Step |
Hyp |
Ref |
Expression |
1 |
|
tfrlem.1 |
|- A = { f | E. x e. On ( f Fn x /\ A. y e. x ( f ` y ) = ( F ` ( f |` y ) ) ) } |
2 |
1
|
tfrlem8 |
|- Ord dom recs ( F ) |
3 |
|
ordzsl |
|- ( Ord dom recs ( F ) <-> ( dom recs ( F ) = (/) \/ E. z e. On dom recs ( F ) = suc z \/ Lim dom recs ( F ) ) ) |
4 |
2 3
|
mpbi |
|- ( dom recs ( F ) = (/) \/ E. z e. On dom recs ( F ) = suc z \/ Lim dom recs ( F ) ) |
5 |
|
res0 |
|- ( recs ( F ) |` (/) ) = (/) |
6 |
|
0ex |
|- (/) e. _V |
7 |
5 6
|
eqeltri |
|- ( recs ( F ) |` (/) ) e. _V |
8 |
|
0elon |
|- (/) e. On |
9 |
1
|
tfrlem15 |
|- ( (/) e. On -> ( (/) e. dom recs ( F ) <-> ( recs ( F ) |` (/) ) e. _V ) ) |
10 |
8 9
|
ax-mp |
|- ( (/) e. dom recs ( F ) <-> ( recs ( F ) |` (/) ) e. _V ) |
11 |
7 10
|
mpbir |
|- (/) e. dom recs ( F ) |
12 |
11
|
n0ii |
|- -. dom recs ( F ) = (/) |
13 |
12
|
pm2.21i |
|- ( dom recs ( F ) = (/) -> Lim dom recs ( F ) ) |
14 |
1
|
tfrlem13 |
|- -. recs ( F ) e. _V |
15 |
|
simpr |
|- ( ( z e. On /\ dom recs ( F ) = suc z ) -> dom recs ( F ) = suc z ) |
16 |
|
df-suc |
|- suc z = ( z u. { z } ) |
17 |
15 16
|
eqtrdi |
|- ( ( z e. On /\ dom recs ( F ) = suc z ) -> dom recs ( F ) = ( z u. { z } ) ) |
18 |
17
|
reseq2d |
|- ( ( z e. On /\ dom recs ( F ) = suc z ) -> ( recs ( F ) |` dom recs ( F ) ) = ( recs ( F ) |` ( z u. { z } ) ) ) |
19 |
1
|
tfrlem6 |
|- Rel recs ( F ) |
20 |
|
resdm |
|- ( Rel recs ( F ) -> ( recs ( F ) |` dom recs ( F ) ) = recs ( F ) ) |
21 |
19 20
|
ax-mp |
|- ( recs ( F ) |` dom recs ( F ) ) = recs ( F ) |
22 |
|
resundi |
|- ( recs ( F ) |` ( z u. { z } ) ) = ( ( recs ( F ) |` z ) u. ( recs ( F ) |` { z } ) ) |
23 |
18 21 22
|
3eqtr3g |
|- ( ( z e. On /\ dom recs ( F ) = suc z ) -> recs ( F ) = ( ( recs ( F ) |` z ) u. ( recs ( F ) |` { z } ) ) ) |
24 |
|
vex |
|- z e. _V |
25 |
24
|
sucid |
|- z e. suc z |
26 |
25 15
|
eleqtrrid |
|- ( ( z e. On /\ dom recs ( F ) = suc z ) -> z e. dom recs ( F ) ) |
27 |
1
|
tfrlem9a |
|- ( z e. dom recs ( F ) -> ( recs ( F ) |` z ) e. _V ) |
28 |
26 27
|
syl |
|- ( ( z e. On /\ dom recs ( F ) = suc z ) -> ( recs ( F ) |` z ) e. _V ) |
29 |
|
snex |
|- { <. z , ( recs ( F ) ` z ) >. } e. _V |
30 |
1
|
tfrlem7 |
|- Fun recs ( F ) |
31 |
|
funressn |
|- ( Fun recs ( F ) -> ( recs ( F ) |` { z } ) C_ { <. z , ( recs ( F ) ` z ) >. } ) |
32 |
30 31
|
ax-mp |
|- ( recs ( F ) |` { z } ) C_ { <. z , ( recs ( F ) ` z ) >. } |
33 |
29 32
|
ssexi |
|- ( recs ( F ) |` { z } ) e. _V |
34 |
|
unexg |
|- ( ( ( recs ( F ) |` z ) e. _V /\ ( recs ( F ) |` { z } ) e. _V ) -> ( ( recs ( F ) |` z ) u. ( recs ( F ) |` { z } ) ) e. _V ) |
35 |
28 33 34
|
sylancl |
|- ( ( z e. On /\ dom recs ( F ) = suc z ) -> ( ( recs ( F ) |` z ) u. ( recs ( F ) |` { z } ) ) e. _V ) |
36 |
23 35
|
eqeltrd |
|- ( ( z e. On /\ dom recs ( F ) = suc z ) -> recs ( F ) e. _V ) |
37 |
36
|
rexlimiva |
|- ( E. z e. On dom recs ( F ) = suc z -> recs ( F ) e. _V ) |
38 |
14 37
|
mto |
|- -. E. z e. On dom recs ( F ) = suc z |
39 |
38
|
pm2.21i |
|- ( E. z e. On dom recs ( F ) = suc z -> Lim dom recs ( F ) ) |
40 |
|
id |
|- ( Lim dom recs ( F ) -> Lim dom recs ( F ) ) |
41 |
13 39 40
|
3jaoi |
|- ( ( dom recs ( F ) = (/) \/ E. z e. On dom recs ( F ) = suc z \/ Lim dom recs ( F ) ) -> Lim dom recs ( F ) ) |
42 |
4 41
|
ax-mp |
|- Lim dom recs ( F ) |