Theorem termoo2 48984

Description: A terminal object is an object in the base set. (Contributed by Zhi Wang, 23-Oct-2025.)

Hypothesis

Ref	Expression
initoo2.b	⊢ 𝐵 = (Base‘𝐶)

Assertion

Ref	Expression
termoo2	⊢ (𝑂 ∈ (TermO‘𝐶) → 𝑂 ∈ 𝐵)

Proof of Theorem termoo2

Dummy variables 𝑏 ℎ are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		initoo2.b	. . . 4 ⊢ 𝐵 = (Base‘𝐶)
2		eqid 2734	. . . 4 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
3		termorcl 18008	. . . 4 ⊢ (𝑂 ∈ (TermO‘𝐶) → 𝐶 ∈ Cat)
4	1, 2, 3	istermoi 18017	. . 3 ⊢ ((𝑂 ∈ (TermO‘𝐶) ∧ 𝑂 ∈ (TermO‘𝐶)) → (𝑂 ∈ 𝐵 ∧ ∀𝑏 ∈ 𝐵 ∃!ℎ ℎ ∈ (𝑏(Hom ‘𝐶)𝑂)))
5	4	anidms 566	. 2 ⊢ (𝑂 ∈ (TermO‘𝐶) → (𝑂 ∈ 𝐵 ∧ ∀𝑏 ∈ 𝐵 ∃!ℎ ℎ ∈ (𝑏(Hom ‘𝐶)𝑂)))
6	5	simpld 494	1 ⊢ (𝑂 ∈ (TermO‘𝐶) → 𝑂 ∈ 𝐵)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1539   ∈ wcel 2107  ∃!weu 2566  ∀wral 3050  ‘cfv 6541  (class class class)co 7413  Basecbs 17230  Hom chom 17285  TermOctermo 17999

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-sep 5276  ax-nul 5286  ax-pr 5412

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-rab 3420  df-v 3465  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4888  df-br 5124  df-opab 5186  df-mpt 5206  df-id 5558  df-xp 5671  df-rel 5672  df-cnv 5673  df-co 5674  df-dm 5675  df-rn 5676  df-res 5677  df-ima 5678  df-iota 6494  df-fun 6543  df-fv 6549  df-ov 7416  df-termo 18002

This theorem is referenced by:  oppcinito  48986  termcterm2  49212