Step |
Hyp |
Ref |
Expression |
1 |
|
symgfixf.p |
|- P = ( Base ` ( SymGrp ` N ) ) |
2 |
|
symgfixf.q |
|- Q = { q e. P | ( q ` K ) = K } |
3 |
|
symgfixf.s |
|- S = ( Base ` ( SymGrp ` ( N \ { K } ) ) ) |
4 |
|
symgfixf.h |
|- H = ( q e. Q |-> ( q |` ( N \ { K } ) ) ) |
5 |
1 2 3 4
|
symgfixf |
|- ( K e. N -> H : Q --> S ) |
6 |
4
|
fvtresfn |
|- ( g e. Q -> ( H ` g ) = ( g |` ( N \ { K } ) ) ) |
7 |
4
|
fvtresfn |
|- ( p e. Q -> ( H ` p ) = ( p |` ( N \ { K } ) ) ) |
8 |
6 7
|
eqeqan12d |
|- ( ( g e. Q /\ p e. Q ) -> ( ( H ` g ) = ( H ` p ) <-> ( g |` ( N \ { K } ) ) = ( p |` ( N \ { K } ) ) ) ) |
9 |
8
|
adantl |
|- ( ( K e. N /\ ( g e. Q /\ p e. Q ) ) -> ( ( H ` g ) = ( H ` p ) <-> ( g |` ( N \ { K } ) ) = ( p |` ( N \ { K } ) ) ) ) |
10 |
1 2
|
symgfixelq |
|- ( g e. _V -> ( g e. Q <-> ( g : N -1-1-onto-> N /\ ( g ` K ) = K ) ) ) |
11 |
10
|
elv |
|- ( g e. Q <-> ( g : N -1-1-onto-> N /\ ( g ` K ) = K ) ) |
12 |
1 2
|
symgfixelq |
|- ( p e. _V -> ( p e. Q <-> ( p : N -1-1-onto-> N /\ ( p ` K ) = K ) ) ) |
13 |
12
|
elv |
|- ( p e. Q <-> ( p : N -1-1-onto-> N /\ ( p ` K ) = K ) ) |
14 |
11 13
|
anbi12i |
|- ( ( g e. Q /\ p e. Q ) <-> ( ( g : N -1-1-onto-> N /\ ( g ` K ) = K ) /\ ( p : N -1-1-onto-> N /\ ( p ` K ) = K ) ) ) |
15 |
|
f1ofn |
|- ( g : N -1-1-onto-> N -> g Fn N ) |
16 |
15
|
adantr |
|- ( ( g : N -1-1-onto-> N /\ ( g ` K ) = K ) -> g Fn N ) |
17 |
|
f1ofn |
|- ( p : N -1-1-onto-> N -> p Fn N ) |
18 |
17
|
adantr |
|- ( ( p : N -1-1-onto-> N /\ ( p ` K ) = K ) -> p Fn N ) |
19 |
16 18
|
anim12i |
|- ( ( ( g : N -1-1-onto-> N /\ ( g ` K ) = K ) /\ ( p : N -1-1-onto-> N /\ ( p ` K ) = K ) ) -> ( g Fn N /\ p Fn N ) ) |
20 |
|
difss |
|- ( N \ { K } ) C_ N |
21 |
19 20
|
jctir |
|- ( ( ( g : N -1-1-onto-> N /\ ( g ` K ) = K ) /\ ( p : N -1-1-onto-> N /\ ( p ` K ) = K ) ) -> ( ( g Fn N /\ p Fn N ) /\ ( N \ { K } ) C_ N ) ) |
22 |
21
|
adantl |
|- ( ( K e. N /\ ( ( g : N -1-1-onto-> N /\ ( g ` K ) = K ) /\ ( p : N -1-1-onto-> N /\ ( p ` K ) = K ) ) ) -> ( ( g Fn N /\ p Fn N ) /\ ( N \ { K } ) C_ N ) ) |
23 |
|
fvreseq |
|- ( ( ( g Fn N /\ p Fn N ) /\ ( N \ { K } ) C_ N ) -> ( ( g |` ( N \ { K } ) ) = ( p |` ( N \ { K } ) ) <-> A. i e. ( N \ { K } ) ( g ` i ) = ( p ` i ) ) ) |
