Step |
Hyp |
Ref |
Expression |
1 |
|
conncn.x |
|- X = U. J |
2 |
|
conncn.j |
|- ( ph -> J e. Conn ) |
3 |
|
conncn.f |
|- ( ph -> F e. ( J Cn K ) ) |
4 |
|
conncn.u |
|- ( ph -> U e. K ) |
5 |
|
conncn.c |
|- ( ph -> U e. ( Clsd ` K ) ) |
6 |
|
conncn.a |
|- ( ph -> A e. X ) |
7 |
|
conncn.1 |
|- ( ph -> ( F ` A ) e. U ) |
8 |
|
eqid |
|- U. K = U. K |
9 |
1 8
|
cnf |
|- ( F e. ( J Cn K ) -> F : X --> U. K ) |
10 |
3 9
|
syl |
|- ( ph -> F : X --> U. K ) |
11 |
10
|
ffnd |
|- ( ph -> F Fn X ) |
12 |
10
|
frnd |
|- ( ph -> ran F C_ U. K ) |
13 |
|
dffn4 |
|- ( F Fn X <-> F : X -onto-> ran F ) |
14 |
11 13
|
sylib |
|- ( ph -> F : X -onto-> ran F ) |
15 |
|
cntop2 |
|- ( F e. ( J Cn K ) -> K e. Top ) |
16 |
3 15
|
syl |
|- ( ph -> K e. Top ) |
17 |
8
|
restuni |
|- ( ( K e. Top /\ ran F C_ U. K ) -> ran F = U. ( K |`t ran F ) ) |
18 |
16 12 17
|
syl2anc |
|- ( ph -> ran F = U. ( K |`t ran F ) ) |
19 |
|
foeq3 |
|- ( ran F = U. ( K |`t ran F ) -> ( F : X -onto-> ran F <-> F : X -onto-> U. ( K |`t ran F ) ) ) |
20 |
18 19
|
syl |
|- ( ph -> ( F : X -onto-> ran F <-> F : X -onto-> U. ( K |`t ran F ) ) ) |
21 |
14 20
|
mpbid |
|- ( ph -> F : X -onto-> U. ( K |`t ran F ) ) |
22 |
|
toptopon2 |
|- ( K e. Top <-> K e. ( TopOn ` U. K ) ) |
23 |
16 22
|
sylib |
|- ( ph -> K e. ( TopOn ` U. K ) ) |
24 |
|
ssidd |
|- ( ph -> ran F C_ ran F ) |
25 |
|
cnrest2 |
|- ( ( K e. ( TopOn ` U. K ) /\ ran F C_ ran F /\ ran F C_ U. K ) -> ( F e. ( J Cn K ) <-> F e. ( J Cn ( K |`t ran F ) ) ) ) |
26 |
23 24 12 25
|
syl3anc |
|- ( ph -> ( F e. ( J Cn K ) <-> F e. ( J Cn ( K |`t ran F ) ) ) ) |
27 |
3 26
|
mpbid |
|- ( ph -> F e. ( J Cn ( K |`t ran F ) ) ) |
28 |
|
eqid |
|- U. ( K |`t ran F ) = U. ( K |`t ran F ) |
29 |
28
|
cnconn |
|- ( ( J e. Conn /\ F : X -onto-> U. ( K |`t ran F ) /\ F e. ( J Cn ( K |`t ran F ) ) ) -> ( K |`t ran F ) e. Conn ) |
30 |
2 21 27 29
|
syl3anc |
|- ( ph -> ( K |`t ran F ) e. Conn ) |
31 |
|
fnfvelrn |
|- ( ( F Fn X /\ A e. X ) -> ( F ` A ) e. ran F ) |
32 |
11 6 31
|
syl2anc |
|- ( ph -> ( F ` A ) e. ran F ) |
33 |
|
inelcm |
|- ( ( ( F ` A ) e. U /\ ( F ` A ) e. ran F ) -> ( U i^i ran F ) =/= (/) ) |
34 |
7 32 33
|
syl2anc |
|- ( ph -> ( U i^i ran F ) =/= (/) ) |
35 |
8 12 30 4 34 5
|
connsubclo |
|- ( ph -> ran F C_ U ) |
36 |
|
df-f |
|- ( F : X --> U <-> ( F Fn X /\ ran F C_ U ) ) |
37 |
11 35 36
|
sylanbrc |
|- ( ph -> F : X --> U ) |