Step |
Hyp |
Ref |
Expression |
1 |
|
sucidg |
|- ( M e. On -> M e. suc M ) |
2 |
|
dff1o3 |
|- ( F : A -1-1-onto-> suc M <-> ( F : A -onto-> suc M /\ Fun `' F ) ) |
3 |
2
|
simprbi |
|- ( F : A -1-1-onto-> suc M -> Fun `' F ) |
4 |
3
|
adantr |
|- ( ( F : A -1-1-onto-> suc M /\ M e. suc M ) -> Fun `' F ) |
5 |
|
f1ofo |
|- ( F : A -1-1-onto-> suc M -> F : A -onto-> suc M ) |
6 |
|
f1ofn |
|- ( F : A -1-1-onto-> suc M -> F Fn A ) |
7 |
|
fnresdm |
|- ( F Fn A -> ( F |` A ) = F ) |
8 |
|
foeq1 |
|- ( ( F |` A ) = F -> ( ( F |` A ) : A -onto-> suc M <-> F : A -onto-> suc M ) ) |
9 |
6 7 8
|
3syl |
|- ( F : A -1-1-onto-> suc M -> ( ( F |` A ) : A -onto-> suc M <-> F : A -onto-> suc M ) ) |
10 |
5 9
|
mpbird |
|- ( F : A -1-1-onto-> suc M -> ( F |` A ) : A -onto-> suc M ) |
11 |
10
|
adantr |
|- ( ( F : A -1-1-onto-> suc M /\ M e. suc M ) -> ( F |` A ) : A -onto-> suc M ) |
12 |
6
|
adantr |
|- ( ( F : A -1-1-onto-> suc M /\ M e. suc M ) -> F Fn A ) |
13 |
|
f1ocnvdm |
|- ( ( F : A -1-1-onto-> suc M /\ M e. suc M ) -> ( `' F ` M ) e. A ) |
14 |
|
f1ocnvfv2 |
|- ( ( F : A -1-1-onto-> suc M /\ M e. suc M ) -> ( F ` ( `' F ` M ) ) = M ) |
15 |
|
snidg |
|- ( M e. suc M -> M e. { M } ) |
16 |
15
|
adantl |
|- ( ( F : A -1-1-onto-> suc M /\ M e. suc M ) -> M e. { M } ) |
17 |
14 16
|
eqeltrd |
|- ( ( F : A -1-1-onto-> suc M /\ M e. suc M ) -> ( F ` ( `' F ` M ) ) e. { M } ) |
18 |
|
fressnfv |
|- ( ( F Fn A /\ ( `' F ` M ) e. A ) -> ( ( F |` { ( `' F ` M ) } ) : { ( `' F ` M ) } --> { M } <-> ( F ` ( `' F ` M ) ) e. { M } ) ) |
19 |
18
|
biimp3ar |
|- ( ( F Fn A /\ ( `' F ` M ) e. A /\ ( F ` ( `' F ` M ) ) e. { M } ) -> ( F |` { ( `' F ` M ) } ) : { ( `' F ` M ) } --> { M } ) |
20 |
12 13 17 19
|
syl3anc |
|- ( ( F : A -1-1-onto-> suc M /\ M e. suc M ) -> ( F |` { ( `' F ` M ) } ) : { ( `' F ` M ) } --> { M } ) |
21 |
|
disjsn |
|- ( ( A i^i { ( `' F ` M ) } ) = (/) <-> -. ( `' F ` M ) e. A ) |
22 |
21
|
con2bii |
|- ( ( `' F ` M ) e. A <-> -. ( A i^i { ( `' F ` M ) } ) = (/) ) |
23 |
13 22
|
sylib |
|- ( ( F : A -1-1-onto-> suc M /\ M e. suc M ) -> -. ( A i^i { ( `' F ` M ) } ) = (/) ) |
24 |
|
fnresdisj |
|- ( F Fn A -> ( ( A i^i { ( `' F ` M ) } ) = (/) <-> ( F |` { ( `' F ` M ) } ) = (/) ) ) |
25 |
6 24
|
syl |
|- ( F : A -1-1-onto-> suc M -> ( ( A i^i { ( `' F ` M ) } ) = (/) <-> ( F |` { ( `' F ` M ) } ) = (/) ) ) |
26 |
25
|
adantr |
|- ( ( F : A -1-1-onto-> suc M /\ M e. suc M ) -> ( ( A i^i { ( `' F ` M ) } ) = (/) <-> ( F |` { ( `' F ` M ) } ) = (/) ) ) |
27 |
23 26
|
mtbid |
|- ( ( F : A -1-1-onto-> suc M /\ M e. suc M ) -> -. ( F |` { ( `' F ` M ) } ) = (/) ) |
28 |
27
|
neqned |
|- ( ( F : A -1-1-onto-> suc M /\ M e. suc M ) -> ( F |` { ( `' F ` M ) } ) =/= (/) ) |
29 |
|
foconst |
