Step |
Hyp |
Ref |
Expression |
1 |
|
noreson |
|- ( ( A e. No /\ X e. On ) -> ( A |` X ) e. No ) |
2 |
1
|
3adant2 |
|- ( ( A e. No /\ B e. No /\ X e. On ) -> ( A |` X ) e. No ) |
3 |
|
noreson |
|- ( ( B e. No /\ X e. On ) -> ( B |` X ) e. No ) |
4 |
3
|
3adant1 |
|- ( ( A e. No /\ B e. No /\ X e. On ) -> ( B |` X ) e. No ) |
5 |
|
sltintdifex |
|- ( ( ( A |` X ) e. No /\ ( B |` X ) e. No ) -> ( ( A |` X ) |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } e. _V ) ) |
6 |
|
onintrab |
|- ( |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } e. _V <-> |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } e. On ) |
7 |
5 6
|
syl6ib |
|- ( ( ( A |` X ) e. No /\ ( B |` X ) e. No ) -> ( ( A |` X ) |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } e. On ) ) |
8 |
2 4 7
|
syl2anc |
|- ( ( A e. No /\ B e. No /\ X e. On ) -> ( ( A |` X ) |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } e. On ) ) |
9 |
8
|
imp |
|- ( ( ( A e. No /\ B e. No /\ X e. On ) /\ ( A |` X ) |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } e. On ) |
10 |
|
simpl3 |
|- ( ( ( A e. No /\ B e. No /\ X e. On ) /\ ( A |` X ) X e. On ) |
11 |
|
sltval2 |
|- ( ( ( A |` X ) e. No /\ ( B |` X ) e. No ) -> ( ( A |` X ) ( ( A |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) { <. 1o , (/) >. , <. 1o , 2o >. , <. (/) , 2o >. } ( ( B |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) ) ) |
12 |
2 4 11
|
syl2anc |
|- ( ( A e. No /\ B e. No /\ X e. On ) -> ( ( A |` X ) ( ( A |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) { <. 1o , (/) >. , <. 1o , 2o >. , <. (/) , 2o >. } ( ( B |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) ) ) |
13 |
|
fvex |
|- ( ( A |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) e. _V |
14 |
|
fvex |
|- ( ( B |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) e. _V |
15 |
13 14
|
brtp |
|- ( ( ( A |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) { <. 1o , (/) >. , <. 1o , 2o >. , <. (/) , 2o >. } ( ( B |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) <-> ( ( ( ( A |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = 1o /\ ( ( B |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = (/) ) \/ ( ( ( A |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = 1o /\ ( ( B |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = 2o ) \/ ( ( ( A |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = (/) /\ ( ( B |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = 2o ) ) ) |
16 |
|
1n0 |
|- 1o =/= (/) |
17 |
16
|
neii |
|- -. 1o = (/) |
18 |
|
eqeq1 |
|- ( ( ( A |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = 1o -> ( ( ( A |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = (/) <-> 1o = (/) ) ) |
19 |
17 18
|
mtbiri |
|- ( ( ( A |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = 1o -> -. ( ( A |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = (/) ) |
20 |
|
ndmfv |
|- ( -. |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } e. dom ( A |` X ) -> ( ( A |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = (/) ) |
21 |
19 20
|
nsyl2 |
|- ( ( ( A |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = 1o -> |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } e. dom ( A |` X ) ) |
22 |
21
|
adantr |
|- ( ( ( ( A |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = 1o /\ ( ( B |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = (/) ) -> |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } e. dom ( A |` X ) ) |
23 |
22
|
orcd |
|- ( ( ( ( A |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = 1o /\ ( ( B |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = (/) ) -> ( |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } e. dom ( A |` X ) \/ |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } e. dom ( B |` X ) ) ) |
24 |
21
|
adantr |
|- ( ( ( ( A |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = 1o /\ ( ( B |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = 2o ) -> |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } e. dom ( A |` X ) ) |
25 |
24
|
orcd |
|- ( ( ( ( A |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = 1o /\ ( ( B |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = 2o ) -> ( |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } e. dom ( A |` X ) \/ |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } e. dom ( B |` X ) ) ) |
26 |
|
2on |
|- 2o e. On |
27 |
26
|
elexi |
|- 2o e. _V |
28 |
27
|
prid2 |
|- 2o e. { 1o , 2o } |
29 |
28
|
nosgnn0i |
|- (/) =/= 2o |
30 |
29
|
neii |
|- -. (/) = 2o |
31 |
|
eqeq1 |
