Theorem termoo2 49854

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 2762	. . . 4 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
3		termorcl 18024	. . . 4 ⊢ (𝑂 ∈ (TermO‘𝐶) → 𝐶 ∈ Cat)
4	1, 2, 3	istermoi 18033	. . 3 ⊢ ((𝑂 ∈ (TermO‘𝐶) ∧ 𝑂 ∈ (TermO‘𝐶)) → (𝑂 ∈ 𝐵 ∧ ∀𝑏 ∈ 𝐵 ∃!ℎ ℎ ∈ (𝑏(Hom ‘𝐶)𝑂)))
5	4	anidms 574	. 2 ⊢ (𝑂 ∈ (TermO‘𝐶) → (𝑂 ∈ 𝐵 ∧ ∀𝑏 ∈ 𝐵 ∃!ℎ ℎ ∈ (𝑏(Hom ‘𝐶)𝑂)))
6	5	simpld 498	1 ⊢ (𝑂 ∈ (TermO‘𝐶) → 𝑂 ∈ 𝐵)

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1560   ∈ wcel 2142  ∃!weu 2595  ∀wral 3076  ‘cfv 6521  (class class class)co 7396  Basecbs 17245  Hom chom 17297  TermOctermo 18015

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-sep 5246  ax-nul 5256  ax-pr 5390

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-nf 1804  df-sb 2091  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3077  df-rex 3087  df-rab 3415  df-v 3456  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4481  df-pw 4557  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5542  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-iota 6477  df-fun 6523  df-fv 6529  df-ov 7399  df-termo 18018

This theorem is referenced by:  oppcinito  49856  termcterm2  50135