Step |
Hyp |
Ref |
Expression |
1 |
|
bnj1523.1 |
|- B = { d | ( d C_ A /\ A. x e. d _pred ( x , A , R ) C_ d ) } |
2 |
|
bnj1523.2 |
|- Y = <. x , ( f |` _pred ( x , A , R ) ) >. |
3 |
|
bnj1523.3 |
|- C = { f | E. d e. B ( f Fn d /\ A. x e. d ( f ` x ) = ( G ` Y ) ) } |
4 |
|
bnj1523.4 |
|- F = U. C |
5 |
|
bnj1523.5 |
|- ( ph <-> ( R _FrSe A /\ H Fn A /\ A. x e. A ( H ` x ) = ( G ` <. x , ( H |` _pred ( x , A , R ) ) >. ) ) ) |
6 |
|
bnj1523.6 |
|- ( ps <-> ( ph /\ F =/= H ) ) |
7 |
|
bnj1523.7 |
|- ( ch <-> ( ps /\ x e. A /\ ( F ` x ) =/= ( H ` x ) ) ) |
8 |
|
bnj1523.8 |
|- D = { x e. A | ( F ` x ) =/= ( H ` x ) } |
9 |
|
bnj1523.9 |
|- ( th <-> ( ch /\ y e. D /\ A. z e. D -. z R y ) ) |
10 |
1 2 3 4
|
bnj60 |
|- ( R _FrSe A -> F Fn A ) |
11 |
5 10
|
bnj835 |
|- ( ph -> F Fn A ) |
12 |
6 11
|
bnj832 |
|- ( ps -> F Fn A ) |
13 |
7 12
|
bnj835 |
|- ( ch -> F Fn A ) |
14 |
9 13
|
bnj835 |
|- ( th -> F Fn A ) |
15 |
5
|
simp2bi |
|- ( ph -> H Fn A ) |
16 |
6 15
|
bnj832 |
|- ( ps -> H Fn A ) |
17 |
7 16
|
bnj835 |
|- ( ch -> H Fn A ) |
18 |
9 17
|
bnj835 |
|- ( th -> H Fn A ) |
19 |
|
bnj213 |
|- _pred ( y , A , R ) C_ A |
20 |
19
|
a1i |
|- ( th -> _pred ( y , A , R ) C_ A ) |
21 |
9
|
simp3bi |
|- ( th -> A. z e. D -. z R y ) |
22 |
21
|
bnj1211 |
|- ( th -> A. z ( z e. D -> -. z R y ) ) |
23 |
|
con2b |
|- ( ( z e. D -> -. z R y ) <-> ( z R y -> -. z e. D ) ) |
24 |
23
|
albii |
|- ( A. z ( z e. D -> -. z R y ) <-> A. z ( z R y -> -. z e. D ) ) |
25 |
22 24
|
sylib |
|- ( th -> A. z ( z R y -> -. z e. D ) ) |
26 |
|
bnj1418 |
|- ( z e. _pred ( y , A , R ) -> z R y ) |
27 |
26
|
imim1i |
|- ( ( z R y -> -. z e. D ) -> ( z e. _pred ( y , A , R ) -> -. z e. D ) ) |
28 |
27
|
alimi |
|- ( A. z ( z R y -> -. z e. D ) -> A. z ( z e. _pred ( y , A , R ) -> -. z e. D ) ) |
29 |
25 28
|
syl |
|- ( th -> A. z ( z e. _pred ( y , A , R ) -> -. z e. D ) ) |
30 |
29
|
bnj1142 |
|- ( th -> A. z e. _pred ( y , A , R ) -. z e. D ) |
31 |
1
|
bnj1309 |
|- ( w e. B -> A. x w e. B ) |
32 |
3 31
|
bnj1307 |
|- ( w e. C -> A. x w e. C ) |
33 |
32
|
nfcii |
|- F/_ x C |
34 |
33
|
nfuni |
|- F/_ x U. C |
35 |
4 34
|
nfcxfr |
|- F/_ x F |
36 |
35
|
nfcrii |
|- ( w e. F -> A. x w e. F ) |
37 |
8 36
|
bnj1534 |
|- D = { z e. A | ( F ` z ) =/= ( H ` z ) } |
38 |
30 19 37
|
bnj1533 |
|- ( th -> A. z e. _pred ( y , A , R ) ( F ` z ) = ( H ` z ) ) |
39 |
14 18 20 38
|
bnj1536 |
|- ( th -> ( F |` _pred ( y , A , R ) ) = ( H |` _pred ( y , A , R ) ) ) |
40 |
39
|
opeq2d |
|- ( th -> <. y , ( F |` _pred ( y , A , R ) ) >. = <. y , ( H |` _pred ( y , A , R ) ) >. ) |
41 |
40
|
fveq2d |
|- ( th -> ( G ` <. y , ( F |` _pred ( y , A , R ) ) >. ) = ( G ` <. y , ( H |` _pred ( y , A , R ) ) >. ) ) |
42 |
1 2 3 4
|
bnj1500 |
|- ( R _FrSe A -> A. x e. A ( F ` x ) = ( G ` <. x , ( F |` _pred ( x , A , R ) ) >. ) ) |
43 |
5 42
|
bnj835 |