|- ( ( ( B |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = 2o -> ( ( ( B |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = (/) <-> 2o = (/) ) ) |
32 |
|
eqcom |
|- ( 2o = (/) <-> (/) = 2o ) |
33 |
31 32
|
bitrdi |
|- ( ( ( B |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = 2o -> ( ( ( B |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = (/) <-> (/) = 2o ) ) |
34 |
30 33
|
mtbiri |
|- ( ( ( B |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = 2o -> -. ( ( B |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = (/) ) |
35 |
|
ndmfv |
|- ( -. |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } e. dom ( B |` X ) -> ( ( B |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = (/) ) |
36 |
34 35
|
nsyl2 |
|- ( ( ( B |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = 2o -> |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } e. dom ( B |` X ) ) |
37 |
36
|
adantl |
|- ( ( ( ( A |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = (/) /\ ( ( B |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = 2o ) -> |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } e. dom ( B |` X ) ) |
38 |
37
|
olcd |
|- ( ( ( ( A |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = (/) /\ ( ( B |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = 2o ) -> ( |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } e. dom ( A |` X ) \/ |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } e. dom ( B |` X ) ) ) |
39 |
23 25 38
|
3jaoi |
|- ( ( ( ( ( A |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = 1o /\ ( ( B |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = (/) ) \/ ( ( ( A |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = 1o /\ ( ( B |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = 2o ) \/ ( ( ( A |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = (/) /\ ( ( B |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = 2o ) ) -> ( |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } e. dom ( A |` X ) \/ |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } e. dom ( B |` X ) ) ) |
40 |
15 39
|
sylbi |
|- ( ( ( A |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) { <. 1o , (/) >. , <. 1o , 2o >. , <. (/) , 2o >. } ( ( B |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) -> ( |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } e. dom ( A |` X ) \/ |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } e. dom ( B |` X ) ) ) |
41 |
12 40
|
syl6bi |
|- ( ( A e. No /\ B e. No /\ X e. On ) -> ( ( A |` X ) ( |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } e. dom ( A |` X ) \/ |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } e. dom ( B |` X ) ) ) ) |
42 |
41
|
imp |
|- ( ( ( A e. No /\ B e. No /\ X e. On ) /\ ( A |` X ) ( |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } e. dom ( A |` X ) \/ |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } e. dom ( B |` X ) ) ) |
43 |
|
dmres |
|- dom ( A |` X ) = ( X i^i dom A ) |
44 |
43
|
elin2 |
|- ( |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } e. dom ( A |` X ) <-> ( |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } e. X /\ |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } e. dom A ) ) |
45 |
44
|
simplbi |
|- ( |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } e. dom ( A |` X ) -> |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } e. X ) |
46 |
|
dmres |
|- dom ( B |` X ) = ( X i^i dom B ) |
47 |
46
|
elin2 |
|- ( |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } e. dom ( B |` X ) <-> ( |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } e. X /\ |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } e. dom B ) ) |
48 |
47
|
simplbi |
|- ( |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } e. dom ( B |` X ) -> |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } e. X ) |
49 |
45 48
|
jaoi |
|- ( ( |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } e. dom ( A |` X ) \/ |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } e. dom ( B |` X ) ) -> |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } e. X ) |
50 |
42 49
|
syl |
|- ( ( ( A e. No /\ B e. No /\ X e. On ) /\ ( A |` X ) |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } e. X ) |
51 |
|
onelss |
|- ( X e. On -> ( |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } e. X -> |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } C_ X ) ) |
52 |
10 50 51
|
sylc |
|- ( ( ( A e. No /\ B e. No /\ X e. On ) /\ ( A |` X ) |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } C_ X ) |
53 |
52
|
sselda |
|- ( ( ( ( A e. No /\ B e. No /\ X e. On ) /\ ( A |` X ) y e. X ) |
54 |
|
onelon |
