Step |
Hyp |
Ref |
Expression |
1 |
|
funrel |
|- ( Fun F -> Rel F ) |
2 |
1
|
3ad2ant1 |
|- ( ( Fun F /\ F e. V /\ Z e. W ) -> Rel F ) |
3 |
|
suppssdm |
|- ( F supp Z ) C_ dom F |
4 |
|
undif |
|- ( ( F supp Z ) C_ dom F <-> ( ( F supp Z ) u. ( dom F \ ( F supp Z ) ) ) = dom F ) |
5 |
4
|
biimpi |
|- ( ( F supp Z ) C_ dom F -> ( ( F supp Z ) u. ( dom F \ ( F supp Z ) ) ) = dom F ) |
6 |
5
|
eqcomd |
|- ( ( F supp Z ) C_ dom F -> dom F = ( ( F supp Z ) u. ( dom F \ ( F supp Z ) ) ) ) |
7 |
3 6
|
mp1i |
|- ( ( Fun F /\ F e. V /\ Z e. W ) -> dom F = ( ( F supp Z ) u. ( dom F \ ( F supp Z ) ) ) ) |
8 |
|
disjdif |
|- ( ( F supp Z ) i^i ( dom F \ ( F supp Z ) ) ) = (/) |
9 |
8
|
a1i |
|- ( ( Fun F /\ F e. V /\ Z e. W ) -> ( ( F supp Z ) i^i ( dom F \ ( F supp Z ) ) ) = (/) ) |
10 |
|
reldisjun |
|- ( ( Rel F /\ dom F = ( ( F supp Z ) u. ( dom F \ ( F supp Z ) ) ) /\ ( ( F supp Z ) i^i ( dom F \ ( F supp Z ) ) ) = (/) ) -> F = ( ( F |` ( F supp Z ) ) u. ( F |` ( dom F \ ( F supp Z ) ) ) ) ) |
11 |
2 7 9 10
|
syl3anc |
|- ( ( Fun F /\ F e. V /\ Z e. W ) -> F = ( ( F |` ( F supp Z ) ) u. ( F |` ( dom F \ ( F supp Z ) ) ) ) ) |
12 |
11
|
difeq1d |
|- ( ( Fun F /\ F e. V /\ Z e. W ) -> ( F \ ( F |` ( dom F \ ( F supp Z ) ) ) ) = ( ( ( F |` ( F supp Z ) ) u. ( F |` ( dom F \ ( F supp Z ) ) ) ) \ ( F |` ( dom F \ ( F supp Z ) ) ) ) ) |
13 |
|
resss |
|- ( F |` ( dom F \ ( F supp Z ) ) ) C_ F |
14 |
|
sseqin2 |
|- ( ( F |` ( dom F \ ( F supp Z ) ) ) C_ F <-> ( F i^i ( F |` ( dom F \ ( F supp Z ) ) ) ) = ( F |` ( dom F \ ( F supp Z ) ) ) ) |
15 |
13 14
|
mpbi |
|- ( F i^i ( F |` ( dom F \ ( F supp Z ) ) ) ) = ( F |` ( dom F \ ( F supp Z ) ) ) |
16 |
|
suppiniseg |
|- ( ( Fun F /\ F e. V /\ Z e. W ) -> ( dom F \ ( F supp Z ) ) = ( `' F " { Z } ) ) |
17 |
16
|
reseq2d |
|- ( ( Fun F /\ F e. V /\ Z e. W ) -> ( F |` ( dom F \ ( F supp Z ) ) ) = ( F |` ( `' F " { Z } ) ) ) |
18 |
|
cnvrescnv |
|- `' ( `' F |` { Z } ) = ( F i^i ( _V X. { Z } ) ) |
19 |
|
funcnvres2 |
|- ( Fun F -> `' ( `' F |` { Z } ) = ( F |` ( `' F " { Z } ) ) ) |
20 |
18 19
|
eqtr3id |
|- ( Fun F -> ( F i^i ( _V X. { Z } ) ) = ( F |` ( `' F " { Z } ) ) ) |
21 |
20
|
3ad2ant1 |
|- ( ( Fun F /\ F e. V /\ Z e. W ) -> ( F i^i ( _V X. { Z } ) ) = ( F |` ( `' F " { Z } ) ) ) |
22 |
17 21
|
eqtr4d |
|- ( ( Fun F /\ F e. V /\ Z e. W ) -> ( F |` ( dom F \ ( F supp Z ) ) ) = ( F i^i ( _V X. { Z } ) ) ) |
23 |
15 22
|
syl5eq |
|- ( ( Fun F /\ F e. V /\ Z e. W ) -> ( F i^i ( F |` ( dom F \ ( F supp Z ) ) ) ) = ( F i^i ( _V X. { Z } ) ) ) |
24 |
|
indifbi |
|- ( ( F i^i ( F |` ( dom F \ ( F supp Z ) ) ) ) = ( F i^i ( _V X. { Z } ) ) <-> ( F \ ( F |` ( dom F \ ( F supp Z ) ) ) ) = ( F \ ( _V X. { Z } ) ) ) |
25 |
23 24
|
sylib |
|- ( ( Fun F /\ F e. V /\ Z e. W ) -> ( F \ ( F |` ( dom F \ ( F supp Z ) ) ) ) = ( F \ ( _V X. { Z } ) ) ) |
26 |
8
|
reseq2i |
|- ( F |` ( ( F supp Z ) i^i ( dom F \ ( F supp Z ) ) ) ) = ( F |` (/) ) |
27 |
|
resindi |
|- ( F |` ( ( F supp Z ) i^i ( dom F \ ( F supp Z ) ) ) ) = ( ( F |` ( F supp Z ) ) i^i ( F |` ( dom F \ ( F supp Z ) ) ) ) |
28 |
|
res0 |
|- ( F |` (/) ) = (/) |
29 |
26 27 28
|
3eqtr3i |
|- ( ( F |` ( F supp Z ) ) i^i ( F |` ( dom F \ ( F supp Z ) ) ) ) = (/) |
30 |
|
undif5 |
|- ( ( ( F |` ( F supp Z ) ) i^i ( F |` ( dom F \ ( F supp Z ) ) ) ) = (/) -> ( ( ( F |` ( F supp Z ) ) u. ( F |` ( dom F \ ( F supp Z ) ) ) ) \ ( F |` ( dom F \ ( F supp Z ) ) ) ) = ( F |` ( F supp Z ) ) ) |
31 |
29 30
|
mp1i |
|- ( ( Fun F /\ F e. V /\ Z e. W ) -> ( ( ( F |` ( F supp Z ) ) u. ( F |` ( dom F \ ( F supp Z ) ) ) ) \ ( F |` ( dom F \ ( F supp Z ) ) ) ) = ( F |` ( F supp Z ) ) ) |
32 |
12 25 31
|
3eqtr3rd |
|- ( ( Fun F /\ F e. V /\ Z e. W ) -> ( F |` ( F supp Z ) ) = ( F \ ( _V X. { Z } ) ) ) |