Theorem pimltpnf2f

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, 15-Dec-2024)

	Ref	Expression
Hypotheses	pimltpnf2f.1	|- F/_ x F
	pimltpnf2f.2	|- F/_ x A
	pimltpnf2f.3	|- ( ph -> F : A --> RR )
Assertion	pimltpnf2f	|- ( ph -> { x e. A | ( F ` x ) < +oo } = A )

Proof

Step	Hyp	Ref	Expression
1		pimltpnf2f.1	|- F/_ x F
2		pimltpnf2f.2	|- F/_ x A
3		pimltpnf2f.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		nfv	|- F/ y ph
15	3	ffvelcdmda	|- ( ( ph /\ y e. A ) -> ( F ` y ) e. RR )
16	14 15	pimltpnf	|- ( ph -> { y e. A | ( F ` y ) < +oo } = A )
17	13 16	eqtrid	|- ( ph -> { x e. A | ( F ` x ) < +oo } = A )