|- ( ( |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } e. On /\ y e. |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) -> y e. On ) |
55 |
9 54
|
sylan |
|- ( ( ( ( A e. No /\ B e. No /\ X e. On ) /\ ( A |` X ) y e. On ) |
56 |
|
intss1 |
|- ( y e. { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } -> |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } C_ y ) |
57 |
|
ontri1 |
|- ( ( |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } e. On /\ y e. On ) -> ( |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } C_ y <-> -. y e. |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) ) |
58 |
56 57
|
syl5ib |
|- ( ( |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } e. On /\ y e. On ) -> ( y e. { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } -> -. y e. |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) ) |
59 |
58
|
con2d |
|- ( ( |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } e. On /\ y e. On ) -> ( y e. |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } -> -. y e. { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) ) |
60 |
9 59
|
sylan |
|- ( ( ( ( A e. No /\ B e. No /\ X e. On ) /\ ( A |` X ) ( y e. |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } -> -. y e. { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) ) |
61 |
60
|
impancom |
|- ( ( ( ( A e. No /\ B e. No /\ X e. On ) /\ ( A |` X ) ( y e. On -> -. y e. { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) ) |
62 |
55 61
|
mpd |
|- ( ( ( ( A e. No /\ B e. No /\ X e. On ) /\ ( A |` X ) -. y e. { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) |
63 |
|
fveq2 |
|- ( a = y -> ( ( A |` X ) ` a ) = ( ( A |` X ) ` y ) ) |
64 |
|
fveq2 |
|- ( a = y -> ( ( B |` X ) ` a ) = ( ( B |` X ) ` y ) ) |
65 |
63 64
|
neeq12d |
|- ( a = y -> ( ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) <-> ( ( A |` X ) ` y ) =/= ( ( B |` X ) ` y ) ) ) |
66 |
65
|
elrab |
|- ( y e. { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } <-> ( y e. On /\ ( ( A |` X ) ` y ) =/= ( ( B |` X ) ` y ) ) ) |
67 |
66
|
simplbi2 |
|- ( y e. On -> ( ( ( A |` X ) ` y ) =/= ( ( B |` X ) ` y ) -> y e. { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) ) |
68 |
67
|
con3d |
|- ( y e. On -> ( -. y e. { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } -> -. ( ( A |` X ) ` y ) =/= ( ( B |` X ) ` y ) ) ) |
69 |
55 62 68
|
sylc |
|- ( ( ( ( A e. No /\ B e. No /\ X e. On ) /\ ( A |` X ) -. ( ( A |` X ) ` y ) =/= ( ( B |` X ) ` y ) ) |
70 |
|
df-ne |
|- ( ( ( A |` X ) ` y ) =/= ( ( B |` X ) ` y ) <-> -. ( ( A |` X ) ` y ) = ( ( B |` X ) ` y ) ) |
71 |
70
|
con2bii |
|- ( ( ( A |` X ) ` y ) = ( ( B |` X ) ` y ) <-> -. ( ( A |` X ) ` y ) =/= ( ( B |` X ) ` y ) ) |
72 |
69 71
|
sylibr |
|- ( ( ( ( A e. No /\ B e. No /\ X e. On ) /\ ( A |` X ) ( ( A |` X ) ` y ) = ( ( B |` X ) ` y ) ) |
73 |
|
fvres |
|- ( y e. X -> ( ( A |` X ) ` y ) = ( A ` y ) ) |
74 |
|
fvres |
|- ( y e. X -> ( ( B |` X ) ` y ) = ( B ` y ) ) |
75 |
73 74
|
eqeq12d |
|- ( y e. X -> ( ( ( A |` X ) ` y ) = ( ( B |` X ) ` y ) <-> ( A ` y ) = ( B ` y ) ) ) |
76 |
75
|
biimpd |
|- ( y e. X -> ( ( ( A |` X ) ` y ) = ( ( B |` X ) ` y ) -> ( A ` y ) = ( B ` y ) ) ) |
77 |
53 72 76
|
sylc |
|- ( ( ( ( A e. No /\ B e. No /\ X e. On ) /\ ( A |` X ) ( A ` y ) = ( B ` y ) ) |
78 |
77
|
ralrimiva |
|- ( ( ( A e. No /\ B e. No /\ X e. On ) /\ ( A |` X ) A. y e. |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ( A ` y ) = ( B ` y ) ) |
79 |
|
fvresval |
|- ( ( ( A |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = ( A ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) \/ ( ( A |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = (/) ) |
80 |
79
|
ori |
|- ( -. ( ( A |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = ( A ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) -> ( ( A |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = (/) ) |
81 |
19 80
|
nsyl2 |
|- ( ( ( A |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = 1o -> ( ( A |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = ( A ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) ) |
82 |
81
|
eqcomd |
|- ( ( ( A |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = 1o -> ( A ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = ( ( A |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) ) |
83 |
|
eqeq2 |
|- ( ( ( A |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = 1o -> ( ( A ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = ( ( A |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) <-> ( A ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = 1o ) ) |
