Theorem termoo 18054

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 2736	. . . 4 ⊢ (Base‘𝐶) = (Base‘𝐶)
2		eqid 2736	. . . 4 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
3		id 22	. . . 4 ⊢ (𝐶 ∈ Cat → 𝐶 ∈ Cat)
4	1, 2, 3	istermoi 18046	. . 3 ⊢ ((𝐶 ∈ Cat ∧ 𝑂 ∈ (TermO‘𝐶)) → (𝑂 ∈ (Base‘𝐶) ∧ ∀𝑏 ∈ (Base‘𝐶)∃!ℎ ℎ ∈ (𝑏(Hom ‘𝐶)𝑂)))
5	4	simpld 494	. 2 ⊢ ((𝐶 ∈ Cat ∧ 𝑂 ∈ (TermO‘𝐶)) → 𝑂 ∈ (Base‘𝐶))
6	5	ex 412	1 ⊢ (𝐶 ∈ Cat → (𝑂 ∈ (TermO‘𝐶) → 𝑂 ∈ (Base‘𝐶)))

Syntax hints:   → wi 4   ∧ wa 395   ∈ wcel 2107  ∃!weu 2567  ∀wral 3060  ‘cfv 6560  (class class class)co 7432  Basecbs 17248  Hom chom 17309  Catccat 17708  TermOctermo 18028

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 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431

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 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-iota 6513  df-fun 6562  df-fv 6568  df-ov 7435  df-termo 18031

This theorem is referenced by:  2termoinv  18063  termoeu1w  18065