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Theorem zerooval 16418
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 16412 . . 3 ZeroO = (𝑐 ∈ Cat ↦ ((InitO‘𝑐) ∩ (TermO‘𝑐)))
21a1i 11 . 2 (𝜑 → ZeroO = (𝑐 ∈ Cat ↦ ((InitO‘𝑐) ∩ (TermO‘𝑐))))
3 fveq2 6088 . . . 4 (𝑐 = 𝐶 → (InitO‘𝑐) = (InitO‘𝐶))
4 fveq2 6088 . . . 4 (𝑐 = 𝐶 → (TermO‘𝑐) = (TermO‘𝐶))
53, 4ineq12d 3776 . . 3 (𝑐 = 𝐶 → ((InitO‘𝑐) ∩ (TermO‘𝑐)) = ((InitO‘𝐶) ∩ (TermO‘𝐶)))
65adantl 480 . 2 ((𝜑𝑐 = 𝐶) → ((InitO‘𝑐) ∩ (TermO‘𝑐)) = ((InitO‘𝐶) ∩ (TermO‘𝐶)))
7 initoval.c . 2 (𝜑𝐶 ∈ Cat)
8 fvex 6098 . . . 4 (InitO‘𝐶) ∈ V
98inex1 4722 . . 3 ((InitO‘𝐶) ∩ (TermO‘𝐶)) ∈ V
109a1i 11 . 2 (𝜑 → ((InitO‘𝐶) ∩ (TermO‘𝐶)) ∈ V)
112, 6, 7, 10fvmptd 6182 1 (𝜑 → (ZeroO‘𝐶) = ((InitO‘𝐶) ∩ (TermO‘𝐶)))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1474  wcel 1976  Vcvv 3172  cin 3538  cmpt 4637  cfv 5790  Basecbs 15641  Hom chom 15725  Catccat 16094  InitOcinito 16407  TermOctermo 16408  ZeroOczeroo 16409
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-sep 4703  ax-nul 4712  ax-pr 4828
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 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ral 2900  df-rex 2901  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-sn 4125  df-pr 4127  df-op 4131  df-uni 4367  df-br 4578  df-opab 4638  df-mpt 4639  df-id 4943  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-iota 5754  df-fun 5792  df-fv 5798  df-zeroo 16412
This theorem is referenced by:  iszeroo  16421  iszeroi  16428
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