84 |
82 83
|
mpbid |
|- ( ( ( A |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = 1o -> ( A ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = 1o ) |
85 |
84
|
adantr |
|- ( ( ( ( A |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = 1o /\ ( ( B |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = (/) ) -> ( A ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = 1o ) |
86 |
85
|
a1i |
|- ( ( A e. No /\ B e. No /\ X e. On ) -> ( ( ( ( A |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = 1o /\ ( ( B |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = (/) ) -> ( A ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = 1o ) ) |
87 |
21
|
ad2antrl |
|- ( ( ( A e. No /\ B e. No /\ X e. On ) /\ ( ( ( A |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = 1o /\ ( ( B |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = (/) ) ) -> |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } e. dom ( A |` X ) ) |
88 |
87 45
|
syl |
|- ( ( ( A e. No /\ B e. No /\ X e. On ) /\ ( ( ( A |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = 1o /\ ( ( B |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = (/) ) ) -> |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } e. X ) |
89 |
|
nofun |
|- ( ( B |` X ) e. No -> Fun ( B |` X ) ) |
90 |
|
fvelrn |
|- ( ( Fun ( B |` X ) /\ |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } e. dom ( B |` X ) ) -> ( ( B |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) e. ran ( B |` X ) ) |
91 |
90
|
ex |
|- ( Fun ( B |` X ) -> ( |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } e. dom ( B |` X ) -> ( ( B |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) e. ran ( B |` X ) ) ) |
92 |
89 91
|
syl |
|- ( ( B |` X ) e. No -> ( |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } e. dom ( B |` X ) -> ( ( B |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) e. ran ( B |` X ) ) ) |
93 |
|
norn |
|- ( ( B |` X ) e. No -> ran ( B |` X ) C_ { 1o , 2o } ) |
94 |
93
|
sseld |
|- ( ( B |` X ) e. No -> ( ( ( B |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) e. ran ( B |` X ) -> ( ( B |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) e. { 1o , 2o } ) ) |
95 |
92 94
|
syld |
|- ( ( B |` X ) e. No -> ( |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } e. dom ( B |` X ) -> ( ( B |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) e. { 1o , 2o } ) ) |
96 |
|
nosgnn0 |
|- -. (/) e. { 1o , 2o } |
97 |
|
eleq1 |
|- ( ( ( B |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = (/) -> ( ( ( B |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) e. { 1o , 2o } <-> (/) e. { 1o , 2o } ) ) |
98 |
96 97
|
mtbiri |
|- ( ( ( B |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = (/) -> -. ( ( B |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) e. { 1o , 2o } ) |
99 |
95 98
|
nsyli |
|- ( ( B |` X ) e. No -> ( ( ( B |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = (/) -> -. |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } e. dom ( B |` X ) ) ) |
100 |
4 99
|
syl |
|- ( ( A e. No /\ B e. No /\ X e. On ) -> ( ( ( B |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = (/) -> -. |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } e. dom ( B |` X ) ) ) |
101 |
100
|
imp |
|- ( ( ( A e. No /\ B e. No /\ X e. On ) /\ ( ( B |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = (/) ) -> -. |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } e. dom ( B |` X ) ) |
102 |
101
|
adantrl |
|- ( ( ( A e. No /\ B e. No /\ X e. On ) /\ ( ( ( A |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = 1o /\ ( ( B |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = (/) ) ) -> -. |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } e. dom ( B |` X ) ) |
103 |
47
|
simplbi2 |
|- ( |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } e. X -> ( |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } e. dom B -> |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } e. dom ( B |` X ) ) ) |
104 |
103
|
con3d |
|- ( |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } e. X -> ( -. |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } e. dom ( B |` X ) -> -. |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } e. dom B ) ) |
105 |
88 102 104
|
sylc |
|- ( ( ( A e. No /\ B e. No /\ X e. On ) /\ ( ( ( A |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = 1o /\ ( ( B |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = (/) ) ) -> -. |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } e. dom B ) |
106 |
|
ndmfv |
