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 |
|
tfrlem.3 |
|- C = ( recs ( F ) u. { <. dom recs ( F ) , ( F ` recs ( F ) ) >. } ) |
3 |
|
elsuci |
|- ( B e. suc dom recs ( F ) -> ( B e. dom recs ( F ) \/ B = dom recs ( F ) ) ) |
4 |
1 2
|
tfrlem10 |
|- ( dom recs ( F ) e. On -> C Fn suc dom recs ( F ) ) |
5 |
|
fnfun |
|- ( C Fn suc dom recs ( F ) -> Fun C ) |
6 |
4 5
|
syl |
|- ( dom recs ( F ) e. On -> Fun C ) |
7 |
|
ssun1 |
|- recs ( F ) C_ ( recs ( F ) u. { <. dom recs ( F ) , ( F ` recs ( F ) ) >. } ) |
8 |
7 2
|
sseqtrri |
|- recs ( F ) C_ C |
9 |
1
|
tfrlem9 |
|- ( B e. dom recs ( F ) -> ( recs ( F ) ` B ) = ( F ` ( recs ( F ) |` B ) ) ) |
10 |
|
funssfv |
|- ( ( Fun C /\ recs ( F ) C_ C /\ B e. dom recs ( F ) ) -> ( C ` B ) = ( recs ( F ) ` B ) ) |
11 |
10
|
3expa |
|- ( ( ( Fun C /\ recs ( F ) C_ C ) /\ B e. dom recs ( F ) ) -> ( C ` B ) = ( recs ( F ) ` B ) ) |
12 |
11
|
adantrl |
|- ( ( ( Fun C /\ recs ( F ) C_ C ) /\ ( dom recs ( F ) e. On /\ B e. dom recs ( F ) ) ) -> ( C ` B ) = ( recs ( F ) ` B ) ) |
13 |
|
onelss |
|- ( dom recs ( F ) e. On -> ( B e. dom recs ( F ) -> B C_ dom recs ( F ) ) ) |
14 |
13
|
imp |
|- ( ( dom recs ( F ) e. On /\ B e. dom recs ( F ) ) -> B C_ dom recs ( F ) ) |
15 |
|
fun2ssres |
|- ( ( Fun C /\ recs ( F ) C_ C /\ B C_ dom recs ( F ) ) -> ( C |` B ) = ( recs ( F ) |` B ) ) |
16 |
15
|
3expa |
|- ( ( ( Fun C /\ recs ( F ) C_ C ) /\ B C_ dom recs ( F ) ) -> ( C |` B ) = ( recs ( F ) |` B ) ) |
17 |
16
|
fveq2d |
|- ( ( ( Fun C /\ recs ( F ) C_ C ) /\ B C_ dom recs ( F ) ) -> ( F ` ( C |` B ) ) = ( F ` ( recs ( F ) |` B ) ) ) |
18 |
14 17
|
sylan2 |
|- ( ( ( Fun C /\ recs ( F ) C_ C ) /\ ( dom recs ( F ) e. On /\ B e. dom recs ( F ) ) ) -> ( F ` ( C |` B ) ) = ( F ` ( recs ( F ) |` B ) ) ) |
19 |
12 18
|
eqeq12d |
|- ( ( ( Fun C /\ recs ( F ) C_ C ) /\ ( dom recs ( F ) e. On /\ B e. dom recs ( F ) ) ) -> ( ( C ` B ) = ( F ` ( C |` B ) ) <-> ( recs ( F ) ` B ) = ( F ` ( recs ( F ) |` B ) ) ) ) |
20 |
9 19
|
syl5ibr |
|- ( ( ( Fun C /\ recs ( F ) C_ C ) /\ ( dom recs ( F ) e. On /\ B e. dom recs ( F ) ) ) -> ( B e. dom recs ( F ) -> ( C ` B ) = ( F ` ( C |` B ) ) ) ) |
21 |
8 20
|
mpanl2 |
