Metamath Proof Explorer


Theorem preimaiocmnf

Description: Preimage of a right-closed interval, unbounded below. (Contributed by Glauco Siliprandi, 23-Oct-2021)

Ref Expression
Hypotheses preimaiocmnf.1
|- ( ph -> F : A --> RR )
preimaiocmnf.2
|- ( ph -> B e. RR* )
Assertion preimaiocmnf
|- ( ph -> ( `' F " ( -oo (,] B ) ) = { x e. A | ( F ` x ) <_ B } )

Proof

Step Hyp Ref Expression
1 preimaiocmnf.1
 |-  ( ph -> F : A --> RR )
2 preimaiocmnf.2
 |-  ( ph -> B e. RR* )
3 1 ffnd
 |-  ( ph -> F Fn A )
4 fncnvima2
 |-  ( F Fn A -> ( `' F " ( -oo (,] B ) ) = { x e. A | ( F ` x ) e. ( -oo (,] B ) } )
5 3 4 syl
 |-  ( ph -> ( `' F " ( -oo (,] B ) ) = { x e. A | ( F ` x ) e. ( -oo (,] B ) } )
6 mnfxr
 |-  -oo e. RR*
7 6 a1i
 |-  ( ( ph /\ ( F ` x ) e. ( -oo (,] B ) ) -> -oo e. RR* )
8 2 adantr
 |-  ( ( ph /\ ( F ` x ) e. ( -oo (,] B ) ) -> B e. RR* )
9 simpr
 |-  ( ( ph /\ ( F ` x ) e. ( -oo (,] B ) ) -> ( F ` x ) e. ( -oo (,] B ) )
10 7 8 9 iocleubd
 |-  ( ( ph /\ ( F ` x ) e. ( -oo (,] B ) ) -> ( F ` x ) <_ B )
11 10 ex
 |-  ( ph -> ( ( F ` x ) e. ( -oo (,] B ) -> ( F ` x ) <_ B ) )
12 11 adantr
 |-  ( ( ph /\ x e. A ) -> ( ( F ` x ) e. ( -oo (,] B ) -> ( F ` x ) <_ B ) )
13 6 a1i
 |-  ( ( ( ph /\ x e. A ) /\ ( F ` x ) <_ B ) -> -oo e. RR* )
14 2 adantr
 |-  ( ( ph /\ ( F ` x ) <_ B ) -> B e. RR* )
15 14 adantlr
 |-  ( ( ( ph /\ x e. A ) /\ ( F ` x ) <_ B ) -> B e. RR* )
16 1 ffvelrnda
 |-  ( ( ph /\ x e. A ) -> ( F ` x ) e. RR )
17 16 rexrd
 |-  ( ( ph /\ x e. A ) -> ( F ` x ) e. RR* )
18 17 adantr
 |-  ( ( ( ph /\ x e. A ) /\ ( F ` x ) <_ B ) -> ( F ` x ) e. RR* )
19 16 mnfltd
 |-  ( ( ph /\ x e. A ) -> -oo < ( F ` x ) )
20 19 adantr
 |-  ( ( ( ph /\ x e. A ) /\ ( F ` x ) <_ B ) -> -oo < ( F ` x ) )
21 simpr
 |-  ( ( ( ph /\ x e. A ) /\ ( F ` x ) <_ B ) -> ( F ` x ) <_ B )
22 13 15 18 20 21 eliocd
 |-  ( ( ( ph /\ x e. A ) /\ ( F ` x ) <_ B ) -> ( F ` x ) e. ( -oo (,] B ) )
23 22 ex
 |-  ( ( ph /\ x e. A ) -> ( ( F ` x ) <_ B -> ( F ` x ) e. ( -oo (,] B ) ) )
24 12 23 impbid
 |-  ( ( ph /\ x e. A ) -> ( ( F ` x ) e. ( -oo (,] B ) <-> ( F ` x ) <_ B ) )
25 24 rabbidva
 |-  ( ph -> { x e. A | ( F ` x ) e. ( -oo (,] B ) } = { x e. A | ( F ` x ) <_ B } )
26 5 25 eqtrd
 |-  ( ph -> ( `' F " ( -oo (,] B ) ) = { x e. A | ( F ` x ) <_ B } )