|- ( -. |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } e. dom B -> ( B ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = (/) ) |
107 |
105 106
|
syl |
|- ( ( ( A e. No /\ B e. No /\ X e. On ) /\ ( ( ( A |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = 1o /\ ( ( B |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = (/) ) ) -> ( B ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = (/) ) |
108 |
107
|
ex |
|- ( ( A e. No /\ B e. No /\ X e. On ) -> ( ( ( ( A |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = 1o /\ ( ( B |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = (/) ) -> ( B ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = (/) ) ) |
109 |
86 108
|
jcad |
|- ( ( A e. No /\ B e. No /\ X e. On ) -> ( ( ( ( A |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = 1o /\ ( ( B |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = (/) ) -> ( ( A ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = 1o /\ ( B ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = (/) ) ) ) |
110 |
|
fvresval |
|- ( ( ( B |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = ( B ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) \/ ( ( B |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = (/) ) |
111 |
110
|
ori |
|- ( -. ( ( B |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = ( B ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) -> ( ( B |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = (/) ) |
112 |
34 111
|
nsyl2 |
|- ( ( ( B |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = 2o -> ( ( B |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = ( B ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) ) |
113 |
112
|
eqcomd |
|- ( ( ( B |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = 2o -> ( B ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = ( ( B |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) ) |
114 |
|
eqeq2 |
|- ( ( ( B |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = 2o -> ( ( B ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = ( ( B |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) <-> ( B ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = 2o ) ) |
115 |
113 114
|
mpbid |
|- ( ( ( B |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = 2o -> ( B ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = 2o ) |
116 |
84 115
|
anim12i |
|- ( ( ( ( A |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = 1o /\ ( ( B |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = 2o ) -> ( ( A ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = 1o /\ ( B ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = 2o ) ) |
117 |
116
|
a1i |
|- ( ( A e. No /\ B e. No /\ X e. On ) -> ( ( ( ( A |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = 1o /\ ( ( B |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = 2o ) -> ( ( A ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = 1o /\ ( B ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = 2o ) ) ) |
118 |
36
|
ad2antll |
|- ( ( ( A e. No /\ B e. No /\ X e. On ) /\ ( ( ( A |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = (/) /\ ( ( B |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = 2o ) ) -> |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } e. dom ( B |` X ) ) |
119 |
118 48
|
syl |
|- ( ( ( A e. No /\ B e. No /\ X e. On ) /\ ( ( ( A |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = (/) /\ ( ( B |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = 2o ) ) -> |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } e. X ) |
120 |
|
nofun |
|- ( ( A |` X ) e. No -> Fun ( A |` X ) ) |
121 |
|
fvelrn |
|- ( ( Fun ( A |` X ) /\ |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } e. dom ( A |` X ) ) -> ( ( A |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) e. ran ( A |` X ) ) |
122 |
121
|
ex |
|- ( Fun ( A |` X ) -> ( |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } e. dom ( A |` X ) -> ( ( A |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) e. ran ( A |` X ) ) ) |
123 |
120 122
|
syl |
|- ( ( A |` X ) e. No -> ( |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } e. dom ( A |` X ) -> ( ( A |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) e. ran ( A |` X ) ) ) |
124 |
|
norn |
|- ( ( A |` X ) e. No -> ran ( A |` X ) C_ { 1o , 2o } ) |
125 |
124
|
sseld |
|- ( ( A |` X ) e. No -> ( ( ( A |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) e. ran ( A |` X ) -> ( ( A |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) e. { 1o , 2o } ) ) |
126 |
123 125
|
syld |
