Step |
Hyp |
Ref |
Expression |
1 |
|
pimconstlt0.x |
|- F/ x ph |
2 |
|
pimconstlt0.b |
|- ( ph -> B e. RR ) |
3 |
|
pimconstlt0.f |
|- F = ( x e. A |-> B ) |
4 |
|
pimconstlt0.c |
|- ( ph -> C e. RR* ) |
5 |
|
pimconstlt0.l |
|- ( ph -> C <_ B ) |
6 |
5
|
adantr |
|- ( ( ph /\ x e. A ) -> C <_ B ) |
7 |
3
|
a1i |
|- ( ph -> F = ( x e. A |-> B ) ) |
8 |
2
|
adantr |
|- ( ( ph /\ x e. A ) -> B e. RR ) |
9 |
7 8
|
fvmpt2d |
|- ( ( ph /\ x e. A ) -> ( F ` x ) = B ) |
10 |
6 9
|
breqtrrd |
|- ( ( ph /\ x e. A ) -> C <_ ( F ` x ) ) |
11 |
4
|
adantr |
|- ( ( ph /\ x e. A ) -> C e. RR* ) |
12 |
9 8
|
eqeltrd |
|- ( ( ph /\ x e. A ) -> ( F ` x ) e. RR ) |
13 |
12
|
rexrd |
|- ( ( ph /\ x e. A ) -> ( F ` x ) e. RR* ) |
14 |
11 13
|
xrlenltd |
|- ( ( ph /\ x e. A ) -> ( C <_ ( F ` x ) <-> -. ( F ` x ) < C ) ) |
15 |
10 14
|
mpbid |
|- ( ( ph /\ x e. A ) -> -. ( F ` x ) < C ) |
16 |
15
|
ex |
|- ( ph -> ( x e. A -> -. ( F ` x ) < C ) ) |
17 |
1 16
|
ralrimi |
|- ( ph -> A. x e. A -. ( F ` x ) < C ) |
18 |
|
rabeq0 |
|- ( { x e. A | ( F ` x ) < C } = (/) <-> A. x e. A -. ( F ` x ) < C ) |
19 |
17 18
|
sylibr |
|- ( ph -> { x e. A | ( F ` x ) < C } = (/) ) |