Step |
Hyp |
Ref |
Expression |
1 |
|
ovex |
|- ( ~P U. ran J ^pm dom J ) e. _V |
2 |
|
brssc |
|- ( h C_cat J <-> E. t ( J Fn ( t X. t ) /\ E. s e. ~P t h e. X_ x e. ( s X. s ) ~P ( J ` x ) ) ) |
3 |
|
simpl |
|- ( ( J Fn ( t X. t ) /\ ( s e. ~P t /\ h e. X_ x e. ( s X. s ) ~P ( J ` x ) ) ) -> J Fn ( t X. t ) ) |
4 |
|
vex |
|- t e. _V |
5 |
4 4
|
xpex |
|- ( t X. t ) e. _V |
6 |
|
fnex |
|- ( ( J Fn ( t X. t ) /\ ( t X. t ) e. _V ) -> J e. _V ) |
7 |
3 5 6
|
sylancl |
|- ( ( J Fn ( t X. t ) /\ ( s e. ~P t /\ h e. X_ x e. ( s X. s ) ~P ( J ` x ) ) ) -> J e. _V ) |
8 |
|
rnexg |
|- ( J e. _V -> ran J e. _V ) |
9 |
|
uniexg |
|- ( ran J e. _V -> U. ran J e. _V ) |
10 |
|
pwexg |
|- ( U. ran J e. _V -> ~P U. ran J e. _V ) |
11 |
7 8 9 10
|
4syl |
|- ( ( J Fn ( t X. t ) /\ ( s e. ~P t /\ h e. X_ x e. ( s X. s ) ~P ( J ` x ) ) ) -> ~P U. ran J e. _V ) |
12 |
|
fndm |
|- ( J Fn ( t X. t ) -> dom J = ( t X. t ) ) |
13 |
12
|
adantr |
|- ( ( J Fn ( t X. t ) /\ ( s e. ~P t /\ h e. X_ x e. ( s X. s ) ~P ( J ` x ) ) ) -> dom J = ( t X. t ) ) |
14 |
13 5
|
eqeltrdi |
|- ( ( J Fn ( t X. t ) /\ ( s e. ~P t /\ h e. X_ x e. ( s X. s ) ~P ( J ` x ) ) ) -> dom J e. _V ) |
15 |
|
ss2ixp |
|- ( A. x e. ( s X. s ) ~P ( J ` x ) C_ ~P U. ran J -> X_ x e. ( s X. s ) ~P ( J ` x ) C_ X_ x e. ( s X. s ) ~P U. ran J ) |
16 |
|
fvssunirn |
|- ( J ` x ) C_ U. ran J |
17 |
16
|
sspwi |
|- ~P ( J ` x ) C_ ~P U. ran J |
18 |
17
|
a1i |
|- ( x e. ( s X. s ) -> ~P ( J ` x ) C_ ~P U. ran J ) |
19 |
15 18
|
mprg |
|- X_ x e. ( s X. s ) ~P ( J ` x ) C_ X_ x e. ( s X. s ) ~P U. ran J |
20 |
|
simprr |
|- ( ( J Fn ( t X. t ) /\ ( s e. ~P t /\ h e. X_ x e. ( s X. s ) ~P ( J ` x ) ) ) -> h e. X_ x e. ( s X. s ) ~P ( J ` x ) ) |
21 |
19 20
|
sselid |
|- ( ( J Fn ( t X. t ) /\ ( s e. ~P t /\ h e. X_ x e. ( s X. s ) ~P ( J ` x ) ) ) -> h e. X_ x e. ( s X. s ) ~P U. ran J ) |
22 |
|
vex |
|- h e. _V |
23 |
22
|
elixpconst |
|- ( h e. X_ x e. ( s X. s ) ~P U. ran J <-> h : ( s X. s ) --> ~P U. ran J ) |
24 |
21 23
|
sylib |
|- ( ( J Fn ( t X. t ) /\ ( s e. ~P t /\ h e. X_ x e. ( s X. s ) ~P ( J ` x ) ) ) -> h : ( s X. s ) --> ~P U. ran J ) |
25 |
|
elpwi |
|- ( s e. ~P t -> s C_ t ) |
26 |
25
|
ad2antrl |
|- ( ( J Fn ( t X. t ) /\ ( s e. ~P t /\ h e. X_ x e. ( s X. s ) ~P ( J ` x ) ) ) -> s C_ t ) |
27 |
|
xpss12 |
|- ( ( s C_ t /\ s C_ t ) -> ( s X. s ) C_ ( t X. t ) ) |
28 |
26 26 27
|
syl2anc |
|- ( ( J Fn ( t X. t ) /\ ( s e. ~P t /\ h e. X_ x e. ( s X. s ) ~P ( J ` x ) ) ) -> ( s X. s ) C_ ( t X. t ) ) |
29 |
28 13
|
sseqtrrd |
|- ( ( J Fn ( t X. t ) /\ ( s e. ~P t /\ h e. X_ x e. ( s X. s ) ~P ( J ` x ) ) ) -> ( s X. s ) C_ dom J ) |
30 |
|
elpm2r |
|- ( ( ( ~P U. ran J e. _V /\ dom J e. _V ) /\ ( h : ( s X. s ) --> ~P U. ran J /\ ( s X. s ) C_ dom J ) ) -> h e. ( ~P U. ran J ^pm dom J ) ) |
31 |
11 14 24 29 30
|
syl22anc |
|- ( ( J Fn ( t X. t ) /\ ( s e. ~P t /\ h e. X_ x e. ( s X. s ) ~P ( J ` x ) ) ) -> h e. ( ~P U. ran J ^pm dom J ) ) |
32 |
31
|
rexlimdvaa |
|- ( J Fn ( t X. t ) -> ( E. s e. ~P t h e. X_ x e. ( s X. s ) ~P ( J ` x ) -> h e. ( ~P U. ran J ^pm dom J ) ) ) |
33 |
32
|
imp |
|- ( ( J Fn ( t X. t ) /\ E. s e. ~P t h e. X_ x e. ( s X. s ) ~P ( J ` x ) ) -> h e. ( ~P U. ran J ^pm dom J ) ) |
34 |
33
|
exlimiv |
|- ( E. t ( J Fn ( t X. t ) /\ E. s e. ~P t h e. X_ x e. ( s X. s ) ~P ( J ` x ) ) -> h e. ( ~P U. ran J ^pm dom J ) ) |
35 |
2 34
|
sylbi |
|- ( h C_cat J -> h e. ( ~P U. ran J ^pm dom J ) ) |
36 |
35
|
abssi |
|- { h | h C_cat J } C_ ( ~P U. ran J ^pm dom J ) |
37 |
1 36
|
ssexi |
|- { h | h C_cat J } e. _V |