Step |
Hyp |
Ref |
Expression |
1 |
|
konigth.1 |
|- A e. _V |
2 |
|
konigth.2 |
|- S = U_ i e. A ( M ` i ) |
3 |
|
konigth.3 |
|- P = X_ i e. A ( N ` i ) |
4 |
|
konigth.4 |
|- D = ( i e. A |-> ( a e. ( M ` i ) |-> ( ( f ` a ) ` i ) ) ) |
5 |
|
konigth.5 |
|- E = ( i e. A |-> ( e ` i ) ) |
6 |
|
fvex |
|- ( M ` i ) e. _V |
7 |
|
fvex |
|- ( ( f ` a ) ` i ) e. _V |
8 |
|
eqid |
|- ( a e. ( M ` i ) |-> ( ( f ` a ) ` i ) ) = ( a e. ( M ` i ) |-> ( ( f ` a ) ` i ) ) |
9 |
7 8
|
fnmpti |
|- ( a e. ( M ` i ) |-> ( ( f ` a ) ` i ) ) Fn ( M ` i ) |
10 |
6
|
mptex |
|- ( a e. ( M ` i ) |-> ( ( f ` a ) ` i ) ) e. _V |
11 |
4
|
fvmpt2 |
|- ( ( i e. A /\ ( a e. ( M ` i ) |-> ( ( f ` a ) ` i ) ) e. _V ) -> ( D ` i ) = ( a e. ( M ` i ) |-> ( ( f ` a ) ` i ) ) ) |
12 |
10 11
|
mpan2 |
|- ( i e. A -> ( D ` i ) = ( a e. ( M ` i ) |-> ( ( f ` a ) ` i ) ) ) |
13 |
12
|
fneq1d |
|- ( i e. A -> ( ( D ` i ) Fn ( M ` i ) <-> ( a e. ( M ` i ) |-> ( ( f ` a ) ` i ) ) Fn ( M ` i ) ) ) |
14 |
9 13
|
mpbiri |
|- ( i e. A -> ( D ` i ) Fn ( M ` i ) ) |
15 |
|
fnrndomg |
|- ( ( M ` i ) e. _V -> ( ( D ` i ) Fn ( M ` i ) -> ran ( D ` i ) ~<_ ( M ` i ) ) ) |
16 |
6 14 15
|
mpsyl |
|- ( i e. A -> ran ( D ` i ) ~<_ ( M ` i ) ) |
17 |
|
domsdomtr |
|- ( ( ran ( D ` i ) ~<_ ( M ` i ) /\ ( M ` i ) ~< ( N ` i ) ) -> ran ( D ` i ) ~< ( N ` i ) ) |
18 |
16 17
|
sylan |
|- ( ( i e. A /\ ( M ` i ) ~< ( N ` i ) ) -> ran ( D ` i ) ~< ( N ` i ) ) |
19 |
|
sdomdif |
|- ( ran ( D ` i ) ~< ( N ` i ) -> ( ( N ` i ) \ ran ( D ` i ) ) =/= (/) ) |
20 |
18 19
|
syl |
|- ( ( i e. A /\ ( M ` i ) ~< ( N ` i ) ) -> ( ( N ` i ) \ ran ( D ` i ) ) =/= (/) ) |
21 |
20
|
ralimiaa |
|- ( A. i e. A ( M ` i ) ~< ( N ` i ) -> A. i e. A ( ( N ` i ) \ ran ( D ` i ) ) =/= (/) ) |
22 |
|
fvex |
|- ( N ` i ) e. _V |
23 |
22
|
difexi |
|- ( ( N ` i ) \ ran ( D ` i ) ) e. _V |
24 |
1 23
|
ac6c5 |
|- ( A. i e. A ( ( N ` i ) \ ran ( D ` i ) ) =/= (/) -> E. e A. i e. A ( e ` i ) e. ( ( N ` i ) \ ran ( D ` i ) ) ) |
25 |
|
equid |
|- f = f |
26 |
|
eldifi |
|- ( ( e ` i ) e. ( ( N ` i ) \ ran ( D ` i ) ) -> ( e ` i ) e. ( N ` i ) ) |
27 |
|
fvex |
|- ( e ` i ) e. _V |
28 |
5
|
fvmpt2 |
|- ( ( i e. A /\ ( e ` i ) e. _V ) -> ( E ` i ) = ( e ` i ) ) |
29 |
27 28
|
mpan2 |
|- ( i e. A -> ( E ` i ) = ( e ` i ) ) |
30 |
29
|
eleq1d |
|- ( i e. A -> ( ( E ` i ) e. ( N ` i ) <-> ( e ` i ) e. ( N ` i ) ) ) |
31 |
26 30
|
syl5ibr |
