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Theorem zerooval 17251
 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 17245 . 2 ZeroO = (𝑐 ∈ Cat ↦ ((InitO‘𝑐) ∩ (TermO‘𝑐)))
2 fveq2 6645 . . 3 (𝑐 = 𝐶 → (InitO‘𝑐) = (InitO‘𝐶))
3 fveq2 6645 . . 3 (𝑐 = 𝐶 → (TermO‘𝑐) = (TermO‘𝐶))
42, 3ineq12d 4140 . 2 (𝑐 = 𝐶 → ((InitO‘𝑐) ∩ (TermO‘𝑐)) = ((InitO‘𝐶) ∩ (TermO‘𝐶)))
5 initoval.c . 2 (𝜑𝐶 ∈ Cat)
6 fvex 6658 . . . 4 (InitO‘𝐶) ∈ V
76inex1 5185 . . 3 ((InitO‘𝐶) ∩ (TermO‘𝐶)) ∈ V
87a1i 11 . 2 (𝜑 → ((InitO‘𝐶) ∩ (TermO‘𝐶)) ∈ V)
91, 4, 5, 8fvmptd3 6768 1 (𝜑 → (ZeroO‘𝐶) = ((InitO‘𝐶) ∩ (TermO‘𝐶)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1538   ∈ wcel 2111  Vcvv 3441   ∩ cin 3880  ‘cfv 6324  Basecbs 16475  Hom chom 16568  Catccat 16927  InitOcinito 17240  TermOctermo 17241  ZeroOczeroo 17242 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pr 5295 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 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-iota 6283  df-fun 6326  df-fv 6332  df-zeroo 17245 This theorem is referenced by:  iszeroo  17254  iszeroi  17261
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