Theorem pimltpnff

Description: Given a real-valued function, the preimage of an open interval, unbounded below, with upper bound +oo , is the whole domain. (Contributed by Glauco Siliprandi, 20-Dec-2024)

	Ref	Expression
Hypotheses	pimltpnff.1	$$⊢Ⅎx\phi$$
	pimltpnff.2	$$⊢\underset{\_}{Ⅎ}xA$$
	pimltpnff.3	$$⊢\left(\phi \wedge x\in A\right)\to B\in ℝ$$
Assertion	pimltpnff	$$⊢\phi \to \left\{x\in A|B<\mathrm{+\infty }\right\}=A$$

Proof

Step	Hyp	Ref	Expression
1		pimltpnff.1	$$⊢Ⅎx\phi$$
2		pimltpnff.2	$$⊢\underset{\_}{Ⅎ}xA$$
3		pimltpnff.3	$$⊢\left(\phi \wedge x\in A\right)\to B\in ℝ$$
4	2	ssrab2f	$$⊢\left\{x\in A|B<\mathrm{+\infty }\right\}\subseteq A$$
5	4	a1i	$$⊢\phi \to \left\{x\in A|B<\mathrm{+\infty }\right\}\subseteq A$$
6		simpr	$$⊢\left(\phi \wedge x\in A\right)\to x\in A$$
7		ltpnf	$$⊢B\in ℝ\to B<\mathrm{+\infty }$$
8	3 7	syl	$$⊢\left(\phi \wedge x\in A\right)\to B<\mathrm{+\infty }$$
9	6 8	jca	$$⊢\left(\phi \wedge x\in A\right)\to \left(x\in A\wedge B<\mathrm{+\infty }\right)$$
10		rabid	$$⊢x\in \left\{x\in A|B<\mathrm{+\infty }\right\}\leftrightarrow \left(x\in A\wedge B<\mathrm{+\infty }\right)$$
11	9 10	sylibr	$$⊢\left(\phi \wedge x\in A\right)\to x\in \left\{x\in A|B<\mathrm{+\infty }\right\}$$
12	11	ex	$$⊢\phi \to \left(x\in A\to x\in \left\{x\in A|B<\mathrm{+\infty }\right\}\right)$$
13	1 12	ralrimi	$$⊢\phi \to \forall x\in Ax\in \left\{x\in A|B<\mathrm{+\infty }\right\}$$
14		nfrab1	$$⊢\underset{\_}{Ⅎ}x\left\{x\in A|B<\mathrm{+\infty }\right\}$$
15	2 14	dfss3f	$$⊢A\subseteq \left\{x\in A|B<\mathrm{+\infty }\right\}\leftrightarrow \forall x\in Ax\in \left\{x\in A|B<\mathrm{+\infty }\right\}$$
16	13 15	sylibr	$$⊢\phi \to A\subseteq \left\{x\in A|B<\mathrm{+\infty }\right\}$$
17	5 16	eqssd	$$⊢\phi \to \left\{x\in A|B<\mathrm{+\infty }\right\}=A$$