24 |
22 23
|
syl |
|- ( ( K e. N /\ ( ( g : N -1-1-onto-> N /\ ( g ` K ) = K ) /\ ( p : N -1-1-onto-> N /\ ( p ` K ) = K ) ) ) -> ( ( g |` ( N \ { K } ) ) = ( p |` ( N \ { K } ) ) <-> A. i e. ( N \ { K } ) ( g ` i ) = ( p ` i ) ) ) |
25 |
|
f1of |
|- ( g : N -1-1-onto-> N -> g : N --> N ) |
26 |
25
|
adantr |
|- ( ( g : N -1-1-onto-> N /\ ( g ` K ) = K ) -> g : N --> N ) |
27 |
|
f1of |
|- ( p : N -1-1-onto-> N -> p : N --> N ) |
28 |
27
|
adantr |
|- ( ( p : N -1-1-onto-> N /\ ( p ` K ) = K ) -> p : N --> N ) |
29 |
|
fdm |
|- ( g : N --> N -> dom g = N ) |
30 |
|
fdm |
|- ( p : N --> N -> dom p = N ) |
31 |
29 30
|
anim12i |
|- ( ( g : N --> N /\ p : N --> N ) -> ( dom g = N /\ dom p = N ) ) |
32 |
26 28 31
|
syl2an |
|- ( ( ( g : N -1-1-onto-> N /\ ( g ` K ) = K ) /\ ( p : N -1-1-onto-> N /\ ( p ` K ) = K ) ) -> ( dom g = N /\ dom p = N ) ) |
33 |
|
eqtr3 |
|- ( ( dom g = N /\ dom p = N ) -> dom g = dom p ) |
34 |
32 33
|
syl |
|- ( ( ( g : N -1-1-onto-> N /\ ( g ` K ) = K ) /\ ( p : N -1-1-onto-> N /\ ( p ` K ) = K ) ) -> dom g = dom p ) |
35 |
34
|
ad2antlr |
|- ( ( ( K e. N /\ ( ( g : N -1-1-onto-> N /\ ( g ` K ) = K ) /\ ( p : N -1-1-onto-> N /\ ( p ` K ) = K ) ) ) /\ A. i e. ( N \ { K } ) ( g ` i ) = ( p ` i ) ) -> dom g = dom p ) |
36 |
|
simpr |
|- ( ( ( K e. N /\ ( ( g : N -1-1-onto-> N /\ ( g ` K ) = K ) /\ ( p : N -1-1-onto-> N /\ ( p ` K ) = K ) ) ) /\ A. i e. ( N \ { K } ) ( g ` i ) = ( p ` i ) ) -> A. i e. ( N \ { K } ) ( g ` i ) = ( p ` i ) ) |
37 |
|
eqtr3 |
|- ( ( ( g ` K ) = K /\ ( p ` K ) = K ) -> ( g ` K ) = ( p ` K ) ) |
38 |
37
|
ad2ant2l |
|- ( ( ( g : N -1-1-onto-> N /\ ( g ` K ) = K ) /\ ( p : N -1-1-onto-> N /\ ( p ` K ) = K ) ) -> ( g ` K ) = ( p ` K ) ) |
39 |
38
|
ad2antlr |
|- ( ( ( K e. N /\ ( ( g : N -1-1-onto-> N /\ ( g ` K ) = K ) /\ ( p : N -1-1-onto-> N /\ ( p ` K ) = K ) ) ) /\ A. i e. ( N \ { K } ) ( g ` i ) = ( p ` i ) ) -> ( g ` K ) = ( p ` K ) ) |
40 |
|
fveq2 |
|- ( i = K -> ( g ` i ) = ( g ` K ) ) |
41 |
|
fveq2 |
|- ( i = K -> ( p ` i ) = ( p ` K ) ) |
42 |
40 41
|
eqeq12d |
|- ( i = K -> ( ( g ` i ) = ( p ` i ) <-> ( g ` K ) = ( p ` K ) ) ) |
43 |
42
|
ralunsn |
