Theorem termoo2 49896

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 2769	. . . 4 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
3		termorcl 18048	. . . 4 ⊢ (𝑂 ∈ (TermO‘𝐶) → 𝐶 ∈ Cat)
4	1, 2, 3	istermoi 18057	. . 3 ⊢ ((𝑂 ∈ (TermO‘𝐶) ∧ 𝑂 ∈ (TermO‘𝐶)) → (𝑂 ∈ 𝐵 ∧ ∀𝑏 ∈ 𝐵 ∃!ℎ ℎ ∈ (𝑏(Hom ‘𝐶)𝑂)))
5	4	anidms 576	. 2 ⊢ (𝑂 ∈ (TermO‘𝐶) → (𝑂 ∈ 𝐵 ∧ ∀𝑏 ∈ 𝐵 ∃!ℎ ℎ ∈ (𝑏(Hom ‘𝐶)𝑂)))
6	5	simpld 499	1 ⊢ (𝑂 ∈ (TermO‘𝐶) → 𝑂 ∈ 𝐵)

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1567   ∈ wcel 2149  ∃!weu 2602  ∀wral 3085  ‘cfv 6537  (class class class)co 7411  Basecbs 17269  Hom chom 17321  TermOctermo 18039

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5261  ax-nul 5271  ax-pr 5405

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-br 5114  df-opab 5178  df-mpt 5197  df-id 5557  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-iota 6493  df-fun 6539  df-fv 6545  df-ov 7414  df-termo 18042

This theorem is referenced by:  oppcinito  49898  termcterm2  50177