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Theorem zerooval 16850
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 16844 . . 3 ZeroO = (𝑐 ∈ Cat ↦ ((InitO‘𝑐) ∩ (TermO‘𝑐)))
21a1i 11 . 2 (𝜑 → ZeroO = (𝑐 ∈ Cat ↦ ((InitO‘𝑐) ∩ (TermO‘𝑐))))
3 fveq2 6352 . . . 4 (𝑐 = 𝐶 → (InitO‘𝑐) = (InitO‘𝐶))
4 fveq2 6352 . . . 4 (𝑐 = 𝐶 → (TermO‘𝑐) = (TermO‘𝐶))
53, 4ineq12d 3958 . . 3 (𝑐 = 𝐶 → ((InitO‘𝑐) ∩ (TermO‘𝑐)) = ((InitO‘𝐶) ∩ (TermO‘𝐶)))
65adantl 473 . 2 ((𝜑𝑐 = 𝐶) → ((InitO‘𝑐) ∩ (TermO‘𝑐)) = ((InitO‘𝐶) ∩ (TermO‘𝐶)))
7 initoval.c . 2 (𝜑𝐶 ∈ Cat)
8 fvex 6362 . . . 4 (InitO‘𝐶) ∈ V
98inex1 4951 . . 3 ((InitO‘𝐶) ∩ (TermO‘𝐶)) ∈ V
109a1i 11 . 2 (𝜑 → ((InitO‘𝐶) ∩ (TermO‘𝐶)) ∈ V)
112, 6, 7, 10fvmptd 6450 1 (𝜑 → (ZeroO‘𝐶) = ((InitO‘𝐶) ∩ (TermO‘𝐶)))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1632  wcel 2139  Vcvv 3340  cin 3714  cmpt 4881  cfv 6049  Basecbs 16059  Hom chom 16154  Catccat 16526  InitOcinito 16839  TermOctermo 16840  ZeroOczeroo 16841
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-iota 6012  df-fun 6051  df-fv 6057  df-zeroo 16844
This theorem is referenced by:  iszeroo  16853  iszeroi  16860
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