Theorem preimaioomnf

Description: Preimage of an open interval, unbounded below. (Contributed by Glauco Siliprandi, 26-Jun-2021)

	Ref	Expression
Hypotheses	preimaioomnf.1	$$⊢\phi \to F:A⟶ℝ$$
	preimaioomnf.2	$$⊢\phi \to B\in {ℝ}^{*}$$
Assertion	preimaioomnf	$$⊢\phi \to F^{-1}\left[\left(\mathrm{-\infty },B\right)\right]=\left\{x\in A|F\left(x\right)<B\right\}$$

Proof

Step	Hyp	Ref	Expression
1		preimaioomnf.1	$$⊢\phi \to F:A⟶ℝ$$
2		preimaioomnf.2	$$⊢\phi \to B\in {ℝ}^{*}$$
3	1	ffund	$$⊢\phi \to \mathrm{Fun}F$$
4	1	frnd	$$⊢\phi \to \mathrm{ran}F\subseteq ℝ$$
5		fimacnvinrn2	$$⊢\left(\mathrm{Fun}F\wedge \mathrm{ran}F\subseteq ℝ\right)\to F^{-1}\left[\left[\mathrm{-\infty },B\right)\right]=F^{-1}\left[\left[\mathrm{-\infty },B\right)\cap ℝ\right]$$
6	3 4 5	syl2anc	$$⊢\phi \to F^{-1}\left[\left[\mathrm{-\infty },B\right)\right]=F^{-1}\left[\left[\mathrm{-\infty },B\right)\cap ℝ\right]$$
7	2	icomnfinre	$$⊢\phi \to \left[\mathrm{-\infty },B\right)\cap ℝ=\left(\mathrm{-\infty },B\right)$$
8	7	imaeq2d	$$⊢\phi \to F^{-1}\left[\left[\mathrm{-\infty },B\right)\cap ℝ\right]=F^{-1}\left[\left(\mathrm{-\infty },B\right)\right]$$
9	6 8	eqtr2d	$$⊢\phi \to F^{-1}\left[\left(\mathrm{-\infty },B\right)\right]=F^{-1}\left[\left[\mathrm{-\infty },B\right)\right]$$
10	1	frexr	$$⊢\phi \to F:A⟶{ℝ}^{*}$$
11	10 2	preimaicomnf	$$⊢\phi \to F^{-1}\left[\left[\mathrm{-\infty },B\right)\right]=\left\{x\in A|F\left(x\right)<B\right\}$$
12	9 11	eqtrd	$$⊢\phi \to F^{-1}\left[\left(\mathrm{-\infty },B\right)\right]=\left\{x\in A|F\left(x\right)<B\right\}$$