Step |
Hyp |
Ref |
Expression |
1 |
|
frrlem5.1 |
|- B = { f | E. x ( f Fn x /\ ( x C_ A /\ A. y e. x Pred ( R , A , y ) C_ x ) /\ A. y e. x ( f ` y ) = ( y G ( f |` Pred ( R , A , y ) ) ) ) } |
2 |
|
frrlem5.2 |
|- F = frecs ( R , A , G ) |
3 |
|
vex |
|- z e. _V |
4 |
3
|
eldm2 |
|- ( z e. dom F <-> E. w <. z , w >. e. F ) |
5 |
1 2
|
frrlem5 |
|- F = U. B |
6 |
1
|
frrlem1 |
|- B = { g | E. a ( g Fn a /\ ( a C_ A /\ A. z e. a Pred ( R , A , z ) C_ a ) /\ A. z e. a ( g ` z ) = ( z G ( g |` Pred ( R , A , z ) ) ) ) } |
7 |
6
|
unieqi |
|- U. B = U. { g | E. a ( g Fn a /\ ( a C_ A /\ A. z e. a Pred ( R , A , z ) C_ a ) /\ A. z e. a ( g ` z ) = ( z G ( g |` Pred ( R , A , z ) ) ) ) } |
8 |
5 7
|
eqtri |
|- F = U. { g | E. a ( g Fn a /\ ( a C_ A /\ A. z e. a Pred ( R , A , z ) C_ a ) /\ A. z e. a ( g ` z ) = ( z G ( g |` Pred ( R , A , z ) ) ) ) } |
9 |
8
|
eleq2i |
|- ( <. z , w >. e. F <-> <. z , w >. e. U. { g | E. a ( g Fn a /\ ( a C_ A /\ A. z e. a Pred ( R , A , z ) C_ a ) /\ A. z e. a ( g ` z ) = ( z G ( g |` Pred ( R , A , z ) ) ) ) } ) |
10 |
|
eluniab |
|- ( <. z , w >. e. U. { g | E. a ( g Fn a /\ ( a C_ A /\ A. z e. a Pred ( R , A , z ) C_ a ) /\ A. z e. a ( g ` z ) = ( z G ( g |` Pred ( R , A , z ) ) ) ) } <-> E. g ( <. z , w >. e. g /\ E. a ( g Fn a /\ ( a C_ A /\ A. z e. a Pred ( R , A , z ) C_ a ) /\ A. z e. a ( g ` z ) = ( z G ( g |` Pred ( R , A , z ) ) ) ) ) ) |
11 |
9 10
|
bitri |
|- ( <. z , w >. e. F <-> E. g ( <. z , w >. e. g /\ E. a ( g Fn a /\ ( a C_ A /\ A. z e. a Pred ( R , A , z ) C_ a ) /\ A. z e. a ( g ` z ) = ( z G ( g |` Pred ( R , A , z ) ) ) ) ) ) |
12 |
|
simpr2r |
|- ( ( <. z , w >. e. g /\ ( g Fn a /\ ( a C_ A /\ A. z e. a Pred ( R , A , z ) C_ a ) /\ A. z e. a ( g ` z ) = ( z G ( g |` Pred ( R , A , z ) ) ) ) ) -> A. z e. a Pred ( R , A , z ) C_ a ) |
13 |
|
vex |
|- w e. _V |
14 |
3 13
|
opeldm |
|- ( <. z , w >. e. g -> z e. dom g ) |
15 |
14
|
adantr |
|- ( ( <. z , w >. e. g /\ ( g Fn a /\ ( a C_ A /\ A. z e. a Pred ( R , A , z ) C_ a ) /\ A. z e. a ( g ` z ) = ( z G ( g |` Pred ( R , A , z ) ) ) ) ) -> z e. dom g ) |
16 |
|
simpr1 |
|- ( ( <. z , w >. e. g /\ ( g Fn a /\ ( a C_ A /\ A. z e. a Pred ( R , A , z ) C_ a ) /\ A. z e. a ( g ` z ) = ( z G ( g |` Pred ( R , A , z ) ) ) ) ) -> g Fn a ) |
17 |
16
|
fndmd |
|- ( ( <. z , w >. e. g /\ ( g Fn a /\ ( a C_ A /\ A. z e. a Pred ( R , A , z ) C_ a ) /\ A. z e. a ( g ` z ) = ( z G ( g |` Pred ( R , A , z ) ) ) ) ) -> dom g = a ) |
18 |
15 17
|
eleqtrd |
|- ( ( <. z , w >. e. g /\ ( g Fn a /\ ( a C_ A /\ A. z e. a Pred ( R , A , z ) C_ a ) /\ A. z e. a ( g ` z ) = ( z G ( g |` Pred ( R , A , z ) ) ) ) ) -> z e. a ) |
19 |
|
rsp |
|- ( A. z e. a Pred ( R , A , z ) C_ a -> ( z e. a -> Pred ( R , A , z ) C_ a ) ) |
20 |
12 18 19
|
sylc |
|- ( ( <. z , w >. e. g /\ ( g Fn a /\ ( a C_ A /\ A. z e. a Pred ( R , A , z ) C_ a ) /\ A. z e. a ( g ` z ) = ( z G ( g |` Pred ( R , A , z ) ) ) ) ) -> Pred ( R , A , z ) C_ a ) |