|- ( ( A |` X ) e. No -> ( |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } e. dom ( A |` X ) -> ( ( A |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) e. { 1o , 2o } ) ) |
127 |
|
eleq1 |
|- ( ( ( A |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = (/) -> ( ( ( A |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) e. { 1o , 2o } <-> (/) e. { 1o , 2o } ) ) |
128 |
96 127
|
mtbiri |
|- ( ( ( A |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = (/) -> -. ( ( A |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) e. { 1o , 2o } ) |
129 |
126 128
|
nsyli |
|- ( ( A |` X ) e. No -> ( ( ( A |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = (/) -> -. |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } e. dom ( A |` X ) ) ) |
130 |
2 129
|
syl |
|- ( ( A e. No /\ B e. No /\ X e. On ) -> ( ( ( A |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = (/) -> -. |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } e. dom ( A |` X ) ) ) |
131 |
130
|
imp |
|- ( ( ( A e. No /\ B e. No /\ X e. On ) /\ ( ( A |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = (/) ) -> -. |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } e. dom ( A |` X ) ) |
132 |
131
|
adantrr |
|- ( ( ( A e. No /\ B e. No /\ X e. On ) /\ ( ( ( A |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = (/) /\ ( ( B |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = 2o ) ) -> -. |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } e. dom ( A |` X ) ) |
133 |
44
|
simplbi2 |
|- ( |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } e. X -> ( |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } e. dom A -> |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } e. dom ( A |` X ) ) ) |
134 |
133
|
con3d |
|- ( |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } e. X -> ( -. |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } e. dom ( A |` X ) -> -. |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } e. dom A ) ) |
135 |
119 132 134
|
sylc |
|- ( ( ( A e. No /\ B e. No /\ X e. On ) /\ ( ( ( A |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = (/) /\ ( ( B |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = 2o ) ) -> -. |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } e. dom A ) |
136 |
135
|
ex |
|- ( ( A e. No /\ B e. No /\ X e. On ) -> ( ( ( ( A |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = (/) /\ ( ( B |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = 2o ) -> -. |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } e. dom A ) ) |
137 |
|
ndmfv |
|- ( -. |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } e. dom A -> ( A ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = (/) ) |
138 |
136 137
|
syl6 |
|- ( ( A e. No /\ B e. No /\ X e. On ) -> ( ( ( ( A |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = (/) /\ ( ( B |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = 2o ) -> ( A ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = (/) ) ) |
139 |
115
|
adantl |
|- ( ( ( ( A |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = (/) /\ ( ( B |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = 2o ) -> ( B ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = 2o ) |
140 |
139
|
a1i |
|- ( ( A e. No /\ B e. No /\ X e. On ) -> ( ( ( ( A |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = (/) /\ ( ( B |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = 2o ) -> ( B ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = 2o ) ) |
141 |
138 140
|
jcad |
|- ( ( A e. No /\ B e. No /\ X e. On ) -> ( ( ( ( A |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = (/) /\ ( ( B |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = 2o ) -> ( ( A ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = (/) /\ ( B ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = 2o ) ) ) |
142 |
109 117 141
|
3orim123d |
|- ( ( A e. No /\ B e. No /\ X e. On ) -> ( ( ( ( ( A |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = 1o /\ ( ( B |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = (/) ) \/ ( ( ( A |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = 1o /\ ( ( B |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = 2o ) \/ ( ( ( A |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = (/) /\ ( ( B |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = 2o ) ) -> ( ( ( A ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = 1o /\ ( B ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = (/) ) \/ ( ( A ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = 1o /\ ( B ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = 2o ) \/ ( ( A ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = (/) /\ ( B ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = 2o ) ) ) ) |
