Theorem zerooval 16472

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	⊢ (𝜑 → 𝐶 ∈ 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.

Step	Hyp	Ref	Expression
1		df-zeroo 16466	. . 3 ⊢ ZeroO = (𝑐 ∈ Cat ↦ ((InitO‘𝑐) ∩ (TermO‘𝑐)))
2	1	a1i 11	. 2 ⊢ (𝜑 → ZeroO = (𝑐 ∈ Cat ↦ ((InitO‘𝑐) ∩ (TermO‘𝑐))))
3		fveq2 6103	. . . 4 ⊢ (𝑐 = 𝐶 → (InitO‘𝑐) = (InitO‘𝐶))
4		fveq2 6103	. . . 4 ⊢ (𝑐 = 𝐶 → (TermO‘𝑐) = (TermO‘𝐶))
5	3, 4	ineq12d 3777	. . 3 ⊢ (𝑐 = 𝐶 → ((InitO‘𝑐) ∩ (TermO‘𝑐)) = ((InitO‘𝐶) ∩ (TermO‘𝐶)))
6	5	adantl 481	. 2 ⊢ ((𝜑 ∧ 𝑐 = 𝐶) → ((InitO‘𝑐) ∩ (TermO‘𝑐)) = ((InitO‘𝐶) ∩ (TermO‘𝐶)))
7		initoval.c	. 2 ⊢ (𝜑 → 𝐶 ∈ Cat)
8		fvex 6113	. . . 4 ⊢ (InitO‘𝐶) ∈ V
9	8	inex1 4727	. . 3 ⊢ ((InitO‘𝐶) ∩ (TermO‘𝐶)) ∈ V
10	9	a1i 11	. 2 ⊢ (𝜑 → ((InitO‘𝐶) ∩ (TermO‘𝐶)) ∈ V)
11	2, 6, 7, 10	fvmptd 6197	1 ⊢ (𝜑 → (ZeroO‘𝐶) = ((InitO‘𝐶) ∩ (TermO‘𝐶)))

Syntax hints:   → wi 4   = wceq 1475   ∈ wcel 1977  Vcvv 3173   ∩ cin 3539   ↦ cmpt 4643  ‘cfv 5804  Basecbs 15695  Hom chom 15779  Catccat 16148  InitOcinito 16461  TermOctermo 16462  ZeroOczeroo 16463

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-iota 5768  df-fun 5806  df-fv 5812  df-zeroo 16466

This theorem is referenced by:  iszeroo  16475  iszeroi  16482