|- ( K e. N -> ( A. i e. ( ( N \ { K } ) u. { K } ) ( g ` i ) = ( p ` i ) <-> ( A. i e. ( N \ { K } ) ( g ` i ) = ( p ` i ) /\ ( g ` K ) = ( p ` K ) ) ) ) |
44 |
43
|
adantr |
|- ( ( K e. N /\ ( ( g : N -1-1-onto-> N /\ ( g ` K ) = K ) /\ ( p : N -1-1-onto-> N /\ ( p ` K ) = K ) ) ) -> ( A. i e. ( ( N \ { K } ) u. { K } ) ( g ` i ) = ( p ` i ) <-> ( A. i e. ( N \ { K } ) ( g ` i ) = ( p ` i ) /\ ( g ` K ) = ( p ` K ) ) ) ) |
45 |
44
|
adantr |
|- ( ( ( K e. N /\ ( ( g : N -1-1-onto-> N /\ ( g ` K ) = K ) /\ ( p : N -1-1-onto-> N /\ ( p ` K ) = K ) ) ) /\ A. i e. ( N \ { K } ) ( g ` i ) = ( p ` i ) ) -> ( A. i e. ( ( N \ { K } ) u. { K } ) ( g ` i ) = ( p ` i ) <-> ( A. i e. ( N \ { K } ) ( g ` i ) = ( p ` i ) /\ ( g ` K ) = ( p ` K ) ) ) ) |
46 |
36 39 45
|
mpbir2and |
|- ( ( ( K e. N /\ ( ( g : N -1-1-onto-> N /\ ( g ` K ) = K ) /\ ( p : N -1-1-onto-> N /\ ( p ` K ) = K ) ) ) /\ A. i e. ( N \ { K } ) ( g ` i ) = ( p ` i ) ) -> A. i e. ( ( N \ { K } ) u. { K } ) ( g ` i ) = ( p ` i ) ) |
47 |
|
f1odm |
|- ( g : N -1-1-onto-> N -> dom g = N ) |
48 |
47
|
adantr |
|- ( ( g : N -1-1-onto-> N /\ ( g ` K ) = K ) -> dom g = N ) |
49 |
48
|
adantr |
|- ( ( ( g : N -1-1-onto-> N /\ ( g ` K ) = K ) /\ ( p : N -1-1-onto-> N /\ ( p ` K ) = K ) ) -> dom g = N ) |
50 |
|
difsnid |
|- ( K e. N -> ( ( N \ { K } ) u. { K } ) = N ) |
51 |
50
|
eqcomd |
|- ( K e. N -> N = ( ( N \ { K } ) u. { K } ) ) |
52 |
49 51
|
sylan9eqr |
|- ( ( K e. N /\ ( ( g : N -1-1-onto-> N /\ ( g ` K ) = K ) /\ ( p : N -1-1-onto-> N /\ ( p ` K ) = K ) ) ) -> dom g = ( ( N \ { K } ) u. { K } ) ) |
53 |
52
|
adantr |
|- ( ( ( K e. N /\ ( ( g : N -1-1-onto-> N /\ ( g ` K ) = K ) /\ ( p : N -1-1-onto-> N /\ ( p ` K ) = K ) ) ) /\ A. i e. ( N \ { K } ) ( g ` i ) = ( p ` i ) ) -> dom g = ( ( N \ { K } ) u. { K } ) ) |
54 |
53
|
raleqdv |
|- ( ( ( K e. N /\ ( ( g : N -1-1-onto-> N /\ ( g ` K ) = K ) /\ ( p : N -1-1-onto-> N /\ ( p ` K ) = K ) ) ) /\ A. i e. ( N \ { K } ) ( g ` i ) = ( p ` i ) ) -> ( A. i e. dom g ( g ` i ) = ( p ` i ) <-> A. i e. ( ( N \ { K } ) u. { K } ) ( g ` i ) = ( p ` i ) ) ) |
55 |
46 54
|
mpbird |