|- ( i e. A -> ( ( e ` i ) e. ( ( N ` i ) \ ran ( D ` i ) ) -> ( E ` i ) e. ( N ` i ) ) ) |
32 |
31
|
ralimia |
|- ( A. i e. A ( e ` i ) e. ( ( N ` i ) \ ran ( D ` i ) ) -> A. i e. A ( E ` i ) e. ( N ` i ) ) |
33 |
27 5
|
fnmpti |
|- E Fn A |
34 |
32 33
|
jctil |
|- ( A. i e. A ( e ` i ) e. ( ( N ` i ) \ ran ( D ` i ) ) -> ( E Fn A /\ A. i e. A ( E ` i ) e. ( N ` i ) ) ) |
35 |
1
|
mptex |
|- ( i e. A |-> ( e ` i ) ) e. _V |
36 |
5 35
|
eqeltri |
|- E e. _V |
37 |
36
|
elixp |
|- ( E e. X_ i e. A ( N ` i ) <-> ( E Fn A /\ A. i e. A ( E ` i ) e. ( N ` i ) ) ) |
38 |
34 37
|
sylibr |
|- ( A. i e. A ( e ` i ) e. ( ( N ` i ) \ ran ( D ` i ) ) -> E e. X_ i e. A ( N ` i ) ) |
39 |
38 3
|
eleqtrrdi |
|- ( A. i e. A ( e ` i ) e. ( ( N ` i ) \ ran ( D ` i ) ) -> E e. P ) |
40 |
|
foelrn |
|- ( ( f : S -onto-> P /\ E e. P ) -> E. a e. S E = ( f ` a ) ) |
41 |
40
|
expcom |
|- ( E e. P -> ( f : S -onto-> P -> E. a e. S E = ( f ` a ) ) ) |
42 |
2
|
eleq2i |
|- ( a e. S <-> a e. U_ i e. A ( M ` i ) ) |
43 |
|
eliun |
|- ( a e. U_ i e. A ( M ` i ) <-> E. i e. A a e. ( M ` i ) ) |
44 |
42 43
|
bitri |
|- ( a e. S <-> E. i e. A a e. ( M ` i ) ) |
45 |
|
nfra1 |
|- F/ i A. i e. A ( e ` i ) e. ( ( N ` i ) \ ran ( D ` i ) ) |
46 |
|
nfv |
|- F/ i E = ( f ` a ) |
47 |
45 46
|
nfan |
|- F/ i ( A. i e. A ( e ` i ) e. ( ( N ` i ) \ ran ( D ` i ) ) /\ E = ( f ` a ) ) |
48 |
|
nfv |
|- F/ i -. f = f |
49 |
29
|
ad2antrl |
|- ( ( E = ( f ` a ) /\ ( i e. A /\ a e. ( M ` i ) ) ) -> ( E ` i ) = ( e ` i ) ) |
50 |
|
fveq1 |
|- ( E = ( f ` a ) -> ( E ` i ) = ( ( f ` a ) ` i ) ) |
51 |
12
|
fveq1d |
|- ( i e. A -> ( ( D ` i ) ` a ) = ( ( a e. ( M ` i ) |-> ( ( f ` a ) ` i ) ) ` a ) ) |
52 |
8
|
fvmpt2 |
|- ( ( a e. ( M ` i ) /\ ( ( f ` a ) ` i ) e. _V ) -> ( ( a e. ( M ` i ) |-> ( ( f ` a ) ` i ) ) ` a ) = ( ( f ` a ) ` i ) ) |
53 |
7 52
|
mpan2 |
|- ( a e. ( M ` i ) -> ( ( a e. ( M ` i ) |-> ( ( f ` a ) ` i ) ) ` a ) = ( ( f ` a ) ` i ) ) |
54 |
51 53
|
sylan9eq |
|- ( ( i e. A /\ a e. ( M ` i ) ) -> ( ( D ` i ) ` a ) = ( ( f ` a ) ` i ) ) |
55 |
54
|
eqcomd |
|- ( ( i e. A /\ a e. ( M ` i ) ) -> ( ( f ` a ) ` i ) = ( ( D ` i ) ` a ) ) |
56 |
50 55
|
sylan9eq |
|- ( ( E = ( f ` a ) /\ ( i e. A /\ a e. ( M ` i ) ) ) -> ( E ` i ) = ( ( D ` i ) ` a ) ) |
57 |
49 56
|
eqtr3d |
|- ( ( E = ( f ` a ) /\ ( i e. A /\ a e. ( M ` i ) ) ) -> ( e ` i ) = ( ( D ` i ) ` a ) ) |
58 |
|
fnfvelrn |
