Theorem pimltmnf2f

Description: Given a real-valued function, the preimage of an open interval, unbounded below, with upper bound -oo , is the empty set. (Contributed by Glauco Siliprandi, 15-Dec-2024)

	Ref	Expression
Hypotheses	pimltmnf2f.1	|- F/_ x F
	pimltmnf2f.2	|- F/_ x A
	pimltmnf2f.3	|- ( ph -> F : A --> RR )
Assertion	pimltmnf2f	|- ( ph -> { x e. A | ( F ` x ) < -oo } = (/) )

Proof

Step	Hyp	Ref	Expression
1		pimltmnf2f.1	|- F/_ x F
2		pimltmnf2f.2	|- F/_ x A
3		pimltmnf2f.3	|- ( ph -> F : A --> RR )
4		nfcv	|- F/_ y A
5		nfv	|- F/ y ( F ` x ) < -oo
6		nfcv	|- F/_ x y
7	1 6	nffv	|- F/_ x ( F ` y )
8		nfcv	|- F/_ x <
9		nfcv	|- F/_ x -oo
10	7 8 9	nfbr	|- F/ x ( F ` y ) < -oo
11		fveq2	|- ( x = y -> ( F ` x ) = ( F ` y ) )
12	11	breq1d	|- ( x = y -> ( ( F ` x ) < -oo <-> ( F ` y ) < -oo ) )
13	2 4 5 10 12	cbvrabw	|- { x e. A | ( F ` x ) < -oo } = { y e. A | ( F ` y ) < -oo }
14	3	ffvelcdmda	|- ( ( ph /\ y e. A ) -> ( F ` y ) e. RR )
15	14	rexrd	|- ( ( ph /\ y e. A ) -> ( F ` y ) e. RR* )
16	15	mnfled	|- ( ( ph /\ y e. A ) -> -oo <_ ( F ` y ) )
17		mnfxr	|- -oo e. RR*
18	17	a1i	|- ( ( ph /\ y e. A ) -> -oo e. RR* )
19	18 15	xrlenltd	|- ( ( ph /\ y e. A ) -> ( -oo <_ ( F ` y ) <-> -. ( F ` y ) < -oo ) )
20	16 19	mpbid	|- ( ( ph /\ y e. A ) -> -. ( F ` y ) < -oo )
21	20	ralrimiva	|- ( ph -> A. y e. A -. ( F ` y ) < -oo )
22		rabeq0	|- ( { y e. A | ( F ` y ) < -oo } = (/) <-> A. y e. A -. ( F ` y ) < -oo )
23	21 22	sylibr	|- ( ph -> { y e. A | ( F ` y ) < -oo } = (/) )
24	13 23	eqtrid	|- ( ph -> { x e. A | ( F ` x ) < -oo } = (/) )