Theorem fcdmnn0fsupp

Description: A function into NN0 is finitely supported iff its support is finite. (Contributed by AV, 8-Jul-2019)

	Ref	Expression
Assertion	fcdmnn0fsupp	$$⊢\left(I\in V\wedge F:I⟶{\mathbb{N}}_0\right)\to \left({finSupp}_0\left(F\right)\leftrightarrow F^{-1}\left[\mathbb{N}\right]\in \mathrm{Fin}\right)$$

Proof

Step	Hyp	Ref	Expression
1		c0ex	$$⊢0\in \mathrm{V}$$
2		ffsuppbi	$$⊢\left(I\in V\wedge 0\in \mathrm{V}\right)\to \left(F:I⟶{\mathbb{N}}_0\to \left({finSupp}_0\left(F\right)\leftrightarrow F^{-1}\left[{\mathbb{N}}_0\setminus \left\{0\right\}\right]\in \mathrm{Fin}\right)\right)$$
3	1 2	mpan2	$$⊢I\in V\to \left(F:I⟶{\mathbb{N}}_0\to \left({finSupp}_0\left(F\right)\leftrightarrow F^{-1}\left[{\mathbb{N}}_0\setminus \left\{0\right\}\right]\in \mathrm{Fin}\right)\right)$$
4	3	imp	$$⊢\left(I\in V\wedge F:I⟶{\mathbb{N}}_0\right)\to \left({finSupp}_0\left(F\right)\leftrightarrow F^{-1}\left[{\mathbb{N}}_0\setminus \left\{0\right\}\right]\in \mathrm{Fin}\right)$$
5		dfn2	$$⊢\mathbb{N}={\mathbb{N}}_0\setminus \left\{0\right\}$$
6	5	imaeq2i	$$⊢F^{-1}\left[\mathbb{N}\right]=F^{-1}\left[{\mathbb{N}}_0\setminus \left\{0\right\}\right]$$
7	6	eleq1i	$$⊢F^{-1}\left[\mathbb{N}\right]\in \mathrm{Fin}\leftrightarrow F^{-1}\left[{\mathbb{N}}_0\setminus \left\{0\right\}\right]\in \mathrm{Fin}$$
8	4 7	bitr4di	$$⊢\left(I\in V\wedge F:I⟶{\mathbb{N}}_0\right)\to \left({finSupp}_0\left(F\right)\leftrightarrow F^{-1}\left[\mathbb{N}\right]\in \mathrm{Fin}\right)$$