Step |
Hyp |
Ref |
Expression |
1 |
|
difss |
|- ( ( ( G o. F ) o. `' G ) \ _I ) C_ ( ( G o. F ) o. `' G ) |
2 |
|
dmss |
|- ( ( ( ( G o. F ) o. `' G ) \ _I ) C_ ( ( G o. F ) o. `' G ) -> dom ( ( ( G o. F ) o. `' G ) \ _I ) C_ dom ( ( G o. F ) o. `' G ) ) |
3 |
1 2
|
ax-mp |
|- dom ( ( ( G o. F ) o. `' G ) \ _I ) C_ dom ( ( G o. F ) o. `' G ) |
4 |
|
dmcoss |
|- dom ( ( G o. F ) o. `' G ) C_ dom `' G |
5 |
3 4
|
sstri |
|- dom ( ( ( G o. F ) o. `' G ) \ _I ) C_ dom `' G |
6 |
|
f1ocnv |
|- ( G : A -1-1-onto-> A -> `' G : A -1-1-onto-> A ) |
7 |
6
|
adantl |
|- ( ( F : A --> A /\ G : A -1-1-onto-> A ) -> `' G : A -1-1-onto-> A ) |
8 |
|
f1odm |
|- ( `' G : A -1-1-onto-> A -> dom `' G = A ) |
9 |
7 8
|
syl |
|- ( ( F : A --> A /\ G : A -1-1-onto-> A ) -> dom `' G = A ) |
10 |
5 9
|
sseqtrid |
|- ( ( F : A --> A /\ G : A -1-1-onto-> A ) -> dom ( ( ( G o. F ) o. `' G ) \ _I ) C_ A ) |
11 |
10
|
sselda |
|- ( ( ( F : A --> A /\ G : A -1-1-onto-> A ) /\ x e. dom ( ( ( G o. F ) o. `' G ) \ _I ) ) -> x e. A ) |
12 |
|
imassrn |
|- ( G " dom ( F \ _I ) ) C_ ran G |
13 |
|
f1of |
|- ( G : A -1-1-onto-> A -> G : A --> A ) |
14 |
13
|
adantl |
|- ( ( F : A --> A /\ G : A -1-1-onto-> A ) -> G : A --> A ) |
15 |
14
|
frnd |
|- ( ( F : A --> A /\ G : A -1-1-onto-> A ) -> ran G C_ A ) |
16 |
12 15
|
sstrid |
|- ( ( F : A --> A /\ G : A -1-1-onto-> A ) -> ( G " dom ( F \ _I ) ) C_ A ) |
17 |
16
|
sselda |
|- ( ( ( F : A --> A /\ G : A -1-1-onto-> A ) /\ x e. ( G " dom ( F \ _I ) ) ) -> x e. A ) |
18 |
|
simpl |
|- ( ( F : A --> A /\ G : A -1-1-onto-> A ) -> F : A --> A ) |
19 |
|
fco |
|- ( ( G : A --> A /\ F : A --> A ) -> ( G o. F ) : A --> A ) |
20 |
14 18 19
|
syl2anc |
|- ( ( F : A --> A /\ G : A -1-1-onto-> A ) -> ( G o. F ) : A --> A ) |
21 |
|
f1of |
|- ( `' G : A -1-1-onto-> A -> `' G : A --> A ) |
22 |
7 21
|
syl |
|- ( ( F : A --> A /\ G : A -1-1-onto-> A ) -> `' G : A --> A ) |
23 |
|
fco |
|- ( ( ( G o. F ) : A --> A /\ `' G : A --> A ) -> ( ( G o. F ) o. `' G ) : A --> A ) |
24 |
20 22 23
|
syl2anc |
|- ( ( F : A --> A /\ G : A -1-1-onto-> A ) -> ( ( G o. F ) o. `' G ) : A --> A ) |
25 |
24
|
ffnd |
|- ( ( F : A --> A /\ G : A -1-1-onto-> A ) -> ( ( G o. F ) o. `' G ) Fn A ) |
26 |
|
fnelnfp |
|- ( ( ( ( G o. F ) o. `' G ) Fn A /\ x e. A ) -> ( x e. dom ( ( ( G o. F ) o. `' G ) \ _I ) <-> ( ( ( G o. F ) o. `' G ) ` x ) =/= x ) ) |
27 |
25 26
|
sylan |
