Step |
Hyp |
Ref |
Expression |
1 |
|
bnj944.1 |
|- ( ph <-> ( f ` (/) ) = _pred ( X , A , R ) ) |
2 |
|
bnj944.2 |
|- ( ps <-> A. i e. _om ( suc i e. n -> ( f ` suc i ) = U_ y e. ( f ` i ) _pred ( y , A , R ) ) ) |
3 |
|
bnj944.3 |
|- ( ch <-> ( n e. D /\ f Fn n /\ ph /\ ps ) ) |
4 |
|
bnj944.4 |
|- ( ph' <-> [. p / n ]. ph ) |
5 |
|
bnj944.7 |
|- ( ph" <-> [. G / f ]. ph' ) |
6 |
|
bnj944.10 |
|- D = ( _om \ { (/) } ) |
7 |
|
bnj944.12 |
|- C = U_ y e. ( f ` m ) _pred ( y , A , R ) |
8 |
|
bnj944.13 |
|- G = ( f u. { <. n , C >. } ) |
9 |
|
bnj944.14 |
|- ( ta <-> ( f Fn n /\ ph /\ ps ) ) |
10 |
|
bnj944.15 |
|- ( si <-> ( n e. D /\ p = suc n /\ m e. n ) ) |
11 |
|
simpl |
|- ( ( ( R _FrSe A /\ X e. A ) /\ ( ch /\ n = suc m /\ p = suc n ) ) -> ( R _FrSe A /\ X e. A ) ) |
12 |
|
bnj667 |
|- ( ( n e. D /\ f Fn n /\ ph /\ ps ) -> ( f Fn n /\ ph /\ ps ) ) |
13 |
3 12
|
sylbi |
|- ( ch -> ( f Fn n /\ ph /\ ps ) ) |
14 |
13 9
|
sylibr |
|- ( ch -> ta ) |
15 |
14
|
3ad2ant1 |
|- ( ( ch /\ n = suc m /\ p = suc n ) -> ta ) |
16 |
15
|
adantl |
|- ( ( ( R _FrSe A /\ X e. A ) /\ ( ch /\ n = suc m /\ p = suc n ) ) -> ta ) |
17 |
3
|
bnj1232 |
|- ( ch -> n e. D ) |
18 |
|
vex |
|- m e. _V |
19 |
18
|
bnj216 |
|- ( n = suc m -> m e. n ) |
20 |
|
id |
|- ( p = suc n -> p = suc n ) |
21 |
17 19 20
|
3anim123i |
|- ( ( ch /\ n = suc m /\ p = suc n ) -> ( n e. D /\ m e. n /\ p = suc n ) ) |
22 |
|
3ancomb |
|- ( ( n e. D /\ p = suc n /\ m e. n ) <-> ( n e. D /\ m e. n /\ p = suc n ) ) |
23 |
10 22
|
bitri |
|- ( si <-> ( n e. D /\ m e. n /\ p = suc n ) ) |
24 |
21 23
|
sylibr |
|- ( ( ch /\ n = suc m /\ p = suc n ) -> si ) |
25 |
24
|
adantl |
|- ( ( ( R _FrSe A /\ X e. A ) /\ ( ch /\ n = suc m /\ p = suc n ) ) -> si ) |
26 |
|
bnj253 |
|- ( ( R _FrSe A /\ X e. A /\ ta /\ si ) <-> ( ( R _FrSe A /\ X e. A ) /\ ta /\ si ) ) |
27 |
11 16 25 26
|
syl3anbrc |
|- ( ( ( R _FrSe A /\ X e. A ) /\ ( ch /\ n = suc m /\ p = suc n ) ) -> ( R _FrSe A /\ X e. A /\ ta /\ si ) ) |
28 |
6 9 10 1 2
|
bnj938 |
|- ( ( R _FrSe A /\ X e. A /\ ta /\ si ) -> U_ y e. ( f ` m ) _pred ( y , A , R ) e. _V ) |
29 |
7 28
|
eqeltrid |
|- ( ( R _FrSe A /\ X e. A /\ ta /\ si ) -> C e. _V ) |
30 |
27 29
|
syl |
|- ( ( ( R _FrSe A /\ X e. A ) /\ ( ch /\ n = suc m /\ p = suc n ) ) -> C e. _V ) |
31 |
|
bnj658 |