|- ( ( ( D ` i ) Fn ( M ` i ) /\ a e. ( M ` i ) ) -> ( ( D ` i ) ` a ) e. ran ( D ` i ) ) |
59 |
14 58
|
sylan |
|- ( ( i e. A /\ a e. ( M ` i ) ) -> ( ( D ` i ) ` a ) e. ran ( D ` i ) ) |
60 |
59
|
adantl |
|- ( ( E = ( f ` a ) /\ ( i e. A /\ a e. ( M ` i ) ) ) -> ( ( D ` i ) ` a ) e. ran ( D ` i ) ) |
61 |
57 60
|
eqeltrd |
|- ( ( E = ( f ` a ) /\ ( i e. A /\ a e. ( M ` i ) ) ) -> ( e ` i ) e. ran ( D ` i ) ) |
62 |
61
|
3adant1 |
|- ( ( A. i e. A ( e ` i ) e. ( ( N ` i ) \ ran ( D ` i ) ) /\ E = ( f ` a ) /\ ( i e. A /\ a e. ( M ` i ) ) ) -> ( e ` i ) e. ran ( D ` i ) ) |
63 |
|
simp1 |
|- ( ( A. i e. A ( e ` i ) e. ( ( N ` i ) \ ran ( D ` i ) ) /\ E = ( f ` a ) /\ ( i e. A /\ a e. ( M ` i ) ) ) -> A. i e. A ( e ` i ) e. ( ( N ` i ) \ ran ( D ` i ) ) ) |
64 |
|
simp3l |
|- ( ( A. i e. A ( e ` i ) e. ( ( N ` i ) \ ran ( D ` i ) ) /\ E = ( f ` a ) /\ ( i e. A /\ a e. ( M ` i ) ) ) -> i e. A ) |
65 |
|
rsp |
|- ( A. i e. A ( e ` i ) e. ( ( N ` i ) \ ran ( D ` i ) ) -> ( i e. A -> ( e ` i ) e. ( ( N ` i ) \ ran ( D ` i ) ) ) ) |
66 |
|
eldifn |
|- ( ( e ` i ) e. ( ( N ` i ) \ ran ( D ` i ) ) -> -. ( e ` i ) e. ran ( D ` i ) ) |
67 |
65 66
|
syl6 |
|- ( A. i e. A ( e ` i ) e. ( ( N ` i ) \ ran ( D ` i ) ) -> ( i e. A -> -. ( e ` i ) e. ran ( D ` i ) ) ) |
68 |
63 64 67
|
sylc |
|- ( ( A. i e. A ( e ` i ) e. ( ( N ` i ) \ ran ( D ` i ) ) /\ E = ( f ` a ) /\ ( i e. A /\ a e. ( M ` i ) ) ) -> -. ( e ` i ) e. ran ( D ` i ) ) |
69 |
62 68
|
pm2.21dd |
|- ( ( A. i e. A ( e ` i ) e. ( ( N ` i ) \ ran ( D ` i ) ) /\ E = ( f ` a ) /\ ( i e. A /\ a e. ( M ` i ) ) ) -> -. f = f ) |
70 |
69
|
3expia |
|- ( ( A. i e. A ( e ` i ) e. ( ( N ` i ) \ ran ( D ` i ) ) /\ E = ( f ` a ) ) -> ( ( i e. A /\ a e. ( M ` i ) ) -> -. f = f ) ) |
71 |
70
|
expd |
|- ( ( A. i e. A ( e ` i ) e. ( ( N ` i ) \ ran ( D ` i ) ) /\ E = ( f ` a ) ) -> ( i e. A -> ( a e. ( M ` i ) -> -. f = f ) ) ) |
72 |
47 48 71
|
rexlimd |
|- ( ( A. i e. A ( e ` i ) e. ( ( N ` i ) \ ran ( D ` i ) ) /\ E = ( f ` a ) ) -> ( E. i e. A a e. ( M ` i ) -> -. f = f ) ) |
73 |
44 72
|
syl5bi |
|- ( ( A. i e. A ( e ` i ) e. ( ( N ` i ) \ ran ( D ` i ) ) /\ E = ( f ` a ) ) -> ( a e. S -> -. f = f ) ) |
74 |
73
|
ex |
|- ( A. i e. A ( e ` i ) e. ( ( N ` i ) \ ran ( D ` i ) ) -> ( E = ( f ` a ) -> ( a e. S -> -. f = f ) ) ) |
75 |
74
|
com23 |
|- ( A. i e. A ( e ` i ) e. ( ( N ` i ) \ ran ( D ` i ) ) -> ( a e. S -> ( E = ( f ` a ) -> -. f = f ) ) ) |
