Metamath Proof Explorer


Theorem pimgtpnf2

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

Ref Expression
Hypotheses pimgtpnf2.1
|- F/_ x F
pimgtpnf2.2
|- ( ph -> F : A --> RR )
Assertion pimgtpnf2
|- ( ph -> { x e. A | +oo < ( F ` x ) } = (/) )

Proof

Step Hyp Ref Expression
1 pimgtpnf2.1
 |-  F/_ x F
2 pimgtpnf2.2
 |-  ( ph -> F : A --> RR )
3 nfcv
 |-  F/_ x A
4 nfcv
 |-  F/_ y A
5 nfv
 |-  F/ y +oo < ( F ` x )
6 nfcv
 |-  F/_ x +oo
7 nfcv
 |-  F/_ x <
8 nfcv
 |-  F/_ x y
9 1 8 nffv
 |-  F/_ x ( F ` y )
10 6 7 9 nfbr
 |-  F/ x +oo < ( F ` y )
11 fveq2
 |-  ( x = y -> ( F ` x ) = ( F ` y ) )
12 11 breq2d
 |-  ( x = y -> ( +oo < ( F ` x ) <-> +oo < ( F ` y ) ) )
13 3 4 5 10 12 cbvrabw
 |-  { x e. A | +oo < ( F ` x ) } = { y e. A | +oo < ( F ` y ) }
14 13 a1i
 |-  ( ph -> { x e. A | +oo < ( F ` x ) } = { y e. A | +oo < ( F ` y ) } )
15 2 ffvelrnda
 |-  ( ( ph /\ y e. A ) -> ( F ` y ) e. RR )
16 15 rexrd
 |-  ( ( ph /\ y e. A ) -> ( F ` y ) e. RR* )
17 pnfxr
 |-  +oo e. RR*
18 17 a1i
 |-  ( ( ph /\ y e. A ) -> +oo e. RR* )
19 15 ltpnfd
 |-  ( ( ph /\ y e. A ) -> ( F ` y ) < +oo )
20 16 18 19 xrltled
 |-  ( ( ph /\ y e. A ) -> ( F ` y ) <_ +oo )
21 16 18 xrlenltd
 |-  ( ( ph /\ y e. A ) -> ( ( F ` y ) <_ +oo <-> -. +oo < ( F ` y ) ) )
22 20 21 mpbid
 |-  ( ( ph /\ y e. A ) -> -. +oo < ( F ` y ) )
23 22 ralrimiva
 |-  ( ph -> A. y e. A -. +oo < ( F ` y ) )
24 rabeq0
 |-  ( { y e. A | +oo < ( F ` y ) } = (/) <-> A. y e. A -. +oo < ( F ` y ) )
25 23 24 sylibr
 |-  ( ph -> { y e. A | +oo < ( F ` y ) } = (/) )
26 14 25 eqtrd
 |-  ( ph -> { x e. A | +oo < ( F ` x ) } = (/) )