Step |
Hyp |
Ref |
Expression |
1 |
|
pimltpnf2.1 |
|- F/_ x F |
2 |
|
pimltpnf2.2 |
|- ( ph -> F : A --> RR ) |
3 |
|
nfcv |
|- F/_ x A |
4 |
|
nfcv |
|- F/_ y A |
5 |
|
nfv |
|- F/ y ( F ` x ) < +oo |
6 |
|
nfcv |
|- F/_ x y |
7 |
1 6
|
nffv |
|- F/_ x ( F ` y ) |
8 |
|
nfcv |
|- F/_ x < |
9 |
|
nfcv |
|- F/_ x +oo |
10 |
7 8 9
|
nfbr |
|- F/ x ( F ` y ) < +oo |
11 |
|
fveq2 |
|- ( x = y -> ( F ` x ) = ( F ` y ) ) |
12 |
11
|
breq1d |
|- ( x = y -> ( ( F ` x ) < +oo <-> ( F ` y ) < +oo ) ) |
13 |
3 4 5 10 12
|
cbvrabw |
|- { x e. A | ( F ` x ) < +oo } = { y e. A | ( F ` y ) < +oo } |
14 |
13
|
a1i |
|- ( ph -> { x e. A | ( F ` x ) < +oo } = { y e. A | ( F ` y ) < +oo } ) |
15 |
|
nfv |
|- F/ y ph |
16 |
2
|
ffvelrnda |
|- ( ( ph /\ y e. A ) -> ( F ` y ) e. RR ) |
17 |
15 16
|
pimltpnf |
|- ( ph -> { y e. A | ( F ` y ) < +oo } = A ) |
18 |
14 17
|
eqtrd |
|- ( ph -> { x e. A | ( F ` x ) < +oo } = A ) |