Theorem pimgtmnf

Description: Given a real-valued function, the preimage of an open interval, unbounded above, with lower bound -oo , is the whole domain. (Contributed by Glauco Siliprandi, 26-Jun-2021)

	Ref	Expression
Hypotheses	pimgtmnf.1	$$⊢Ⅎx\phi$$
	pimgtmnf.2	$$⊢\left(\phi \wedge x\in A\right)\to B\in ℝ$$
Assertion	pimgtmnf	$$⊢\phi \to \left\{x\in A|\mathrm{-\infty }<B\right\}=A$$

Proof

Step	Hyp	Ref	Expression
1		pimgtmnf.1	$$⊢Ⅎx\phi$$
2		pimgtmnf.2	$$⊢\left(\phi \wedge x\in A\right)\to B\in ℝ$$
3		eqidd	$$⊢\phi \to \left(x\in A⟼B\right)=\left(x\in A⟼B\right)$$
4	3 2	fvmpt2d	$$⊢\left(\phi \wedge x\in A\right)\to \left(x\in A⟼B\right)\left(x\right)=B$$
5	4	eqcomd	$$⊢\left(\phi \wedge x\in A\right)\to B=\left(x\in A⟼B\right)\left(x\right)$$
6	5	breq2d	$$⊢\left(\phi \wedge x\in A\right)\to \left(\mathrm{-\infty }<B\leftrightarrow \mathrm{-\infty }<\left(x\in A⟼B\right)\left(x\right)\right)$$
7	1 6	rabbida	$$⊢\phi \to \left\{x\in A|\mathrm{-\infty }<B\right\}=\left\{x\in A|\mathrm{-\infty }<\left(x\in A⟼B\right)\left(x\right)\right\}$$
8		nfmpt1	$$⊢\underset{\_}{Ⅎ}x\left(x\in A⟼B\right)$$
9		eqid	$$⊢\left(x\in A⟼B\right)=\left(x\in A⟼B\right)$$
10	1 2 9	fmptdf	$$⊢\phi \to \left(x\in A⟼B\right):A⟶ℝ$$
11	8 10	pimgtmnf2	$$⊢\phi \to \left\{x\in A|\mathrm{-\infty }<\left(x\in A⟼B\right)\left(x\right)\right\}=A$$
12	7 11	eqtrd	$$⊢\phi \to \left\{x\in A|\mathrm{-\infty }<B\right\}=A$$