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 ) } = (/) ) |