Theorem termoo 18062

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 2735	. . . 4 ⊢ (Base‘𝐶) = (Base‘𝐶)
2		eqid 2735	. . . 4 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
3		id 22	. . . 4 ⊢ (𝐶 ∈ Cat → 𝐶 ∈ Cat)
4	1, 2, 3	istermoi 18054	. . 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 2106  ∃!weu 2566  ∀wral 3059  ‘cfv 6563  (class class class)co 7431  Basecbs 17245  Hom chom 17309  Catccat 17709  TermOctermo 18036

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-iota 6516  df-fun 6565  df-fv 6571  df-ov 7434  df-termo 18039

This theorem is referenced by:  2termoinv  18071  termoeu1w  18073