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 } ) |