|- ( ( n e. D /\ f Fn n /\ ph /\ ps ) -> ( n e. D /\ f Fn n /\ ph ) ) |
32 |
3 31
|
sylbi |
|- ( ch -> ( n e. D /\ f Fn n /\ ph ) ) |
33 |
32
|
3ad2ant1 |
|- ( ( ch /\ n = suc m /\ p = suc n ) -> ( n e. D /\ f Fn n /\ ph ) ) |
34 |
|
simp3 |
|- ( ( ch /\ n = suc m /\ p = suc n ) -> p = suc n ) |
35 |
|
bnj291 |
|- ( ( n e. D /\ p = suc n /\ f Fn n /\ ph ) <-> ( ( n e. D /\ f Fn n /\ ph ) /\ p = suc n ) ) |
36 |
33 34 35
|
sylanbrc |
|- ( ( ch /\ n = suc m /\ p = suc n ) -> ( n e. D /\ p = suc n /\ f Fn n /\ ph ) ) |
37 |
36
|
adantl |
|- ( ( ( R _FrSe A /\ X e. A ) /\ ( ch /\ n = suc m /\ p = suc n ) ) -> ( n e. D /\ p = suc n /\ f Fn n /\ ph ) ) |
38 |
|
opeq2 |
|- ( C = if ( C e. _V , C , (/) ) -> <. n , C >. = <. n , if ( C e. _V , C , (/) ) >. ) |
39 |
38
|
sneqd |
|- ( C = if ( C e. _V , C , (/) ) -> { <. n , C >. } = { <. n , if ( C e. _V , C , (/) ) >. } ) |
40 |
39
|
uneq2d |
|- ( C = if ( C e. _V , C , (/) ) -> ( f u. { <. n , C >. } ) = ( f u. { <. n , if ( C e. _V , C , (/) ) >. } ) ) |
41 |
8 40
|
syl5eq |
|- ( C = if ( C e. _V , C , (/) ) -> G = ( f u. { <. n , if ( C e. _V , C , (/) ) >. } ) ) |
42 |
41
|
sbceq1d |
|- ( C = if ( C e. _V , C , (/) ) -> ( [. G / f ]. ph' <-> [. ( f u. { <. n , if ( C e. _V , C , (/) ) >. } ) / f ]. ph' ) ) |
43 |
5 42
|
syl5bb |
|- ( C = if ( C e. _V , C , (/) ) -> ( ph" <-> [. ( f u. { <. n , if ( C e. _V , C , (/) ) >. } ) / f ]. ph' ) ) |
44 |
43
|
imbi2d |
|- ( C = if ( C e. _V , C , (/) ) -> ( ( ( n e. D /\ p = suc n /\ f Fn n /\ ph ) -> ph" ) <-> ( ( n e. D /\ p = suc n /\ f Fn n /\ ph ) -> [. ( f u. { <. n , if ( C e. _V , C , (/) ) >. } ) / f ]. ph' ) ) ) |
45 |
|
biid |
|- ( [. ( f u. { <. n , if ( C e. _V , C , (/) ) >. } ) / f ]. ph' <-> [. ( f u. { <. n , if ( C e. _V , C , (/) ) >. } ) / f ]. ph' ) |
46 |
|
eqid |
|- ( f u. { <. n , if ( C e. _V , C , (/) ) >. } ) = ( f u. { <. n , if ( C e. _V , C , (/) ) >. } ) |
47 |
|
0ex |
|- (/) e. _V |
48 |
47
|
elimel |
|- if ( C e. _V , C , (/) ) e. _V |
49 |
1 4 45 6 46 48
|
bnj929 |
|- ( ( n e. D /\ p = suc n /\ f Fn n /\ ph ) -> [. ( f u. { <. n , if ( C e. _V , C , (/) ) >. } ) / f ]. ph' ) |
50 |
44 49
|
dedth |
|- ( C e. _V -> ( ( n e. D /\ p = suc n /\ f Fn n /\ ph ) -> ph" ) ) |
51 |
30 37 50
|
sylc |
|- ( ( ( R _FrSe A /\ X e. A ) /\ ( ch /\ n = suc m /\ p = suc n ) ) -> ph" ) |