Step |
Hyp |
Ref |
Expression |
1 |
|
alexsub.1 |
|- ( ph -> X e. UFL ) |
2 |
|
alexsub.2 |
|- ( ph -> X = U. B ) |
3 |
|
alexsub.3 |
|- ( ph -> J = ( topGen ` ( fi ` B ) ) ) |
4 |
|
alexsub.4 |
|- ( ( ph /\ ( x C_ B /\ X = U. x ) ) -> E. y e. ( ~P x i^i Fin ) X = U. y ) |
5 |
1
|
adantr |
|- ( ( ph /\ ( f e. ( UFil ` X ) /\ ( J fLim f ) = (/) ) ) -> X e. UFL ) |
6 |
2
|
adantr |
|- ( ( ph /\ ( f e. ( UFil ` X ) /\ ( J fLim f ) = (/) ) ) -> X = U. B ) |
7 |
3
|
adantr |
|- ( ( ph /\ ( f e. ( UFil ` X ) /\ ( J fLim f ) = (/) ) ) -> J = ( topGen ` ( fi ` B ) ) ) |
8 |
4
|
adantlr |
|- ( ( ( ph /\ ( f e. ( UFil ` X ) /\ ( J fLim f ) = (/) ) ) /\ ( x C_ B /\ X = U. x ) ) -> E. y e. ( ~P x i^i Fin ) X = U. y ) |
9 |
|
simprl |
|- ( ( ph /\ ( f e. ( UFil ` X ) /\ ( J fLim f ) = (/) ) ) -> f e. ( UFil ` X ) ) |
10 |
|
simprr |
|- ( ( ph /\ ( f e. ( UFil ` X ) /\ ( J fLim f ) = (/) ) ) -> ( J fLim f ) = (/) ) |
11 |
5 6 7 8 9 10
|
alexsublem |
|- -. ( ph /\ ( f e. ( UFil ` X ) /\ ( J fLim f ) = (/) ) ) |
12 |
11
|
pm2.21i |
|- ( ( ph /\ ( f e. ( UFil ` X ) /\ ( J fLim f ) = (/) ) ) -> -. ( J fLim f ) = (/) ) |
13 |
12
|
expr |
|- ( ( ph /\ f e. ( UFil ` X ) ) -> ( ( J fLim f ) = (/) -> -. ( J fLim f ) = (/) ) ) |
14 |
13
|
pm2.01d |
|- ( ( ph /\ f e. ( UFil ` X ) ) -> -. ( J fLim f ) = (/) ) |
15 |
14
|
neqned |
|- ( ( ph /\ f e. ( UFil ` X ) ) -> ( J fLim f ) =/= (/) ) |
16 |
15
|
ralrimiva |
|- ( ph -> A. f e. ( UFil ` X ) ( J fLim f ) =/= (/) ) |
17 |
|
fibas |
|- ( fi ` B ) e. TopBases |
18 |
|
tgtopon |
|- ( ( fi ` B ) e. TopBases -> ( topGen ` ( fi ` B ) ) e. ( TopOn ` U. ( fi ` B ) ) ) |
19 |
17 18
|
ax-mp |
|- ( topGen ` ( fi ` B ) ) e. ( TopOn ` U. ( fi ` B ) ) |
20 |
3 19
|
eqeltrdi |
|- ( ph -> J e. ( TopOn ` U. ( fi ` B ) ) ) |
21 |
1
|
elexd |
|- ( ph -> X e. _V ) |
22 |
2 21
|
eqeltrrd |
|- ( ph -> U. B e. _V ) |
23 |
|
uniexb |
|- ( B e. _V <-> U. B e. _V ) |
24 |
22 23
|
sylibr |
|- ( ph -> B e. _V ) |
25 |
|
fiuni |
|- ( B e. _V -> U. B = U. ( fi ` B ) ) |
26 |
24 25
|
syl |
|- ( ph -> U. B = U. ( fi ` B ) ) |
27 |
2 26
|
eqtrd |
|- ( ph -> X = U. ( fi ` B ) ) |
28 |
27
|
fveq2d |
|- ( ph -> ( TopOn ` X ) = ( TopOn ` U. ( fi ` B ) ) ) |
29 |
20 28
|
eleqtrrd |
|- ( ph -> J e. ( TopOn ` X ) ) |
30 |
|
ufilcmp |
|- ( ( X e. UFL /\ J e. ( TopOn ` X ) ) -> ( J e. Comp <-> A. f e. ( UFil ` X ) ( J fLim f ) =/= (/) ) ) |
31 |
1 29 30
|
syl2anc |
|- ( ph -> ( J e. Comp <-> A. f e. ( UFil ` X ) ( J fLim f ) =/= (/) ) ) |
32 |
16 31
|
mpbird |
|- ( ph -> J e. Comp ) |