-pr 4602  df-op 4606  df-uni 4882  df-iun 4967  df-br 5118  df-opab 5180  df-mpt 5200  df-id 5546  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-rn 5663  df-res 5664  df-ima 5665  df-iota 6481  df-fun 6530  df-fn 6531  df-f 6532  df-f1 6533  df-fo 6534  df-f1o 6535  df-fv 6536  df-riota 7357  df-ov 7403  df-cat 17667  df-cid 17668  df-thinc 49167  df-termc 49220

This theorem is referenced by:  diag1f1olem  49279