|- ( ph -> A. x e. A ( F ` x ) = ( G ` <. x , ( F |` _pred ( x , A , R ) ) >. ) ) |
44 |
6 43
|
bnj832 |
|- ( ps -> A. x e. A ( F ` x ) = ( G ` <. x , ( F |` _pred ( x , A , R ) ) >. ) ) |
45 |
7 44
|
bnj835 |
|- ( ch -> A. x e. A ( F ` x ) = ( G ` <. x , ( F |` _pred ( x , A , R ) ) >. ) ) |
46 |
45 36
|
bnj1529 |
|- ( ch -> A. y e. A ( F ` y ) = ( G ` <. y , ( F |` _pred ( y , A , R ) ) >. ) ) |
47 |
9 46
|
bnj835 |
|- ( th -> A. y e. A ( F ` y ) = ( G ` <. y , ( F |` _pred ( y , A , R ) ) >. ) ) |
48 |
8
|
ssrab3 |
|- D C_ A |
49 |
9
|
simp2bi |
|- ( th -> y e. D ) |
50 |
48 49
|
bnj1213 |
|- ( th -> y e. A ) |
51 |
47 50
|
bnj1294 |
|- ( th -> ( F ` y ) = ( G ` <. y , ( F |` _pred ( y , A , R ) ) >. ) ) |
52 |
5
|
simp3bi |
|- ( ph -> A. x e. A ( H ` x ) = ( G ` <. x , ( H |` _pred ( x , A , R ) ) >. ) ) |
53 |
6 52
|
bnj832 |
|- ( ps -> A. x e. A ( H ` x ) = ( G ` <. x , ( H |` _pred ( x , A , R ) ) >. ) ) |
54 |
7 53
|
bnj835 |
|- ( ch -> A. x e. A ( H ` x ) = ( G ` <. x , ( H |` _pred ( x , A , R ) ) >. ) ) |
55 |
|
ax-5 |
|- ( v e. H -> A. x v e. H ) |
56 |
54 55
|
bnj1529 |
|- ( ch -> A. y e. A ( H ` y ) = ( G ` <. y , ( H |` _pred ( y , A , R ) ) >. ) ) |
57 |
9 56
|
bnj835 |
|- ( th -> A. y e. A ( H ` y ) = ( G ` <. y , ( H |` _pred ( y , A , R ) ) >. ) ) |
58 |
57 50
|
bnj1294 |
|- ( th -> ( H ` y ) = ( G ` <. y , ( H |` _pred ( y , A , R ) ) >. ) ) |
59 |
41 51 58
|
3eqtr4d |
|- ( th -> ( F ` y ) = ( H ` y ) ) |
60 |
8 36
|
bnj1534 |
|- D = { y e. A | ( F ` y ) =/= ( H ` y ) } |
61 |
60
|
bnj1538 |
|- ( y e. D -> ( F ` y ) =/= ( H ` y ) ) |
62 |
9 61
|
bnj836 |
|- ( th -> ( F ` y ) =/= ( H ` y ) ) |
63 |
62
|
neneqd |
|- ( th -> -. ( F ` y ) = ( H ` y ) ) |
64 |
59 63
|
pm2.65i |
|- -. th |
65 |
64
|
nex |
|- -. E. y th |
66 |
5
|
simp1bi |
|- ( ph -> R _FrSe A ) |
67 |
6 66
|
bnj832 |
|- ( ps -> R _FrSe A ) |
68 |
7 67
|
bnj835 |
|- ( ch -> R _FrSe A ) |
69 |
48
|
a1i |
|- ( ch -> D C_ A ) |
70 |
7
|
simp2bi |
|- ( ch -> x e. A ) |
71 |
7
|
simp3bi |
|- ( ch -> ( F ` x ) =/= ( H ` x ) ) |
72 |
8
|
rabeq2i |
|- ( x e. D <-> ( x e. A /\ ( F ` x ) =/= ( H ` x ) ) ) |
73 |
70 71 72
|
sylanbrc |
|- ( ch -> x e. D ) |
74 |
73
|
ne0d |
|- ( ch -> D =/= (/) ) |
75 |
|
bnj69 |
|- ( ( R _FrSe A /\ D C_ A /\ D =/= (/) ) -> E. y e. D A. z e. D -. z R y ) |
76 |
68 69 74 75
|
syl3anc |
|- ( ch -> E. y e. D A. z e. D -. z R y ) |
77 |
76 9
|
bnj1209 |
|- ( ch -> E. y th ) |
78 |
65 77
|
mto |
|- -. ch |
79 |
78
|
nex |
|- -. E. x ch |
80 |
6
|
simprbi |
|- ( ps -> F =/= H ) |
81 |
12 16 80 36
|
bnj1542 |
|- ( ps -> E. x e. A ( F ` x ) =/= ( H ` x ) ) |
82 |
1 2 3 4 5 6
|
bnj1525 |
|- ( ps -> A. x ps ) |
83 |
81 7 82
|
bnj1521 |
|- ( ps -> E. x ch ) |
84 |
79 83
|
mto |
|- -. ps |
85 |
6 84
|
bnj1541 |
|- ( ph -> F = H ) |
86 |
5 85
|
sylbir |
|- ( ( R _FrSe A /\ H Fn A /\ A. x e. A ( H ` x ) = ( G ` <. x , ( H |` _pred ( x , A , R ) ) >. ) ) -> F = H ) |