|- ( ( ( F |` { ( `' F ` M ) } ) : { ( `' F ` M ) } --> { M } /\ ( F |` { ( `' F ` M ) } ) =/= (/) ) -> ( F |` { ( `' F ` M ) } ) : { ( `' F ` M ) } -onto-> { M } ) |
30 |
20 28 29
|
syl2anc |
|- ( ( F : A -1-1-onto-> suc M /\ M e. suc M ) -> ( F |` { ( `' F ` M ) } ) : { ( `' F ` M ) } -onto-> { M } ) |
31 |
|
resdif |
|- ( ( Fun `' F /\ ( F |` A ) : A -onto-> suc M /\ ( F |` { ( `' F ` M ) } ) : { ( `' F ` M ) } -onto-> { M } ) -> ( F |` ( A \ { ( `' F ` M ) } ) ) : ( A \ { ( `' F ` M ) } ) -1-1-onto-> ( suc M \ { M } ) ) |
32 |
4 11 30 31
|
syl3anc |
|- ( ( F : A -1-1-onto-> suc M /\ M e. suc M ) -> ( F |` ( A \ { ( `' F ` M ) } ) ) : ( A \ { ( `' F ` M ) } ) -1-1-onto-> ( suc M \ { M } ) ) |
33 |
1 32
|
sylan2 |
|- ( ( F : A -1-1-onto-> suc M /\ M e. On ) -> ( F |` ( A \ { ( `' F ` M ) } ) ) : ( A \ { ( `' F ` M ) } ) -1-1-onto-> ( suc M \ { M } ) ) |
34 |
|
eloni |
|- ( M e. On -> Ord M ) |
35 |
|
orddif |
|- ( Ord M -> M = ( suc M \ { M } ) ) |
36 |
34 35
|
syl |
|- ( M e. On -> M = ( suc M \ { M } ) ) |
37 |
36
|
f1oeq3d |
|- ( M e. On -> ( ( F |` ( A \ { ( `' F ` M ) } ) ) : ( A \ { ( `' F ` M ) } ) -1-1-onto-> M <-> ( F |` ( A \ { ( `' F ` M ) } ) ) : ( A \ { ( `' F ` M ) } ) -1-1-onto-> ( suc M \ { M } ) ) ) |
38 |
37
|
adantl |
|- ( ( F : A -1-1-onto-> suc M /\ M e. On ) -> ( ( F |` ( A \ { ( `' F ` M ) } ) ) : ( A \ { ( `' F ` M ) } ) -1-1-onto-> M <-> ( F |` ( A \ { ( `' F ` M ) } ) ) : ( A \ { ( `' F ` M ) } ) -1-1-onto-> ( suc M \ { M } ) ) ) |
39 |
33 38
|
mpbird |
|- ( ( F : A -1-1-onto-> suc M /\ M e. On ) -> ( F |` ( A \ { ( `' F ` M ) } ) ) : ( A \ { ( `' F ` M ) } ) -1-1-onto-> M ) |
40 |
39
|
ancoms |
|- ( ( M e. On /\ F : A -1-1-onto-> suc M ) -> ( F |` ( A \ { ( `' F ` M ) } ) ) : ( A \ { ( `' F ` M ) } ) -1-1-onto-> M ) |
41 |
40
|
3ad2antl3 |
|- ( ( ( F e. V /\ A e. W /\ M e. On ) /\ F : A -1-1-onto-> suc M ) -> ( F |` ( A \ { ( `' F ` M ) } ) ) : ( A \ { ( `' F ` M ) } ) -1-1-onto-> M ) |
42 |
|
difexg |
|- ( A e. W -> ( A \ { ( `' F ` M ) } ) e. _V ) |
43 |
|
resexg |
|- ( F e. V -> ( F |` ( A \ { ( `' F ` M ) } ) ) e. _V ) |
44 |
|
f1oen4g |
|- ( ( ( ( F |` ( A \ { ( `' F ` M ) } ) ) e. _V /\ ( A \ { ( `' F ` M ) } ) e. _V /\ M e. On ) /\ ( F |` ( A \ { ( `' F ` M ) } ) ) : ( A \ { ( `' F ` M ) } ) -1-1-onto-> M ) -> ( A \ { ( `' F ` M ) } ) ~~ M ) |
45 |
43 44
|
syl3anl1 |
|- ( ( ( F e. V /\ ( A \ { ( `' F ` M ) } ) e. _V /\ M e. On ) /\ ( F |` ( A \ { ( `' F ` M ) } ) ) : ( A \ { ( `' F ` M ) } ) -1-1-onto-> M ) -> ( A \ { ( `' F ` M ) } ) ~~ M ) |
46 |
42 45
|
syl3anl2 |
|- ( ( ( F e. V /\ A e. W /\ M e. On ) /\ ( F |` ( A \ { ( `' F ` M ) } ) ) : ( A \ { ( `' F ` M ) } ) -1-1-onto-> M ) -> ( A \ { ( `' F ` M ) } ) ~~ M ) |
47 |
41 46
|
syldan |
|- ( ( ( F e. V /\ A e. W /\ M e. On ) /\ F : A -1-1-onto-> suc M ) -> ( A \ { ( `' F ` M ) } ) ~~ M ) |