Theorem frnnn0supp

Description: Two ways to write the support of a function on NN0 . (Contributed by Mario Carneiro, 29-Dec-2014) (Revised by AV, 7-Jul-2019)

	Ref	Expression
Assertion	frnnn0supp	$$⊢\left(I\in V\wedge F:I⟶{\mathbb{N}}_0\right)\to F\mathrm{supp}0=F^{-1}\left[\mathbb{N}\right]$$

Proof

Step	Hyp	Ref	Expression
1		c0ex	$$⊢0\in \mathrm{V}$$
2		frnsuppeq	$$⊢\left(I\in V\wedge 0\in \mathrm{V}\right)\to \left(F:I⟶{\mathbb{N}}_0\to F\mathrm{supp}0=F^{-1}\left[{\mathbb{N}}_0\setminus \left\{0\right\}\right]\right)$$
3	2	imp	$$⊢\left(\left(I\in V\wedge 0\in \mathrm{V}\right)\wedge F:I⟶{\mathbb{N}}_0\right)\to F\mathrm{supp}0=F^{-1}\left[{\mathbb{N}}_0\setminus \left\{0\right\}\right]$$
4	1 3	mpanl2	$$⊢\left(I\in V\wedge F:I⟶{\mathbb{N}}_0\right)\to F\mathrm{supp}0=F^{-1}\left[{\mathbb{N}}_0\setminus \left\{0\right\}\right]$$
5		dfn2	$$⊢\mathbb{N}={\mathbb{N}}_0\setminus \left\{0\right\}$$
6	5	eqcomi	$$⊢{\mathbb{N}}_0\setminus \left\{0\right\}=\mathbb{N}$$
7	6	imaeq2i	$$⊢F^{-1}\left[{\mathbb{N}}_0\setminus \left\{0\right\}\right]=F^{-1}\left[\mathbb{N}\right]$$
8	4 7	syl6eq	$$⊢\left(I\in V\wedge F:I⟶{\mathbb{N}}_0\right)\to F\mathrm{supp}0=F^{-1}\left[\mathbb{N}\right]$$