Theorem frnnn0suppg

Description: Version of frnnn0supp avoiding ax-rep by assuming F is a set rather than its domain I . (Contributed by SN, 5-Aug-2024)

	Ref	Expression
Assertion	frnnn0suppg	$$⊢\left(F\in V\wedge F:I⟶{\mathbb{N}}_0\right)\to F\mathrm{supp}0=F^{-1}\left[\mathbb{N}\right]$$

Proof

Step	Hyp	Ref	Expression
1		c0ex	$$⊢0\in \mathrm{V}$$
2		frnsuppeqg	$$⊢\left(F\in V\wedge 0\in \mathrm{V}\right)\to \left(F:I⟶{\mathbb{N}}_0\to F\mathrm{supp}0=F^{-1}\left[{\mathbb{N}}_0\setminus \left\{0\right\}\right]\right)$$
3	1 2	mpan2	$$⊢F\in V\to \left(F:I⟶{\mathbb{N}}_0\to F\mathrm{supp}0=F^{-1}\left[{\mathbb{N}}_0\setminus \left\{0\right\}\right]\right)$$
4	3	imp	$$⊢\left(F\in V\wedge F:I⟶{\mathbb{N}}_0\right)\to F\mathrm{supp}0=F^{-1}\left[{\mathbb{N}}_0\setminus \left\{0\right\}\right]$$
5		dfn2	$$⊢\mathbb{N}={\mathbb{N}}_0\setminus \left\{0\right\}$$
6	5	imaeq2i	$$⊢F^{-1}\left[\mathbb{N}\right]=F^{-1}\left[{\mathbb{N}}_0\setminus \left\{0\right\}\right]$$
7	4 6	eqtr4di	$$⊢\left(F\in V\wedge F:I⟶{\mathbb{N}}_0\right)\to F\mathrm{supp}0=F^{-1}\left[\mathbb{N}\right]$$