zerooval - Metamath Proof Explorer

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Theorem zerooval 17960

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 17951	. 2 ⊢ ZeroO = (𝑐 ∈ Cat ↦ ((InitO‘𝑐) ∩ (TermO‘𝑐)))
2		fveq2 6834	. . 3 ⊢ (𝑐 = 𝐶 → (InitO‘𝑐) = (InitO‘𝐶))
3		fveq2 6834	. . 3 ⊢ (𝑐 = 𝐶 → (TermO‘𝑐) = (TermO‘𝐶))
4	2, 3	ineq12d 4157	. 2 ⊢ (𝑐 = 𝐶 → ((InitO‘𝑐) ∩ (TermO‘𝑐)) = ((InitO‘𝐶) ∩ (TermO‘𝐶)))
5		initoval.c	. 2 ⊢ (𝜑 → 𝐶 ∈ Cat)
6		fvex 6847	. . . 4 ⊢ (InitO‘𝐶) ∈ V
7	6	inex1 5252	. . 3 ⊢ ((InitO‘𝐶) ∩ (TermO‘𝐶)) ∈ V
8	7	a1i 11	. 2 ⊢ (𝜑 → ((InitO‘𝐶) ∩ (TermO‘𝐶)) ∈ V)
9	1, 4, 5, 8	fvmptd3 6966	1 ⊢ (𝜑 → (ZeroO‘𝐶) = ((InitO‘𝐶) ∩ (TermO‘𝐶)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1547 ∈ wcel 2119 Vcvv 3432 ∩ cin 3889 ‘cfv 6492 Basecbs 17177 Hom chom 17229 Catccat 17628 InitOcinito 17946 TermOctermo 17947 ZeroOczeroo 17948

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1974 ax-7 2015 ax-8 2121 ax-9 2129 ax-10 2152 ax-11 2168 ax-12 2189 ax-ext 2712 ax-sep 5225 ax-nul 5235 ax-pr 5369

This theorem depends on definitions: df-bi 208 df-an 397 df-or 854 df-3an 1094 df-tru 1550 df-fal 1560 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2543 df-eu 2573 df-clab 2719 df-cleq 2732 df-clel 2815 df-nfc 2889 df-ne 2936 df-ral 3055 df-rex 3065 df-rab 3393 df-v 3434 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-nul 4269 df-if 4462 df-sn 4563 df-pr 4565 df-op 4569 df-uni 4846 df-br 5080 df-opab 5142 df-mpt 5161 df-id 5520 df-xp 5631 df-rel 5632 df-cnv 5633 df-co 5634 df-dm 5635 df-iota 6448 df-fun 6494 df-fv 6500 df-zeroo 17951

This theorem is referenced by: iszeroo 17963 iszeroi 17974 oppczeroo 49734 zeroopropdlem 49739 zeroopropd 49742

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