Theorem termchom2 49958

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 49957	. 2 ⊢ (𝜑 → (𝑋𝐻𝑌) = {( 1 ‘𝑋)})
8		termchom2.z	. . . . 5 ⊢ (𝜑 → 𝑍 ∈ 𝐵)
9	1, 2, 3, 8	termcbasmo 49952	. . . 4 ⊢ (𝜑 → 𝑋 = 𝑍)
10	9	fveq2d 6845	. . 3 ⊢ (𝜑 → ( 1 ‘𝑋) = ( 1 ‘𝑍))
11	10	sneqd 4580	. 2 ⊢ (𝜑 → {( 1 ‘𝑋)} = {( 1 ‘𝑍)})
12	7, 11	eqtrd 2772	1 ⊢ (𝜑 → (𝑋𝐻𝑌) = {( 1 ‘𝑍)})

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2114  {csn 4568  ‘cfv 6499  (class class class)co 7367  Basecbs 17179  Hom chom 17231  Idccid 17631  TermCatctermc 49941

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5213  ax-sep 5232  ax-nul 5242  ax-pr 5376

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rmo 3343  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6455  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-riota 7324  df-ov 7370  df-cat 17634  df-cid 17635  df-thinc 49887  df-termc 49942

This theorem is referenced by:  diag1f1olem  50002