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Theorem iszeroo 16417
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 16414 . . 3 (𝜑 → (ZeroO‘𝐶) = ((InitO‘𝐶) ∩ (TermO‘𝐶)))
54eleq2d 2668 . 2 (𝜑 → (𝐼 ∈ (ZeroO‘𝐶) ↔ 𝐼 ∈ ((InitO‘𝐶) ∩ (TermO‘𝐶))))
6 elin 3753 . 2 (𝐼 ∈ ((InitO‘𝐶) ∩ (TermO‘𝐶)) ↔ (𝐼 ∈ (InitO‘𝐶) ∧ 𝐼 ∈ (TermO‘𝐶)))
75, 6syl6bb 274 1 (𝜑 → (𝐼 ∈ (ZeroO‘𝐶) ↔ (𝐼 ∈ (InitO‘𝐶) ∧ 𝐼 ∈ (TermO‘𝐶))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wa 382   = wceq 1474  wcel 1975  cin 3534  cfv 5786  Basecbs 15637  Hom chom 15721  Catccat 16090  InitOcinito 16403  TermOctermo 16404  ZeroOczeroo 16405
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-9 1984  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2228  ax-ext 2585  ax-sep 4699  ax-nul 4708  ax-pr 4824
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 1866  df-eu 2457  df-mo 2458  df-clab 2592  df-cleq 2598  df-clel 2601  df-nfc 2735  df-ral 2896  df-rex 2897  df-rab 2900  df-v 3170  df-sbc 3398  df-csb 3495  df-dif 3538  df-un 3540  df-in 3542  df-ss 3549  df-nul 3870  df-if 4032  df-sn 4121  df-pr 4123  df-op 4127  df-uni 4363  df-br 4574  df-opab 4634  df-mpt 4635  df-id 4939  df-xp 5030  df-rel 5031  df-cnv 5032  df-co 5033  df-dm 5034  df-iota 5750  df-fun 5788  df-fv 5794  df-zeroo 16408
This theorem is referenced by:  iszeroi  16424  zrzeroorngc  41792
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