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Theorem termoo 17126
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 2779 . . . 4 (Base‘𝐶) = (Base‘𝐶)
2 eqid 2779 . . . 4 (Hom ‘𝐶) = (Hom ‘𝐶)
3 id 22 . . . 4 (𝐶 ∈ Cat → 𝐶 ∈ Cat)
41, 2, 3istermoi 17122 . . 3 ((𝐶 ∈ Cat ∧ 𝑂 ∈ (TermO‘𝐶)) → (𝑂 ∈ (Base‘𝐶) ∧ ∀𝑏 ∈ (Base‘𝐶)∃! ∈ (𝑏(Hom ‘𝐶)𝑂)))
54simpld 487 . 2 ((𝐶 ∈ Cat ∧ 𝑂 ∈ (TermO‘𝐶)) → 𝑂 ∈ (Base‘𝐶))
65ex 405 1 (𝐶 ∈ Cat → (𝑂 ∈ (TermO‘𝐶) → 𝑂 ∈ (Base‘𝐶)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 387  wcel 2050  ∃!weu 2583  wral 3089  cfv 6188  (class class class)co 6976  Basecbs 16339  Hom chom 16432  Catccat 16793  TermOctermo 17107
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2751  ax-sep 5060  ax-nul 5067  ax-pr 5186
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2584  df-clab 2760  df-cleq 2772  df-clel 2847  df-nfc 2919  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3418  df-sbc 3683  df-dif 3833  df-un 3835  df-in 3837  df-ss 3844  df-nul 4180  df-if 4351  df-sn 4442  df-pr 4444  df-op 4448  df-uni 4713  df-br 4930  df-opab 4992  df-mpt 5009  df-id 5312  df-xp 5413  df-rel 5414  df-cnv 5415  df-co 5416  df-dm 5417  df-iota 6152  df-fun 6190  df-fv 6196  df-ov 6979  df-termo 17110
This theorem is referenced by:  2termoinv  17135  termoeu1w  17137
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