Theorem termoo2 50494

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 2763	. . . 4 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
3		termorcl 18178	. . . 4 ⊢ (𝑂 ∈ (TermO‘𝐶) → 𝐶 ∈ Cat)
4	1, 2, 3	istermoi 18187	. . 3 ⊢ ((𝑂 ∈ (TermO‘𝐶) ∧ 𝑂 ∈ (TermO‘𝐶)) → (𝑂 ∈ 𝐵 ∧ ∀𝑏 ∈ 𝐵 ∃!ℎ ℎ ∈ (𝑏(Hom ‘𝐶)𝑂)))
5	4	anidms 577	. 2 ⊢ (𝑂 ∈ (TermO‘𝐶) → (𝑂 ∈ 𝐵 ∧ ∀𝑏 ∈ 𝐵 ∃!ℎ ℎ ∈ (𝑏(Hom ‘𝐶)𝑂)))
6	5	simpld 500	1 ⊢ (𝑂 ∈ (TermO‘𝐶) → 𝑂 ∈ 𝐵)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1569   ∈ wcel 2146  ∃!weu 2596  ∀wral 3079  ‘cfv 6546  (class class class)co 7427  Basecbs 17399  Hom chom 17451  TermOctermo 18169

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2196  ax-12 2215  ax-ext 2735  ax-sep 5251  ax-nul 5262  ax-pr 5394

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2101  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3416  df-v 3455  df-ss 3900  df-dif 4049  df-in 4093  df-un 4153  df-nul 4281  df-if 4485  df-pw 4561  df-sn 4588  df-pr 4590  df-op 4838  df-uni 4870  df-br 5106  df-opab 5170  df-mpt 5189  df-id 5549  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-iota 6500  df-fun 6548  df-fv 6554  df-ov 7430  df-termo 18172

This theorem is used by:  oppcinito  50496  termcterm2  50788