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Theorem zerooval 18049
Description: The value of the zero object function, i.e. the set of all zero objects of a category. (Contributed by AV, 3-Apr-2020.)
Hypotheses
Ref Expression
initoval.c (𝜑𝐶 ∈ Cat)
initoval.b 𝐵 = (Base‘𝐶)
initoval.h 𝐻 = (Hom ‘𝐶)
Assertion
Ref Expression
zerooval (𝜑 → (ZeroO‘𝐶) = ((InitO‘𝐶) ∩ (TermO‘𝐶)))

Proof of Theorem zerooval
Dummy variable 𝑐 is distinct from all other variables.
StepHypRef Expression
1 df-zeroo 18040 . 2 ZeroO = (𝑐 ∈ Cat ↦ ((InitO‘𝑐) ∩ (TermO‘𝑐)))
2 fveq2 6907 . . 3 (𝑐 = 𝐶 → (InitO‘𝑐) = (InitO‘𝐶))
3 fveq2 6907 . . 3 (𝑐 = 𝐶 → (TermO‘𝑐) = (TermO‘𝐶))
42, 3ineq12d 4229 . 2 (𝑐 = 𝐶 → ((InitO‘𝑐) ∩ (TermO‘𝑐)) = ((InitO‘𝐶) ∩ (TermO‘𝐶)))
5 initoval.c . 2 (𝜑𝐶 ∈ Cat)
6 fvex 6920 . . . 4 (InitO‘𝐶) ∈ V
76inex1 5323 . . 3 ((InitO‘𝐶) ∩ (TermO‘𝐶)) ∈ V
87a1i 11 . 2 (𝜑 → ((InitO‘𝐶) ∩ (TermO‘𝐶)) ∈ V)
91, 4, 5, 8fvmptd3 7039 1 (𝜑 → (ZeroO‘𝐶) = ((InitO‘𝐶) ∩ (TermO‘𝐶)))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1537  wcel 2106  Vcvv 3478  cin 3962  cfv 6563  Basecbs 17245  Hom chom 17309  Catccat 17709  InitOcinito 18035  TermOctermo 18036  ZeroOczeroo 18037
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-iota 6516  df-fun 6565  df-fv 6571  df-zeroo 18040
This theorem is referenced by:  iszeroo  18052  iszeroi  18063
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