|- ( ( Fun C /\ ( dom recs ( F ) e. On /\ B e. dom recs ( F ) ) ) -> ( B e. dom recs ( F ) -> ( C ` B ) = ( F ` ( C |` B ) ) ) ) |
22 |
6 21
|
sylan |
|- ( ( dom recs ( F ) e. On /\ ( dom recs ( F ) e. On /\ B e. dom recs ( F ) ) ) -> ( B e. dom recs ( F ) -> ( C ` B ) = ( F ` ( C |` B ) ) ) ) |
23 |
22
|
exp32 |
|- ( dom recs ( F ) e. On -> ( dom recs ( F ) e. On -> ( B e. dom recs ( F ) -> ( B e. dom recs ( F ) -> ( C ` B ) = ( F ` ( C |` B ) ) ) ) ) ) |
24 |
23
|
pm2.43i |
|- ( dom recs ( F ) e. On -> ( B e. dom recs ( F ) -> ( B e. dom recs ( F ) -> ( C ` B ) = ( F ` ( C |` B ) ) ) ) ) |
25 |
24
|
pm2.43d |
|- ( dom recs ( F ) e. On -> ( B e. dom recs ( F ) -> ( C ` B ) = ( F ` ( C |` B ) ) ) ) |
26 |
|
opex |
|- <. B , ( F ` ( C |` B ) ) >. e. _V |
27 |
26
|
snid |
|- <. B , ( F ` ( C |` B ) ) >. e. { <. B , ( F ` ( C |` B ) ) >. } |
28 |
|
opeq1 |
|- ( B = dom recs ( F ) -> <. B , ( F ` ( C |` B ) ) >. = <. dom recs ( F ) , ( F ` ( C |` B ) ) >. ) |
29 |
28
|
adantl |
|- ( ( dom recs ( F ) e. On /\ B = dom recs ( F ) ) -> <. B , ( F ` ( C |` B ) ) >. = <. dom recs ( F ) , ( F ` ( C |` B ) ) >. ) |
30 |
|
eqimss |
|- ( B = dom recs ( F ) -> B C_ dom recs ( F ) ) |
31 |
8 15
|
mp3an2 |
|- ( ( Fun C /\ B C_ dom recs ( F ) ) -> ( C |` B ) = ( recs ( F ) |` B ) ) |
32 |
6 30 31
|
syl2an |
|- ( ( dom recs ( F ) e. On /\ B = dom recs ( F ) ) -> ( C |` B ) = ( recs ( F ) |` B ) ) |
33 |
|
reseq2 |
|- ( B = dom recs ( F ) -> ( recs ( F ) |` B ) = ( recs ( F ) |` dom recs ( F ) ) ) |
34 |
1
|
tfrlem6 |
|- Rel recs ( F ) |
35 |
|
resdm |
|- ( Rel recs ( F ) -> ( recs ( F ) |` dom recs ( F ) ) = recs ( F ) ) |
36 |
34 35
|
ax-mp |
|- ( recs ( F ) |` dom recs ( F ) ) = recs ( F ) |
37 |
33 36
|
eqtrdi |
|- ( B = dom recs ( F ) -> ( recs ( F ) |` B ) = recs ( F ) ) |
38 |
37
|
adantl |
|- ( ( dom recs ( F ) e. On /\ B = dom recs ( F ) ) -> ( recs ( F ) |` B ) = recs ( F ) ) |
39 |
32 38
|
eqtrd |
|- ( ( dom recs ( F ) e. On /\ B = dom recs ( F ) ) -> ( C |` B ) = recs ( F ) ) |
40 |
39
|
fveq2d |
|- ( ( dom recs ( F ) e. On /\ B = dom recs ( F ) ) -> ( F ` ( C |` B ) ) = ( F ` recs ( F ) ) ) |
41 |
40
|
opeq2d |
|- ( ( dom recs ( F ) e. On /\ B = dom recs ( F ) ) -> <. dom recs ( F ) , ( F ` ( C |` B ) ) >. = <. dom recs ( F ) , ( F ` recs ( F ) ) >. ) |
