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Theorem termoo 17260
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 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2798 . . . 4 (Base‘𝐶) = (Base‘𝐶)
2 eqid 2798 . . . 4 (Hom ‘𝐶) = (Hom ‘𝐶)
3 id 22 . . . 4 (𝐶 ∈ Cat → 𝐶 ∈ Cat)
41, 2, 3istermoi 17256 . . 3 ((𝐶 ∈ Cat ∧ 𝑂 ∈ (TermO‘𝐶)) → (𝑂 ∈ (Base‘𝐶) ∧ ∀𝑏 ∈ (Base‘𝐶)∃! ∈ (𝑏(Hom ‘𝐶)𝑂)))
54simpld 498 . 2 ((𝐶 ∈ Cat ∧ 𝑂 ∈ (TermO‘𝐶)) → 𝑂 ∈ (Base‘𝐶))
65ex 416 1 (𝐶 ∈ Cat → (𝑂 ∈ (TermO‘𝐶) → 𝑂 ∈ (Base‘𝐶)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  wcel 2111  ∃!weu 2628  wral 3106  cfv 6324  (class class class)co 7135  Basecbs 16475  Hom chom 16568  Catccat 16927  TermOctermo 17241
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pr 5295
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-iota 6283  df-fun 6326  df-fv 6332  df-ov 7138  df-termo 17244
This theorem is referenced by:  2termoinv  17269  termoeu1w  17271
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