Theorem termoo 18100

Description: A terminal object is an object. (Contributed by AV, 18-Apr-2020.)

Assertion

Ref	Expression
termoo	⊢ (𝐶 ∈ Cat → (𝑂 ∈ (TermO‘𝐶) → 𝑂 ∈ (Base‘𝐶)))

Proof of Theorem termoo

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

Step	Hyp	Ref	Expression
1		eqid 2760	. . . 4 ⊢ (Base‘𝐶) = (Base‘𝐶)
2		eqid 2760	. . . 4 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
3		id 23	. . . 4 ⊢ (𝐶 ∈ Cat → 𝐶 ∈ Cat)
4	1, 2, 3	istermoi 18092	. . 3 ⊢ ((𝐶 ∈ Cat ∧ 𝑂 ∈ (TermO‘𝐶)) → (𝑂 ∈ (Base‘𝐶) ∧ ∀𝑏 ∈ (Base‘𝐶)∃!ℎ ℎ ∈ (𝑏(Hom ‘𝐶)𝑂)))
5	4	simpld 500	. 2 ⊢ ((𝐶 ∈ Cat ∧ 𝑂 ∈ (TermO‘𝐶)) → 𝑂 ∈ (Base‘𝐶))
6	5	ex 418	1 ⊢ (𝐶 ∈ Cat → (𝑂 ∈ (TermO‘𝐶) → 𝑂 ∈ (Base‘𝐶)))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∈ wcel 2145  ∃!weu 2593  ∀wral 3076  ‘cfv 6533  (class class class)co 7414  Basecbs 17304  Hom chom 17356  Catccat 17755  TermOctermo 18074

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-sep 5251  ax-nul 5263  ax-pr 5398

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-iota 6489  df-fun 6535  df-fv 6541  df-ov 7417  df-termo 18077

This theorem is used by:  2termoinv  18109  termoeu1w  18111