76 |
75
|
rexlimdv |
|- ( A. i e. A ( e ` i ) e. ( ( N ` i ) \ ran ( D ` i ) ) -> ( E. a e. S E = ( f ` a ) -> -. f = f ) ) |
77 |
41 76
|
syl9r |
|- ( A. i e. A ( e ` i ) e. ( ( N ` i ) \ ran ( D ` i ) ) -> ( E e. P -> ( f : S -onto-> P -> -. f = f ) ) ) |
78 |
39 77
|
mpd |
|- ( A. i e. A ( e ` i ) e. ( ( N ` i ) \ ran ( D ` i ) ) -> ( f : S -onto-> P -> -. f = f ) ) |
79 |
25 78
|
mt2i |
|- ( A. i e. A ( e ` i ) e. ( ( N ` i ) \ ran ( D ` i ) ) -> -. f : S -onto-> P ) |
80 |
79
|
exlimiv |
|- ( E. e A. i e. A ( e ` i ) e. ( ( N ` i ) \ ran ( D ` i ) ) -> -. f : S -onto-> P ) |
81 |
21 24 80
|
3syl |
|- ( A. i e. A ( M ` i ) ~< ( N ` i ) -> -. f : S -onto-> P ) |
82 |
81
|
nexdv |
|- ( A. i e. A ( M ` i ) ~< ( N ` i ) -> -. E. f f : S -onto-> P ) |
83 |
6
|
0dom |
|- (/) ~<_ ( M ` i ) |
84 |
|
domsdomtr |
|- ( ( (/) ~<_ ( M ` i ) /\ ( M ` i ) ~< ( N ` i ) ) -> (/) ~< ( N ` i ) ) |
85 |
83 84
|
mpan |
|- ( ( M ` i ) ~< ( N ` i ) -> (/) ~< ( N ` i ) ) |
86 |
22
|
0sdom |
|- ( (/) ~< ( N ` i ) <-> ( N ` i ) =/= (/) ) |
87 |
85 86
|
sylib |
|- ( ( M ` i ) ~< ( N ` i ) -> ( N ` i ) =/= (/) ) |
88 |
87
|
ralimi |
|- ( A. i e. A ( M ` i ) ~< ( N ` i ) -> A. i e. A ( N ` i ) =/= (/) ) |
89 |
3
|
neeq1i |
|- ( P =/= (/) <-> X_ i e. A ( N ` i ) =/= (/) ) |
90 |
22
|
rgenw |
|- A. i e. A ( N ` i ) e. _V |
91 |
|
ixpexg |
|- ( A. i e. A ( N ` i ) e. _V -> X_ i e. A ( N ` i ) e. _V ) |
92 |
90 91
|
ax-mp |
|- X_ i e. A ( N ` i ) e. _V |
93 |
3 92
|
eqeltri |
|- P e. _V |
94 |
93
|
0sdom |
|- ( (/) ~< P <-> P =/= (/) ) |
95 |
1 22
|
ac9 |
|- ( A. i e. A ( N ` i ) =/= (/) <-> X_ i e. A ( N ` i ) =/= (/) ) |
96 |
89 94 95
|
3bitr4i |
|- ( (/) ~< P <-> A. i e. A ( N ` i ) =/= (/) ) |
97 |
88 96
|
sylibr |
|- ( A. i e. A ( M ` i ) ~< ( N ` i ) -> (/) ~< P ) |
98 |
1 6
|
iunex |
|- U_ i e. A ( M ` i ) e. _V |
99 |
2 98
|
eqeltri |
|- S e. _V |
100 |
|
domtri |
|- ( ( P e. _V /\ S e. _V ) -> ( P ~<_ S <-> -. S ~< P ) ) |
101 |
93 99 100
|
mp2an |
|- ( P ~<_ S <-> -. S ~< P ) |
102 |
101
|
biimpri |
|- ( -. S ~< P -> P ~<_ S ) |
103 |
|
fodomr |
|- ( ( (/) ~< P /\ P ~<_ S ) -> E. f f : S -onto-> P ) |
104 |
97 102 103
|
syl2an |
|- ( ( A. i e. A ( M ` i ) ~< ( N ` i ) /\ -. S ~< P ) -> E. f f : S -onto-> P ) |
105 |
82 104
|
mtand |
|- ( A. i e. A ( M ` i ) ~< ( N ` i ) -> -. -. S ~< P ) |
106 |
105
|
notnotrd |
|- ( A. i e. A ( M ` i ) ~< ( N ` i ) -> S ~< P ) |