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Theorem initoo 16428
Description: An initial object is an object. (Contributed by AV, 14-Apr-2020.)
Assertion
Ref Expression
initoo (𝐶 ∈ Cat → (𝑂 ∈ (InitO‘𝐶) → 𝑂 ∈ (Base‘𝐶)))

Proof of Theorem initoo
Dummy variables 𝑏 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2609 . . . 4 (Base‘𝐶) = (Base‘𝐶)
2 eqid 2609 . . . 4 (Hom ‘𝐶) = (Hom ‘𝐶)
3 id 22 . . . 4 (𝐶 ∈ Cat → 𝐶 ∈ Cat)
41, 2, 3isinitoi 16424 . . 3 ((𝐶 ∈ Cat ∧ 𝑂 ∈ (InitO‘𝐶)) → (𝑂 ∈ (Base‘𝐶) ∧ ∀𝑏 ∈ (Base‘𝐶)∃! ∈ (𝑂(Hom ‘𝐶)𝑏)))
54simpld 473 . 2 ((𝐶 ∈ Cat ∧ 𝑂 ∈ (InitO‘𝐶)) → 𝑂 ∈ (Base‘𝐶))
65ex 448 1 (𝐶 ∈ Cat → (𝑂 ∈ (InitO‘𝐶) → 𝑂 ∈ (Base‘𝐶)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382  wcel 1976  ∃!weu 2457  wral 2895  cfv 5789  (class class class)co 6526  Basecbs 15643  Hom chom 15727  Catccat 16096  InitOcinito 16409
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-sep 4703  ax-nul 4711  ax-pr 4827
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ral 2900  df-rex 2901  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-sn 4125  df-pr 4127  df-op 4131  df-uni 4367  df-br 4578  df-opab 4638  df-mpt 4639  df-id 4942  df-xp 5033  df-rel 5034  df-cnv 5035  df-co 5036  df-dm 5037  df-iota 5753  df-fun 5791  df-fv 5797  df-ov 6529  df-inito 16412
This theorem is referenced by:  iszeroi  16430  2initoinv  16431  initoeu1w  16433
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