42 |
29 41
|
eqtrd |
|- ( ( dom recs ( F ) e. On /\ B = dom recs ( F ) ) -> <. B , ( F ` ( C |` B ) ) >. = <. dom recs ( F ) , ( F ` recs ( F ) ) >. ) |
43 |
42
|
sneqd |
|- ( ( dom recs ( F ) e. On /\ B = dom recs ( F ) ) -> { <. B , ( F ` ( C |` B ) ) >. } = { <. dom recs ( F ) , ( F ` recs ( F ) ) >. } ) |
44 |
27 43
|
eleqtrid |
|- ( ( dom recs ( F ) e. On /\ B = dom recs ( F ) ) -> <. B , ( F ` ( C |` B ) ) >. e. { <. dom recs ( F ) , ( F ` recs ( F ) ) >. } ) |
45 |
|
elun2 |
|- ( <. B , ( F ` ( C |` B ) ) >. e. { <. dom recs ( F ) , ( F ` recs ( F ) ) >. } -> <. B , ( F ` ( C |` B ) ) >. e. ( recs ( F ) u. { <. dom recs ( F ) , ( F ` recs ( F ) ) >. } ) ) |
46 |
44 45
|
syl |
|- ( ( dom recs ( F ) e. On /\ B = dom recs ( F ) ) -> <. B , ( F ` ( C |` B ) ) >. e. ( recs ( F ) u. { <. dom recs ( F ) , ( F ` recs ( F ) ) >. } ) ) |
47 |
46 2
|
eleqtrrdi |
|- ( ( dom recs ( F ) e. On /\ B = dom recs ( F ) ) -> <. B , ( F ` ( C |` B ) ) >. e. C ) |
48 |
|
simpr |
|- ( ( dom recs ( F ) e. On /\ B = dom recs ( F ) ) -> B = dom recs ( F ) ) |
49 |
|
sucidg |
|- ( dom recs ( F ) e. On -> dom recs ( F ) e. suc dom recs ( F ) ) |
50 |
49
|
adantr |
|- ( ( dom recs ( F ) e. On /\ B = dom recs ( F ) ) -> dom recs ( F ) e. suc dom recs ( F ) ) |
51 |
48 50
|
eqeltrd |
|- ( ( dom recs ( F ) e. On /\ B = dom recs ( F ) ) -> B e. suc dom recs ( F ) ) |
52 |
|
fnopfvb |
|- ( ( C Fn suc dom recs ( F ) /\ B e. suc dom recs ( F ) ) -> ( ( C ` B ) = ( F ` ( C |` B ) ) <-> <. B , ( F ` ( C |` B ) ) >. e. C ) ) |
53 |
4 51 52
|
syl2an2r |
|- ( ( dom recs ( F ) e. On /\ B = dom recs ( F ) ) -> ( ( C ` B ) = ( F ` ( C |` B ) ) <-> <. B , ( F ` ( C |` B ) ) >. e. C ) ) |
54 |
47 53
|
mpbird |
|- ( ( dom recs ( F ) e. On /\ B = dom recs ( F ) ) -> ( C ` B ) = ( F ` ( C |` B ) ) ) |
55 |
54
|
ex |
|- ( dom recs ( F ) e. On -> ( B = dom recs ( F ) -> ( C ` B ) = ( F ` ( C |` B ) ) ) ) |
56 |
25 55
|
jaod |
|- ( dom recs ( F ) e. On -> ( ( B e. dom recs ( F ) \/ B = dom recs ( F ) ) -> ( C ` B ) = ( F ` ( C |` B ) ) ) ) |
57 |
3 56
|
syl5 |
|- ( dom recs ( F ) e. On -> ( B e. suc dom recs ( F ) -> ( C ` B ) = ( F ` ( C |` B ) ) ) ) |