21 |
20 17
|
sseqtrrd |
|- ( ( <. z , w >. e. g /\ ( g Fn a /\ ( a C_ A /\ A. z e. a Pred ( R , A , z ) C_ a ) /\ A. z e. a ( g ` z ) = ( z G ( g |` Pred ( R , A , z ) ) ) ) ) -> Pred ( R , A , z ) C_ dom g ) |
22 |
|
19.8a |
|- ( ( g Fn a /\ ( a C_ A /\ A. z e. a Pred ( R , A , z ) C_ a ) /\ A. z e. a ( g ` z ) = ( z G ( g |` Pred ( R , A , z ) ) ) ) -> E. a ( g Fn a /\ ( a C_ A /\ A. z e. a Pred ( R , A , z ) C_ a ) /\ A. z e. a ( g ` z ) = ( z G ( g |` Pred ( R , A , z ) ) ) ) ) |
23 |
6
|
abeq2i |
|- ( g e. B <-> E. a ( g Fn a /\ ( a C_ A /\ A. z e. a Pred ( R , A , z ) C_ a ) /\ A. z e. a ( g ` z ) = ( z G ( g |` Pred ( R , A , z ) ) ) ) ) |
24 |
22 23
|
sylibr |
|- ( ( g Fn a /\ ( a C_ A /\ A. z e. a Pred ( R , A , z ) C_ a ) /\ A. z e. a ( g ` z ) = ( z G ( g |` Pred ( R , A , z ) ) ) ) -> g e. B ) |
25 |
24
|
adantl |
|- ( ( <. z , w >. e. g /\ ( g Fn a /\ ( a C_ A /\ A. z e. a Pred ( R , A , z ) C_ a ) /\ A. z e. a ( g ` z ) = ( z G ( g |` Pred ( R , A , z ) ) ) ) ) -> g e. B ) |
26 |
|
elssuni |
|- ( g e. B -> g C_ U. B ) |
27 |
25 26
|
syl |
|- ( ( <. z , w >. e. g /\ ( g Fn a /\ ( a C_ A /\ A. z e. a Pred ( R , A , z ) C_ a ) /\ A. z e. a ( g ` z ) = ( z G ( g |` Pred ( R , A , z ) ) ) ) ) -> g C_ U. B ) |
28 |
27 5
|
sseqtrrdi |
|- ( ( <. z , w >. e. g /\ ( g Fn a /\ ( a C_ A /\ A. z e. a Pred ( R , A , z ) C_ a ) /\ A. z e. a ( g ` z ) = ( z G ( g |` Pred ( R , A , z ) ) ) ) ) -> g C_ F ) |
29 |
|
dmss |
|- ( g C_ F -> dom g C_ dom F ) |
30 |
28 29
|
syl |
|- ( ( <. z , w >. e. g /\ ( g Fn a /\ ( a C_ A /\ A. z e. a Pred ( R , A , z ) C_ a ) /\ A. z e. a ( g ` z ) = ( z G ( g |` Pred ( R , A , z ) ) ) ) ) -> dom g C_ dom F ) |
31 |
21 30
|
sstrd |
|- ( ( <. z , w >. e. g /\ ( g Fn a /\ ( a C_ A /\ A. z e. a Pred ( R , A , z ) C_ a ) /\ A. z e. a ( g ` z ) = ( z G ( g |` Pred ( R , A , z ) ) ) ) ) -> Pred ( R , A , z ) C_ dom F ) |
32 |
31
|
expcom |
|- ( ( g Fn a /\ ( a C_ A /\ A. z e. a Pred ( R , A , z ) C_ a ) /\ A. z e. a ( g ` z ) = ( z G ( g |` Pred ( R , A , z ) ) ) ) -> ( <. z , w >. e. g -> Pred ( R , A , z ) C_ dom F ) ) |
33 |
32
|
exlimiv |
|- ( E. a ( g Fn a /\ ( a C_ A /\ A. z e. a Pred ( R , A , z ) C_ a ) /\ A. z e. a ( g ` z ) = ( z G ( g |` Pred ( R , A , z ) ) ) ) -> ( <. z , w >. e. g -> Pred ( R , A , z ) C_ dom F ) ) |
34 |
33
|
impcom |
|- ( ( <. z , w >. e. g /\ E. a ( g Fn a /\ ( a C_ A /\ A. z e. a Pred ( R , A , z ) C_ a ) /\ A. z e. a ( g ` z ) = ( z G ( g |` Pred ( R , A , z ) ) ) ) ) -> Pred ( R , A , z ) C_ dom F ) |
35 |
34
|
exlimiv |
|- ( E. g ( <. z , w >. e. g /\ E. a ( g Fn a /\ ( a C_ A /\ A. z e. a Pred ( R , A , z ) C_ a ) /\ A. z e. a ( g ` z ) = ( z G ( g |` Pred ( R , A , z ) ) ) ) ) -> Pred ( R , A , z ) C_ dom F ) |
36 |
11 35
|
sylbi |
|- ( <. z , w >. e. F -> Pred ( R , A , z ) C_ dom F ) |
37 |
36
|
exlimiv |
|- ( E. w <. z , w >. e. F -> Pred ( R , A , z ) C_ dom F ) |
38 |
4 37
|
sylbi |
|- ( z e. dom F -> Pred ( R , A , z ) C_ dom F ) |