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Theorem iszeroo 17262
Description: The predicate "is a zero object" of a category. (Contributed by AV, 3-Apr-2020.)
Hypotheses
Ref Expression
isinito.b 𝐵 = (Base‘𝐶)
isinito.h 𝐻 = (Hom ‘𝐶)
isinito.c (𝜑𝐶 ∈ Cat)
isinito.i (𝜑𝐼𝐵)
Assertion
Ref Expression
iszeroo (𝜑 → (𝐼 ∈ (ZeroO‘𝐶) ↔ (𝐼 ∈ (InitO‘𝐶) ∧ 𝐼 ∈ (TermO‘𝐶))))

Proof of Theorem iszeroo
StepHypRef Expression
1 isinito.c . . . 4 (𝜑𝐶 ∈ Cat)
2 isinito.b . . . 4 𝐵 = (Base‘𝐶)
3 isinito.h . . . 4 𝐻 = (Hom ‘𝐶)
41, 2, 3zerooval 17259 . . 3 (𝜑 → (ZeroO‘𝐶) = ((InitO‘𝐶) ∩ (TermO‘𝐶)))
54eleq2d 2901 . 2 (𝜑 → (𝐼 ∈ (ZeroO‘𝐶) ↔ 𝐼 ∈ ((InitO‘𝐶) ∩ (TermO‘𝐶))))
6 elin 3935 . 2 (𝐼 ∈ ((InitO‘𝐶) ∩ (TermO‘𝐶)) ↔ (𝐼 ∈ (InitO‘𝐶) ∧ 𝐼 ∈ (TermO‘𝐶)))
75, 6syl6bb 290 1 (𝜑 → (𝐼 ∈ (ZeroO‘𝐶) ↔ (𝐼 ∈ (InitO‘𝐶) ∧ 𝐼 ∈ (TermO‘𝐶))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1538  wcel 2115  cin 3918  cfv 6343  Basecbs 16483  Hom chom 16576  Catccat 16935  InitOcinito 17248  TermOctermo 17249  ZeroOczeroo 17250
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pr 5317
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 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-sbc 3759  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-br 5053  df-opab 5115  df-mpt 5133  df-id 5447  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-iota 6302  df-fun 6345  df-fv 6351  df-zeroo 17253
This theorem is referenced by:  iszeroi  17269  zrzeroorngc  44557
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