143 |
|
fvex |
|- ( A ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) e. _V |
144 |
|
fvex |
|- ( B ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) e. _V |
145 |
143 144
|
brtp |
|- ( ( A ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) { <. 1o , (/) >. , <. 1o , 2o >. , <. (/) , 2o >. } ( B ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) <-> ( ( ( A ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = 1o /\ ( B ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = (/) ) \/ ( ( A ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = 1o /\ ( B ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = 2o ) \/ ( ( A ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = (/) /\ ( B ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = 2o ) ) ) |
146 |
142 15 145
|
3imtr4g |
|- ( ( A e. No /\ B e. No /\ X e. On ) -> ( ( ( A |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) { <. 1o , (/) >. , <. 1o , 2o >. , <. (/) , 2o >. } ( ( B |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) -> ( A ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) { <. 1o , (/) >. , <. 1o , 2o >. , <. (/) , 2o >. } ( B ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) ) ) |
147 |
12 146
|
sylbid |
|- ( ( A e. No /\ B e. No /\ X e. On ) -> ( ( A |` X ) ( A ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) { <. 1o , (/) >. , <. 1o , 2o >. , <. (/) , 2o >. } ( B ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) ) ) |
148 |
147
|
imp |
|- ( ( ( A e. No /\ B e. No /\ X e. On ) /\ ( A |` X ) ( A ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) { <. 1o , (/) >. , <. 1o , 2o >. , <. (/) , 2o >. } ( B ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) ) |
149 |
|
raleq |
|- ( x = |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } -> ( A. y e. x ( A ` y ) = ( B ` y ) <-> A. y e. |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ( A ` y ) = ( B ` y ) ) ) |
150 |
|
fveq2 |
|- ( x = |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } -> ( A ` x ) = ( A ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) ) |
151 |
|
fveq2 |
|- ( x = |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } -> ( B ` x ) = ( B ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) ) |
152 |
150 151
|
breq12d |
|- ( x = |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } -> ( ( A ` x ) { <. 1o , (/) >. , <. 1o , 2o >. , <. (/) , 2o >. } ( B ` x ) <-> ( A ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) { <. 1o , (/) >. , <. 1o , 2o >. , <. (/) , 2o >. } ( B ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) ) ) |
153 |
149 152
|
anbi12d |
|- ( x = |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } -> ( ( A. y e. x ( A ` y ) = ( B ` y ) /\ ( A ` x ) { <. 1o , (/) >. , <. 1o , 2o >. , <. (/) , 2o >. } ( B ` x ) ) <-> ( A. y e. |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ( A ` y ) = ( B ` y ) /\ ( A ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) { <. 1o , (/) >. , <. 1o , 2o >. , <. (/) , 2o >. } ( B ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) ) ) ) |
154 |
153
|
rspcev |
|- ( ( |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } e. On /\ ( A. y e. |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ( A ` y ) = ( B ` y ) /\ ( A ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) { <. 1o , (/) >. , <. 1o , 2o >. , <. (/) , 2o >. } ( B ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) ) ) -> E. x e. On ( A. y e. x ( A ` y ) = ( B ` y ) /\ ( A ` x ) { <. 1o , (/) >. , <. 1o , 2o >. , <. (/) , 2o >. } ( B ` x ) ) ) |
155 |
9 78 148 154
|
syl12anc |
|- ( ( ( A e. No /\ B e. No /\ X e. On ) /\ ( A |` X ) E. x e. On ( A. y e. x ( A ` y ) = ( B ` y ) /\ ( A ` x ) { <. 1o , (/) >. , <. 1o , 2o >. , <. (/) , 2o >. } ( B ` x ) ) ) |
156 |
|
sltval |
|- ( ( A e. No /\ B e. No ) -> ( A E. x e. On ( A. y e. x ( A ` y ) = ( B ` y ) /\ ( A ` x ) { <. 1o , (/) >. , <. 1o , 2o >. , <. (/) , 2o >. } ( B ` x ) ) ) ) |
157 |
156
|
3adant3 |
|- ( ( A e. No /\ B e. No /\ X e. On ) -> ( A E. x e. On ( A. y e. x ( A ` y ) = ( B ` y ) /\ ( A ` x ) { <. 1o , (/) >. , <. 1o , 2o >. , <. (/) , 2o >. } ( B ` x ) ) ) ) |
158 |
157
|
adantr |
|- ( ( ( A e. No /\ B e. No /\ X e. On ) /\ ( A |` X ) ( A E. x e. On ( A. y e. x ( A ` y ) = ( B ` y ) /\ ( A ` x ) { <. 1o , (/) >. , <. 1o , 2o >. , <. (/) , 2o >. } ( B ` x ) ) ) ) |
159 |
155 158
|
mpbird |
|- ( ( ( A e. No /\ B e. No /\ X e. On ) /\ ( A |` X ) A |
160 |
159
|
ex |
|- ( ( A e. No /\ B e. No /\ X e. On ) -> ( ( A |` X ) A |