Step |
Hyp |
Ref |
Expression |
1 |
|
bnj1321.1 |
|- B = { d | ( d C_ A /\ A. x e. d _pred ( x , A , R ) C_ d ) } |
2 |
|
bnj1321.2 |
|- Y = <. x , ( f |` _pred ( x , A , R ) ) >. |
3 |
|
bnj1321.3 |
|- C = { f | E. d e. B ( f Fn d /\ A. x e. d ( f ` x ) = ( G ` Y ) ) } |
4 |
|
bnj1321.4 |
|- ( ta <-> ( f e. C /\ dom f = ( { x } u. _trCl ( x , A , R ) ) ) ) |
5 |
|
simpr |
|- ( ( R _FrSe A /\ E. f ta ) -> E. f ta ) |
6 |
|
simp1 |
|- ( ( R _FrSe A /\ ta /\ [ g / f ] ta ) -> R _FrSe A ) |
7 |
4
|
simplbi |
|- ( ta -> f e. C ) |
8 |
7
|
3ad2ant2 |
|- ( ( R _FrSe A /\ ta /\ [ g / f ] ta ) -> f e. C ) |
9 |
|
nfab1 |
|- F/_ f { f | E. d e. B ( f Fn d /\ A. x e. d ( f ` x ) = ( G ` Y ) ) } |
10 |
3 9
|
nfcxfr |
|- F/_ f C |
11 |
10
|
nfcri |
|- F/ f g e. C |
12 |
|
nfv |
|- F/ f dom g = ( { x } u. _trCl ( x , A , R ) ) |
13 |
11 12
|
nfan |
|- F/ f ( g e. C /\ dom g = ( { x } u. _trCl ( x , A , R ) ) ) |
14 |
|
eleq1w |
|- ( f = g -> ( f e. C <-> g e. C ) ) |
15 |
|
dmeq |
|- ( f = g -> dom f = dom g ) |
16 |
15
|
eqeq1d |
|- ( f = g -> ( dom f = ( { x } u. _trCl ( x , A , R ) ) <-> dom g = ( { x } u. _trCl ( x , A , R ) ) ) ) |
17 |
14 16
|
anbi12d |
|- ( f = g -> ( ( f e. C /\ dom f = ( { x } u. _trCl ( x , A , R ) ) ) <-> ( g e. C /\ dom g = ( { x } u. _trCl ( x , A , R ) ) ) ) ) |
18 |
4 17
|
syl5bb |
|- ( f = g -> ( ta <-> ( g e. C /\ dom g = ( { x } u. _trCl ( x , A , R ) ) ) ) ) |
19 |
13 18
|
sbiev |
|- ( [ g / f ] ta <-> ( g e. C /\ dom g = ( { x } u. _trCl ( x , A , R ) ) ) ) |
20 |
19
|
simplbi |
|- ( [ g / f ] ta -> g e. C ) |
21 |
20
|
3ad2ant3 |
|- ( ( R _FrSe A /\ ta /\ [ g / f ] ta ) -> g e. C ) |
22 |
|
eqid |
|- ( dom f i^i dom g ) = ( dom f i^i dom g ) |
23 |
1 2 3 22
|
bnj1326 |
|- ( ( R _FrSe A /\ f e. C /\ g e. C ) -> ( f |` ( dom f i^i dom g ) ) = ( g |` ( dom f i^i dom g ) ) ) |
24 |
6 8 21 23
|
syl3anc |
|- ( ( R _FrSe A /\ ta /\ [ g / f ] ta ) -> ( f |` ( dom f i^i dom g ) ) = ( g |` ( dom f i^i dom g ) ) ) |
25 |
4
|
simprbi |
|- ( ta -> dom f = ( { x } u. _trCl ( x , A , R ) ) ) |
26 |
25
|
3ad2ant2 |
|- ( ( R _FrSe A /\ ta /\ [ g / f ] ta ) -> dom f = ( { x } u. _trCl ( x , A , R ) ) ) |
27 |
19
