Step |
Hyp |
Ref |
Expression |
0 |
|
cefmnd |
|- EndoFMnd |
1 |
|
vx |
|- x |
2 |
|
cvv |
|- _V |
3 |
1
|
cv |
|- x |
4 |
|
cmap |
|- ^m |
5 |
3 3 4
|
co |
|- ( x ^m x ) |
6 |
|
vb |
|- b |
7 |
|
cbs |
|- Base |
8 |
|
cnx |
|- ndx |
9 |
8 7
|
cfv |
|- ( Base ` ndx ) |
10 |
6
|
cv |
|- b |
11 |
9 10
|
cop |
|- <. ( Base ` ndx ) , b >. |
12 |
|
cplusg |
|- +g |
13 |
8 12
|
cfv |
|- ( +g ` ndx ) |
14 |
|
vf |
|- f |
15 |
|
vg |
|- g |
16 |
14
|
cv |
|- f |
17 |
15
|
cv |
|- g |
18 |
16 17
|
ccom |
|- ( f o. g ) |
19 |
14 15 10 10 18
|
cmpo |
|- ( f e. b , g e. b |-> ( f o. g ) ) |
20 |
13 19
|
cop |
|- <. ( +g ` ndx ) , ( f e. b , g e. b |-> ( f o. g ) ) >. |
21 |
|
cts |
|- TopSet |
22 |
8 21
|
cfv |
|- ( TopSet ` ndx ) |
23 |
|
cpt |
|- Xt_ |
24 |
3
|
cpw |
|- ~P x |
25 |
24
|
csn |
|- { ~P x } |
26 |
3 25
|
cxp |
|- ( x X. { ~P x } ) |
27 |
26 23
|
cfv |
|- ( Xt_ ` ( x X. { ~P x } ) ) |
28 |
22 27
|
cop |
|- <. ( TopSet ` ndx ) , ( Xt_ ` ( x X. { ~P x } ) ) >. |
29 |
11 20 28
|
ctp |
|- { <. ( Base ` ndx ) , b >. , <. ( +g ` ndx ) , ( f e. b , g e. b |-> ( f o. g ) ) >. , <. ( TopSet ` ndx ) , ( Xt_ ` ( x X. { ~P x } ) ) >. } |
30 |
6 5 29
|
csb |
|- [_ ( x ^m x ) / b ]_ { <. ( Base ` ndx ) , b >. , <. ( +g ` ndx ) , ( f e. b , g e. b |-> ( f o. g ) ) >. , <. ( TopSet ` ndx ) , ( Xt_ ` ( x X. { ~P x } ) ) >. } |
31 |
1 2 30
|
cmpt |
|- ( x e. _V |-> [_ ( x ^m x ) / b ]_ { <. ( Base ` ndx ) , b >. , <. ( +g ` ndx ) , ( f e. b , g e. b |-> ( f o. g ) ) >. , <. ( TopSet ` ndx ) , ( Xt_ ` ( x X. { ~P x } ) ) >. } ) |
32 |
0 31
|
wceq |
|- EndoFMnd = ( x e. _V |-> [_ ( x ^m x ) / b ]_ { <. ( Base ` ndx ) , b >. , <. ( +g ` ndx ) , ( f e. b , g e. b |-> ( f o. g ) ) >. , <. ( TopSet ` ndx ) , ( Xt_ ` ( x X. { ~P x } ) ) >. } ) |