|- ( ( ( F : A --> A /\ G : A -1-1-onto-> A ) /\ x e. A ) -> ( x e. dom ( ( ( G o. F ) o. `' G ) \ _I ) <-> ( ( ( G o. F ) o. `' G ) ` x ) =/= x ) ) |
28 |
|
f1ofn |
|- ( `' G : A -1-1-onto-> A -> `' G Fn A ) |
29 |
7 28
|
syl |
|- ( ( F : A --> A /\ G : A -1-1-onto-> A ) -> `' G Fn A ) |
30 |
|
fvco2 |
|- ( ( `' G Fn A /\ x e. A ) -> ( ( ( G o. F ) o. `' G ) ` x ) = ( ( G o. F ) ` ( `' G ` x ) ) ) |
31 |
29 30
|
sylan |
|- ( ( ( F : A --> A /\ G : A -1-1-onto-> A ) /\ x e. A ) -> ( ( ( G o. F ) o. `' G ) ` x ) = ( ( G o. F ) ` ( `' G ` x ) ) ) |
32 |
|
ffn |
|- ( F : A --> A -> F Fn A ) |
33 |
32
|
ad2antrr |
|- ( ( ( F : A --> A /\ G : A -1-1-onto-> A ) /\ x e. A ) -> F Fn A ) |
34 |
|
ffvelrn |
|- ( ( `' G : A --> A /\ x e. A ) -> ( `' G ` x ) e. A ) |
35 |
22 34
|
sylan |
|- ( ( ( F : A --> A /\ G : A -1-1-onto-> A ) /\ x e. A ) -> ( `' G ` x ) e. A ) |
36 |
|
fvco2 |
|- ( ( F Fn A /\ ( `' G ` x ) e. A ) -> ( ( G o. F ) ` ( `' G ` x ) ) = ( G ` ( F ` ( `' G ` x ) ) ) ) |
37 |
33 35 36
|
syl2anc |
|- ( ( ( F : A --> A /\ G : A -1-1-onto-> A ) /\ x e. A ) -> ( ( G o. F ) ` ( `' G ` x ) ) = ( G ` ( F ` ( `' G ` x ) ) ) ) |
38 |
31 37
|
eqtrd |
|- ( ( ( F : A --> A /\ G : A -1-1-onto-> A ) /\ x e. A ) -> ( ( ( G o. F ) o. `' G ) ` x ) = ( G ` ( F ` ( `' G ` x ) ) ) ) |
39 |
38
|
eqeq1d |
|- ( ( ( F : A --> A /\ G : A -1-1-onto-> A ) /\ x e. A ) -> ( ( ( ( G o. F ) o. `' G ) ` x ) = x <-> ( G ` ( F ` ( `' G ` x ) ) ) = x ) ) |
40 |
|
simplr |
|- ( ( ( F : A --> A /\ G : A -1-1-onto-> A ) /\ x e. A ) -> G : A -1-1-onto-> A ) |
41 |
|
simpll |
|- ( ( ( F : A --> A /\ G : A -1-1-onto-> A ) /\ x e. A ) -> F : A --> A ) |
42 |
|
ffvelrn |
|- ( ( F : A --> A /\ ( `' G ` x ) e. A ) -> ( F ` ( `' G ` x ) ) e. A ) |
43 |
41 35 42
|
syl2anc |
|- ( ( ( F : A --> A /\ G : A -1-1-onto-> A ) /\ x e. A ) -> ( F ` ( `' G ` x ) ) e. A ) |
44 |
|
simpr |
|- ( ( ( F : A --> A /\ G : A -1-1-onto-> A ) /\ x e. A ) -> x e. A ) |
45 |
|
f1ocnvfvb |
|- ( ( G : A -1-1-onto-> A /\ ( F ` ( `' G ` x ) ) e. A /\ x e. A ) -> ( ( G ` ( F ` ( `' G ` x ) ) ) = x <-> ( `' G ` x ) = ( F ` ( `' G ` x ) ) ) ) |
46 |
40 43 44 45
|
syl3anc |
|- ( ( ( F : A --> A /\ G : A -1-1-onto-> A ) /\ x e. A ) -> ( ( G ` ( F ` ( `' G ` x ) ) ) = x <-> ( `' G ` x ) = ( F ` ( `' G ` x ) ) ) ) |
47 |
39 46
|
bitrd |
|- ( ( ( F : A --> A /\ G : A -1-1-onto-> A ) /\ x e. A ) -> ( ( ( ( G o. F ) o. `' G ) ` x ) = x <-> ( `' G ` x ) = ( F ` ( `' G ` x ) ) ) ) |
48 |
47
|
necon3bid |