|
simprbi |
|- ( [ g / f ] ta -> dom g = ( { x } u. _trCl ( x , A , R ) ) ) |
28 |
27
|
3ad2ant3 |
|- ( ( R _FrSe A /\ ta /\ [ g / f ] ta ) -> dom g = ( { x } u. _trCl ( x , A , R ) ) ) |
29 |
26 28
|
eqtr4d |
|- ( ( R _FrSe A /\ ta /\ [ g / f ] ta ) -> dom f = dom g ) |
30 |
|
bnj1322 |
|- ( dom f = dom g -> ( dom f i^i dom g ) = dom f ) |
31 |
30
|
reseq2d |
|- ( dom f = dom g -> ( f |` ( dom f i^i dom g ) ) = ( f |` dom f ) ) |
32 |
29 31
|
syl |
|- ( ( R _FrSe A /\ ta /\ [ g / f ] ta ) -> ( f |` ( dom f i^i dom g ) ) = ( f |` dom f ) ) |
33 |
|
releq |
|- ( z = f -> ( Rel z <-> Rel f ) ) |
34 |
1 2 3
|
bnj66 |
|- ( z e. C -> Rel z ) |
35 |
33 34
|
vtoclga |
|- ( f e. C -> Rel f ) |
36 |
|
resdm |
|- ( Rel f -> ( f |` dom f ) = f ) |
37 |
8 35 36
|
3syl |
|- ( ( R _FrSe A /\ ta /\ [ g / f ] ta ) -> ( f |` dom f ) = f ) |
38 |
32 37
|
eqtrd |
|- ( ( R _FrSe A /\ ta /\ [ g / f ] ta ) -> ( f |` ( dom f i^i dom g ) ) = f ) |
39 |
|
eqeq2 |
|- ( dom f = dom g -> ( ( dom f i^i dom g ) = dom f <-> ( dom f i^i dom g ) = dom g ) ) |
40 |
30 39
|
mpbid |
|- ( dom f = dom g -> ( dom f i^i dom g ) = dom g ) |
41 |
40
|
reseq2d |
|- ( dom f = dom g -> ( g |` ( dom f i^i dom g ) ) = ( g |` dom g ) ) |
42 |
29 41
|
syl |
|- ( ( R _FrSe A /\ ta /\ [ g / f ] ta ) -> ( g |` ( dom f i^i dom g ) ) = ( g |` dom g ) ) |
43 |
1 2 3
|
bnj66 |
|- ( g e. C -> Rel g ) |
44 |
|
resdm |
|- ( Rel g -> ( g |` dom g ) = g ) |
45 |
21 43 44
|
3syl |
|- ( ( R _FrSe A /\ ta /\ [ g / f ] ta ) -> ( g |` dom g ) = g ) |
46 |
42 45
|
eqtrd |
|- ( ( R _FrSe A /\ ta /\ [ g / f ] ta ) -> ( g |` ( dom f i^i dom g ) ) = g ) |
47 |
24 38 46
|
3eqtr3d |
|- ( ( R _FrSe A /\ ta /\ [ g / f ] ta ) -> f = g ) |
48 |
47
|
3expib |
|- ( R _FrSe A -> ( ( ta /\ [ g / f ] ta ) -> f = g ) ) |
49 |
48
|
alrimivv |
|- ( R _FrSe A -> A. f A. g ( ( ta /\ [ g / f ] ta ) -> f = g ) ) |
50 |
49
|
adantr |
|- ( ( R _FrSe A /\ E. f ta ) -> A. f A. g ( ( ta /\ [ g / f ] ta ) -> f = g ) ) |
51 |
|
nfv |
|- F/ g ta |
52 |
51
|
eu2 |
|- ( E! f ta <-> ( E. f ta /\ A. f A. g ( ( ta /\ [ g / f ] ta ) -> f = g ) ) ) |
53 |
5 50 52
|
sylanbrc |
|- ( ( R _FrSe A /\ E. f ta ) -> E! f ta ) |