Theorem pimltpnf

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, 26-Jun-2021)

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

Proof

Step	Hyp	Ref	Expression
1		pimltpnf.1	$$⊢Ⅎx\phi$$
2		pimltpnf.2	$$⊢\left(\phi \wedge x\in A\right)\to B\in ℝ$$
3		ssrab2	$$⊢\left\{x\in A|B<\mathrm{+\infty }\right\}\subseteq A$$
4	3	a1i	$$⊢\phi \to \left\{x\in A|B<\mathrm{+\infty }\right\}\subseteq A$$
5		simpr	$$⊢\left(\phi \wedge x\in A\right)\to x\in A$$
6		ltpnf	$$⊢B\in ℝ\to B<\mathrm{+\infty }$$
7	2 6	syl	$$⊢\left(\phi \wedge x\in A\right)\to B<\mathrm{+\infty }$$
8	5 7	jca	$$⊢\left(\phi \wedge x\in A\right)\to \left(x\in A\wedge B<\mathrm{+\infty }\right)$$
9		rabid	$$⊢x\in \left\{x\in A|B<\mathrm{+\infty }\right\}\leftrightarrow \left(x\in A\wedge B<\mathrm{+\infty }\right)$$
10	8 9	sylibr	$$⊢\left(\phi \wedge x\in A\right)\to x\in \left\{x\in A|B<\mathrm{+\infty }\right\}$$
11	10	ex	$$⊢\phi \to \left(x\in A\to x\in \left\{x\in A|B<\mathrm{+\infty }\right\}\right)$$
12	1 11	ralrimi	$$⊢\phi \to \forall x\in Ax\in \left\{x\in A|B<\mathrm{+\infty }\right\}$$
13		nfcv	$$⊢\underset{\_}{Ⅎ}xA$$
14		nfrab1	$$⊢\underset{\_}{Ⅎ}x\left\{x\in A|B<\mathrm{+\infty }\right\}$$
15	13 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	12 15	sylibr	$$⊢\phi \to A\subseteq \left\{x\in A|B<\mathrm{+\infty }\right\}$$
17	4 16	eqssd	$$⊢\phi \to \left\{x\in A|B<\mathrm{+\infty }\right\}=A$$