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Theorem zerooval 17130
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 17124 . 2 ZeroO = (𝑐 ∈ Cat ↦ ((InitO‘𝑐) ∩ (TermO‘𝑐)))
2 fveq2 6497 . . 3 (𝑐 = 𝐶 → (InitO‘𝑐) = (InitO‘𝐶))
3 fveq2 6497 . . 3 (𝑐 = 𝐶 → (TermO‘𝑐) = (TermO‘𝐶))
42, 3ineq12d 4072 . 2 (𝑐 = 𝐶 → ((InitO‘𝑐) ∩ (TermO‘𝑐)) = ((InitO‘𝐶) ∩ (TermO‘𝐶)))
5 initoval.c . 2 (𝜑𝐶 ∈ Cat)
6 fvex 6510 . . . 4 (InitO‘𝐶) ∈ V
76inex1 5075 . . 3 ((InitO‘𝐶) ∩ (TermO‘𝐶)) ∈ V
87a1i 11 . 2 (𝜑 → ((InitO‘𝐶) ∩ (TermO‘𝐶)) ∈ V)
91, 4, 5, 8fvmptd3 6616 1 (𝜑 → (ZeroO‘𝐶) = ((InitO‘𝐶) ∩ (TermO‘𝐶)))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1508  wcel 2051  Vcvv 3410  cin 3823  cfv 6186  Basecbs 16338  Hom chom 16431  Catccat 16806  InitOcinito 17119  TermOctermo 17120  ZeroOczeroo 17121
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1759  ax-4 1773  ax-5 1870  ax-6 1929  ax-7 1966  ax-8 2053  ax-9 2060  ax-10 2080  ax-11 2094  ax-12 2107  ax-13 2302  ax-ext 2745  ax-sep 5057  ax-nul 5064  ax-pr 5183
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 835  df-3an 1071  df-tru 1511  df-ex 1744  df-nf 1748  df-sb 2017  df-mo 2548  df-eu 2585  df-clab 2754  df-cleq 2766  df-clel 2841  df-nfc 2913  df-ral 3088  df-rex 3089  df-rab 3092  df-v 3412  df-sbc 3677  df-dif 3827  df-un 3829  df-in 3831  df-ss 3838  df-nul 4174  df-if 4346  df-sn 4437  df-pr 4439  df-op 4443  df-uni 4710  df-br 4927  df-opab 4989  df-mpt 5006  df-id 5309  df-xp 5410  df-rel 5411  df-cnv 5412  df-co 5413  df-dm 5414  df-iota 6150  df-fun 6188  df-fv 6194  df-zeroo 17124
This theorem is referenced by:  iszeroo  17133  iszeroi  17140
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