|- ( ( ( F : A --> A /\ G : A -1-1-onto-> A ) /\ x e. A ) -> ( ( ( ( G o. F ) o. `' G ) ` x ) =/= x <-> ( `' G ` x ) =/= ( F ` ( `' G ` x ) ) ) ) |
49 |
|
necom |
|- ( ( `' G ` x ) =/= ( F ` ( `' G ` x ) ) <-> ( F ` ( `' G ` x ) ) =/= ( `' G ` x ) ) |
50 |
|
f1of1 |
|- ( G : A -1-1-onto-> A -> G : A -1-1-> A ) |
51 |
50
|
ad2antlr |
|- ( ( ( F : A --> A /\ G : A -1-1-onto-> A ) /\ x e. A ) -> G : A -1-1-> A ) |
52 |
|
difss |
|- ( F \ _I ) C_ F |
53 |
|
dmss |
|- ( ( F \ _I ) C_ F -> dom ( F \ _I ) C_ dom F ) |
54 |
52 53
|
ax-mp |
|- dom ( F \ _I ) C_ dom F |
55 |
|
fdm |
|- ( F : A --> A -> dom F = A ) |
56 |
54 55
|
sseqtrid |
|- ( F : A --> A -> dom ( F \ _I ) C_ A ) |
57 |
56
|
ad2antrr |
|- ( ( ( F : A --> A /\ G : A -1-1-onto-> A ) /\ x e. A ) -> dom ( F \ _I ) C_ A ) |
58 |
|
f1elima |
|- ( ( G : A -1-1-> A /\ ( `' G ` x ) e. A /\ dom ( F \ _I ) C_ A ) -> ( ( G ` ( `' G ` x ) ) e. ( G " dom ( F \ _I ) ) <-> ( `' G ` x ) e. dom ( F \ _I ) ) ) |
59 |
51 35 57 58
|
syl3anc |
|- ( ( ( F : A --> A /\ G : A -1-1-onto-> A ) /\ x e. A ) -> ( ( G ` ( `' G ` x ) ) e. ( G " dom ( F \ _I ) ) <-> ( `' G ` x ) e. dom ( F \ _I ) ) ) |
60 |
|
f1ocnvfv2 |
|- ( ( G : A -1-1-onto-> A /\ x e. A ) -> ( G ` ( `' G ` x ) ) = x ) |
61 |
60
|
adantll |
|- ( ( ( F : A --> A /\ G : A -1-1-onto-> A ) /\ x e. A ) -> ( G ` ( `' G ` x ) ) = x ) |
62 |
61
|
eleq1d |
|- ( ( ( F : A --> A /\ G : A -1-1-onto-> A ) /\ x e. A ) -> ( ( G ` ( `' G ` x ) ) e. ( G " dom ( F \ _I ) ) <-> x e. ( G " dom ( F \ _I ) ) ) ) |
63 |
|
fnelnfp |
|- ( ( F Fn A /\ ( `' G ` x ) e. A ) -> ( ( `' G ` x ) e. dom ( F \ _I ) <-> ( F ` ( `' G ` x ) ) =/= ( `' G ` x ) ) ) |
64 |
33 35 63
|
syl2anc |
|- ( ( ( F : A --> A /\ G : A -1-1-onto-> A ) /\ x e. A ) -> ( ( `' G ` x ) e. dom ( F \ _I ) <-> ( F ` ( `' G ` x ) ) =/= ( `' G ` x ) ) ) |
65 |
59 62 64
|
3bitr3rd |
|- ( ( ( F : A --> A /\ G : A -1-1-onto-> A ) /\ x e. A ) -> ( ( F ` ( `' G ` x ) ) =/= ( `' G ` x ) <-> x e. ( G " dom ( F \ _I ) ) ) ) |
66 |
49 65
|
syl5bb |
|- ( ( ( F : A --> A /\ G : A -1-1-onto-> A ) /\ x e. A ) -> ( ( `' G ` x ) =/= ( F ` ( `' G ` x ) ) <-> x e. ( G " dom ( F \ _I ) ) ) ) |
67 |
27 48 66
|
3bitrd |
|- ( ( ( F : A --> A /\ G : A -1-1-onto-> A ) /\ x e. A ) -> ( x e. dom ( ( ( G o. F ) o. `' G ) \ _I ) <-> x e. ( G " dom ( F \ _I ) ) ) ) |
68 |
11 17 67
|
eqrdav |
|- ( ( F : A --> A /\ G : A -1-1-onto-> A ) -> dom ( ( ( G o. F ) o. `' G ) \ _I ) = ( G " dom ( F \ _I ) ) ) |