|- ( ( ( K e. N /\ ( ( g : N -1-1-onto-> N /\ ( g ` K ) = K ) /\ ( p : N -1-1-onto-> N /\ ( p ` K ) = K ) ) ) /\ A. i e. ( N \ { K } ) ( g ` i ) = ( p ` i ) ) -> A. i e. dom g ( g ` i ) = ( p ` i ) ) |
56 |
|
f1ofun |
|- ( g : N -1-1-onto-> N -> Fun g ) |
57 |
56
|
adantr |
|- ( ( g : N -1-1-onto-> N /\ ( g ` K ) = K ) -> Fun g ) |
58 |
|
f1ofun |
|- ( p : N -1-1-onto-> N -> Fun p ) |
59 |
58
|
adantr |
|- ( ( p : N -1-1-onto-> N /\ ( p ` K ) = K ) -> Fun p ) |
60 |
57 59
|
anim12i |
|- ( ( ( g : N -1-1-onto-> N /\ ( g ` K ) = K ) /\ ( p : N -1-1-onto-> N /\ ( p ` K ) = K ) ) -> ( Fun g /\ Fun p ) ) |
61 |
60
|
ad2antlr |
|- ( ( ( K e. N /\ ( ( g : N -1-1-onto-> N /\ ( g ` K ) = K ) /\ ( p : N -1-1-onto-> N /\ ( p ` K ) = K ) ) ) /\ A. i e. ( N \ { K } ) ( g ` i ) = ( p ` i ) ) -> ( Fun g /\ Fun p ) ) |
62 |
|
eqfunfv |
|- ( ( Fun g /\ Fun p ) -> ( g = p <-> ( dom g = dom p /\ A. i e. dom g ( g ` i ) = ( p ` i ) ) ) ) |
63 |
61 62
|
syl |
|- ( ( ( K e. N /\ ( ( g : N -1-1-onto-> N /\ ( g ` K ) = K ) /\ ( p : N -1-1-onto-> N /\ ( p ` K ) = K ) ) ) /\ A. i e. ( N \ { K } ) ( g ` i ) = ( p ` i ) ) -> ( g = p <-> ( dom g = dom p /\ A. i e. dom g ( g ` i ) = ( p ` i ) ) ) ) |
64 |
35 55 63
|
mpbir2and |
|- ( ( ( K e. N /\ ( ( g : N -1-1-onto-> N /\ ( g ` K ) = K ) /\ ( p : N -1-1-onto-> N /\ ( p ` K ) = K ) ) ) /\ A. i e. ( N \ { K } ) ( g ` i ) = ( p ` i ) ) -> g = p ) |
65 |
64
|
ex |
|- ( ( K e. N /\ ( ( g : N -1-1-onto-> N /\ ( g ` K ) = K ) /\ ( p : N -1-1-onto-> N /\ ( p ` K ) = K ) ) ) -> ( A. i e. ( N \ { K } ) ( g ` i ) = ( p ` i ) -> g = p ) ) |
66 |
24 65
|
sylbid |
|- ( ( K e. N /\ ( ( g : N -1-1-onto-> N /\ ( g ` K ) = K ) /\ ( p : N -1-1-onto-> N /\ ( p ` K ) = K ) ) ) -> ( ( g |` ( N \ { K } ) ) = ( p |` ( N \ { K } ) ) -> g = p ) ) |
67 |
14 66
|
sylan2b |
|- ( ( K e. N /\ ( g e. Q /\ p e. Q ) ) -> ( ( g |` ( N \ { K } ) ) = ( p |` ( N \ { K } ) ) -> g = p ) ) |
68 |
9 67
|
sylbid |
|- ( ( K e. N /\ ( g e. Q /\ p e. Q ) ) -> ( ( H ` g ) = ( H ` p ) -> g = p ) ) |
69 |
68
|
ralrimivva |
|- ( K e. N -> A. g e. Q A. p e. Q ( ( H ` g ) = ( H ` p ) -> g = p ) ) |
70 |
|
dff13 |
|- ( H : Q -1-1-> S <-> ( H : Q --> S /\ A. g e. Q A. p e. Q ( ( H ` g ) = ( H ` p ) -> g = p ) ) ) |
71 |
5 69 70
|
sylanbrc |
|- ( K e. N -> H : Q -1-1-> S ) |