Step |
Hyp |
Ref |
Expression |
1 |
|
simprl |
|- ( ( N C_ ( ( nei ` J ) ` S ) /\ ( N e. Fin /\ N =/= (/) ) ) -> N e. Fin ) |
2 |
|
innei |
|- ( ( J e. Top /\ x e. ( ( nei ` J ) ` S ) /\ y e. ( ( nei ` J ) ` S ) ) -> ( x i^i y ) e. ( ( nei ` J ) ` S ) ) |
3 |
2
|
3expib |
|- ( J e. Top -> ( ( x e. ( ( nei ` J ) ` S ) /\ y e. ( ( nei ` J ) ` S ) ) -> ( x i^i y ) e. ( ( nei ` J ) ` S ) ) ) |
4 |
3
|
ralrimivv |
|- ( J e. Top -> A. x e. ( ( nei ` J ) ` S ) A. y e. ( ( nei ` J ) ` S ) ( x i^i y ) e. ( ( nei ` J ) ` S ) ) |
5 |
|
fiint |
|- ( A. x e. ( ( nei ` J ) ` S ) A. y e. ( ( nei ` J ) ` S ) ( x i^i y ) e. ( ( nei ` J ) ` S ) <-> A. x ( ( x C_ ( ( nei ` J ) ` S ) /\ x =/= (/) /\ x e. Fin ) -> |^| x e. ( ( nei ` J ) ` S ) ) ) |
6 |
4 5
|
sylib |
|- ( J e. Top -> A. x ( ( x C_ ( ( nei ` J ) ` S ) /\ x =/= (/) /\ x e. Fin ) -> |^| x e. ( ( nei ` J ) ` S ) ) ) |
7 |
|
sseq1 |
|- ( x = N -> ( x C_ ( ( nei ` J ) ` S ) <-> N C_ ( ( nei ` J ) ` S ) ) ) |
8 |
|
neeq1 |
|- ( x = N -> ( x =/= (/) <-> N =/= (/) ) ) |
9 |
|
eleq1 |
|- ( x = N -> ( x e. Fin <-> N e. Fin ) ) |
10 |
7 8 9
|
3anbi123d |
|- ( x = N -> ( ( x C_ ( ( nei ` J ) ` S ) /\ x =/= (/) /\ x e. Fin ) <-> ( N C_ ( ( nei ` J ) ` S ) /\ N =/= (/) /\ N e. Fin ) ) ) |
11 |
|
3ancomb |
|- ( ( N C_ ( ( nei ` J ) ` S ) /\ N =/= (/) /\ N e. Fin ) <-> ( N C_ ( ( nei ` J ) ` S ) /\ N e. Fin /\ N =/= (/) ) ) |
12 |
|
3anass |
|- ( ( N C_ ( ( nei ` J ) ` S ) /\ N e. Fin /\ N =/= (/) ) <-> ( N C_ ( ( nei ` J ) ` S ) /\ ( N e. Fin /\ N =/= (/) ) ) ) |
13 |
11 12
|
bitri |
|- ( ( N C_ ( ( nei ` J ) ` S ) /\ N =/= (/) /\ N e. Fin ) <-> ( N C_ ( ( nei ` J ) ` S ) /\ ( N e. Fin /\ N =/= (/) ) ) ) |
14 |
10 13
|
syl6bb |
|- ( x = N -> ( ( x C_ ( ( nei ` J ) ` S ) /\ x =/= (/) /\ x e. Fin ) <-> ( N C_ ( ( nei ` J ) ` S ) /\ ( N e. Fin /\ N =/= (/) ) ) ) ) |
15 |
|
inteq |
|- ( x = N -> |^| x = |^| N ) |
16 |
15
|
eleq1d |
|- ( x = N -> ( |^| x e. ( ( nei ` J ) ` S ) <-> |^| N e. ( ( nei ` J ) ` S ) ) ) |
17 |
14 16
|
imbi12d |
|- ( x = N -> ( ( ( x C_ ( ( nei ` J ) ` S ) /\ x =/= (/) /\ x e. Fin ) -> |^| x e. ( ( nei ` J ) ` S ) ) <-> ( ( N C_ ( ( nei ` J ) ` S ) /\ ( N e. Fin /\ N =/= (/) ) ) -> |^| N e. ( ( nei ` J ) ` S ) ) ) ) |
18 |
17
|
spcgv |
|- ( N e. Fin -> ( A. x ( ( x C_ ( ( nei ` J ) ` S ) /\ x =/= (/) /\ x e. Fin ) -> |^| x e. ( ( nei ` J ) ` S ) ) -> ( ( N C_ ( ( nei ` J ) ` S ) /\ ( N e. Fin /\ N =/= (/) ) ) -> |^| N e. ( ( nei ` J ) ` S ) ) ) ) |
19 |
6 18
|
syl5 |
|- ( N e. Fin -> ( J e. Top -> ( ( N C_ ( ( nei ` J ) ` S ) /\ ( N e. Fin /\ N =/= (/) ) ) -> |^| N e. ( ( nei ` J ) ` S ) ) ) ) |
20 |
19
|
com3l |
|- ( J e. Top -> ( ( N C_ ( ( nei ` J ) ` S ) /\ ( N e. Fin /\ N =/= (/) ) ) -> ( N e. Fin -> |^| N e. ( ( nei ` J ) ` S ) ) ) ) |
21 |
1 20
|
mpdi |
|- ( J e. Top -> ( ( N C_ ( ( nei ` J ) ` S ) /\ ( N e. Fin /\ N =/= (/) ) ) -> |^| N e. ( ( nei ` J ) ` S ) ) ) |
22 |
21
|
impl |
|- ( ( ( J e. Top /\ N C_ ( ( nei ` J ) ` S ) ) /\ ( N e. Fin /\ N =/= (/) ) ) -> |^| N e. ( ( nei ` J ) ` S ) ) |