Step |
Hyp |
Ref |
Expression |
1 |
|
bnj580.1 |
|- ( ph <-> ( f ` (/) ) = _pred ( x , A , R ) ) |
2 |
|
bnj580.2 |
|- ( ps <-> A. i e. _om ( suc i e. n -> ( f ` suc i ) = U_ y e. ( f ` i ) _pred ( y , A , R ) ) ) |
3 |
|
bnj580.3 |
|- ( ch <-> ( f Fn n /\ ph /\ ps ) ) |
4 |
|
bnj580.4 |
|- ( ph' <-> [. g / f ]. ph ) |
5 |
|
bnj580.5 |
|- ( ps' <-> [. g / f ]. ps ) |
6 |
|
bnj580.6 |
|- ( ch' <-> [. g / f ]. ch ) |
7 |
|
bnj580.7 |
|- D = ( _om \ { (/) } ) |
8 |
|
bnj580.8 |
|- ( th <-> ( ( n e. D /\ ch /\ ch' ) -> ( f ` j ) = ( g ` j ) ) ) |
9 |
|
bnj580.9 |
|- ( ta <-> A. k e. n ( k _E j -> [. k / j ]. th ) ) |
10 |
3
|
simp1bi |
|- ( ch -> f Fn n ) |
11 |
3 4 5 6
|
bnj581 |
|- ( ch' <-> ( g Fn n /\ ph' /\ ps' ) ) |
12 |
11
|
simp1bi |
|- ( ch' -> g Fn n ) |
13 |
10 12
|
bnj240 |
|- ( ( n e. D /\ ch /\ ch' ) -> ( f Fn n /\ g Fn n ) ) |
14 |
4 1
|
bnj154 |
|- ( ph' <-> ( g ` (/) ) = _pred ( x , A , R ) ) |
15 |
|
vex |
|- g e. _V |
16 |
2 5 15
|
bnj540 |
|- ( ps' <-> A. i e. _om ( suc i e. n -> ( g ` suc i ) = U_ y e. ( g ` i ) _pred ( y , A , R ) ) ) |
17 |
8
|
bnj591 |
|- ( [. k / j ]. th <-> ( ( n e. D /\ ch /\ ch' ) -> ( f ` k ) = ( g ` k ) ) ) |
18 |
1 2 3 7 14 16 11 8 17 9
|
bnj594 |
|- ( ( j e. n /\ ta ) -> th ) |
19 |
18
|
ex |
|- ( j e. n -> ( ta -> th ) ) |
20 |
19
|
rgen |
|- A. j e. n ( ta -> th ) |
21 |
|
vex |
|- n e. _V |
22 |
21 9
|
bnj110 |
|- ( ( _E Fr n /\ A. j e. n ( ta -> th ) ) -> A. j e. n th ) |
23 |
20 22
|
mpan2 |
|- ( _E Fr n -> A. j e. n th ) |
24 |
8
|
ralbii |
|- ( A. j e. n th <-> A. j e. n ( ( n e. D /\ ch /\ ch' ) -> ( f ` j ) = ( g ` j ) ) ) |
25 |
23 24
|
sylib |
|- ( _E Fr n -> A. j e. n ( ( n e. D /\ ch /\ ch' ) -> ( f ` j ) = ( g ` j ) ) ) |
26 |
25
|
r19.21be |
|- A. j e. n ( _E Fr n -> ( ( n e. D /\ ch /\ ch' ) -> ( f ` j ) = ( g ` j ) ) ) |
27 |
7
|
bnj923 |
|- ( n e. D -> n e. _om ) |
28 |
|
nnord |
|- ( n e. _om -> Ord n ) |
29 |
|
ordfr |
|- ( Ord n -> _E Fr n ) |
30 |
27 28 29
|
3syl |
|- ( n e. D -> _E Fr n ) |
31 |
30
|
3ad2ant1 |
|- ( ( n e. D /\ ch /\ ch' ) -> _E Fr n ) |
32 |
31
|
pm4.71ri |
|- ( ( n e. D /\ ch /\ ch' ) <-> ( _E Fr n /\ ( n e. D /\ ch /\ ch' ) ) ) |
33 |
32
|
imbi1i |
|- ( ( ( n e. D /\ ch /\ ch' ) -> ( f ` j ) = ( g ` j ) ) <-> ( ( _E Fr n /\ ( n e. D /\ ch /\ ch' ) ) -> ( f ` j ) = ( g ` j ) ) ) |
34 |
|
impexp |
|- ( ( ( _E Fr n /\ ( n e. D /\ ch /\ ch' ) ) -> ( f ` j ) = ( g ` j ) ) <-> ( _E Fr n -> ( ( n e. D /\ ch /\ ch' ) -> ( f ` j ) = ( g ` j ) ) ) ) |
35 |
33 34
|
bitri |
|- ( ( ( n e. D /\ ch /\ ch' ) -> ( f ` j ) = ( g ` j ) ) <-> ( _E Fr n -> ( ( n e. D /\ ch /\ ch' ) -> ( f ` j ) = ( g ` j ) ) ) ) |
36 |
35
|
ralbii |
|- ( A. j e. n ( ( n e. D /\ ch /\ ch' ) -> ( f ` j ) = ( g ` j ) ) <-> A. j e. n ( _E Fr n -> ( ( n e. D /\ ch /\ ch' ) -> ( f ` j ) = ( g ` j ) ) ) ) |
37 |
26 36
|
mpbir |
|- A. j e. n ( ( n e. D /\ ch /\ ch' ) -> ( f ` j ) = ( g ` j ) ) |
38 |
|
r19.21v |
|- ( A. j e. n ( ( n e. D /\ ch /\ ch' ) -> ( f ` j ) = ( g ` j ) ) <-> ( ( n e. D /\ ch /\ ch' ) -> A. j e. n ( f ` j ) = ( g ` j ) ) ) |
39 |
37 38
|
mpbi |
|- ( ( n e. D /\ ch /\ ch' ) -> A. j e. n ( f ` j ) = ( g ` j ) ) |
40 |
|
eqfnfv |
|- ( ( f Fn n /\ g Fn n ) -> ( f = g <-> A. j e. n ( f ` j ) = ( g ` j ) ) ) |
41 |
40
|
biimprd |
|- ( ( f Fn n /\ g Fn n ) -> ( A. j e. n ( f ` j ) = ( g ` j ) -> f = g ) ) |
42 |
13 39 41
|
sylc |
|- ( ( n e. D /\ ch /\ ch' ) -> f = g ) |
43 |
42
|
3expib |
|- ( n e. D -> ( ( ch /\ ch' ) -> f = g ) ) |
44 |
43
|
alrimivv |
|- ( n e. D -> A. f A. g ( ( ch /\ ch' ) -> f = g ) ) |
45 |
|
sbsbc |
|- ( [ g / f ] ch <-> [. g / f ]. ch ) |
46 |
45
|
anbi2i |
|- ( ( ch /\ [ g / f ] ch ) <-> ( ch /\ [. g / f ]. ch ) ) |
47 |
46
|
imbi1i |
|- ( ( ( ch /\ [ g / f ] ch ) -> f = g ) <-> ( ( ch /\ [. g / f ]. ch ) -> f = g ) ) |
48 |
47
|
2albii |
|- ( A. f A. g ( ( ch /\ [ g / f ] ch ) -> f = g ) <-> A. f A. g ( ( ch /\ [. g / f ]. ch ) -> f = g ) ) |
49 |
|
nfv |
|- F/ g ch |
50 |
49
|
mo3 |
|- ( E* f ch <-> A. f A. g ( ( ch /\ [ g / f ] ch ) -> f = g ) ) |
51 |
6
|
anbi2i |
|- ( ( ch /\ ch' ) <-> ( ch /\ [. g / f ]. ch ) ) |
52 |
51
|
imbi1i |
|- ( ( ( ch /\ ch' ) -> f = g ) <-> ( ( ch /\ [. g / f ]. ch ) -> f = g ) ) |
53 |
52
|
2albii |
|- ( A. f A. g ( ( ch /\ ch' ) -> f = g ) <-> A. f A. g ( ( ch /\ [. g / f ]. ch ) -> f = g ) ) |
54 |
48 50 53
|
3bitr4i |
|- ( E* f ch <-> A. f A. g ( ( ch /\ ch' ) -> f = g ) ) |
55 |
44 54
|
sylibr |
|- ( n e. D -> E* f ch ) |