Step |
Hyp |
Ref |
Expression |
1 |
|
cncfuni.acn |
|- ( ph -> A C_ CC ) |
2 |
|
cncfuni.f |
|- ( ph -> F : A --> CC ) |
3 |
|
cncfuni.auni |
|- ( ph -> A C_ U. B ) |
4 |
|
cncfuni.opn |
|- ( ( ph /\ b e. B ) -> ( A i^i b ) e. ( ( TopOpen ` CCfld ) |`t A ) ) |
5 |
|
cncfuni.fcn |
|- ( ( ph /\ b e. B ) -> ( F |` b ) e. ( ( A i^i b ) -cn-> CC ) ) |
6 |
3
|
sselda |
|- ( ( ph /\ x e. A ) -> x e. U. B ) |
7 |
|
eluni2 |
|- ( x e. U. B <-> E. b e. B x e. b ) |
8 |
6 7
|
sylib |
|- ( ( ph /\ x e. A ) -> E. b e. B x e. b ) |
9 |
|
simp1l |
|- ( ( ( ph /\ x e. A ) /\ b e. B /\ x e. b ) -> ph ) |
10 |
|
simp2 |
|- ( ( ( ph /\ x e. A ) /\ b e. B /\ x e. b ) -> b e. B ) |
11 |
|
elin |
|- ( x e. ( A i^i b ) <-> ( x e. A /\ x e. b ) ) |
12 |
11
|
biimpri |
|- ( ( x e. A /\ x e. b ) -> x e. ( A i^i b ) ) |
13 |
12
|
adantll |
|- ( ( ( ph /\ x e. A ) /\ x e. b ) -> x e. ( A i^i b ) ) |
14 |
13
|
3adant2 |
|- ( ( ( ph /\ x e. A ) /\ b e. B /\ x e. b ) -> x e. ( A i^i b ) ) |
15 |
2
|
fdmd |
|- ( ph -> dom F = A ) |
16 |
15
|
ineq2d |
|- ( ph -> ( b i^i dom F ) = ( b i^i A ) ) |
17 |
|
incom |
|- ( b i^i A ) = ( A i^i b ) |
18 |
16 17
|
eqtr2di |
|- ( ph -> ( A i^i b ) = ( b i^i dom F ) ) |
19 |
18
|
reseq2d |
|- ( ph -> ( F |` ( A i^i b ) ) = ( F |` ( b i^i dom F ) ) ) |
20 |
|
frel |
|- ( F : A --> CC -> Rel F ) |
21 |
2 20
|
syl |
|- ( ph -> Rel F ) |
22 |
|
resindm |
|- ( Rel F -> ( F |` ( b i^i dom F ) ) = ( F |` b ) ) |
23 |
21 22
|
syl |
|- ( ph -> ( F |` ( b i^i dom F ) ) = ( F |` b ) ) |
24 |
19 23
|
eqtrd |
|- ( ph -> ( F |` ( A i^i b ) ) = ( F |` b ) ) |
25 |
|
inss1 |
|- ( A i^i b ) C_ A |
26 |
25
|
a1i |
|- ( ph -> ( A i^i b ) C_ A ) |
27 |
26 1
|
sstrd |
|- ( ph -> ( A i^i b ) C_ CC ) |
28 |
|
ssidd |
|- ( ph -> CC C_ CC ) |
29 |
|
eqid |
|- ( TopOpen ` CCfld ) = ( TopOpen ` CCfld ) |
30 |
|
eqid |
|- ( ( TopOpen ` CCfld ) |`t ( A i^i b ) ) = ( ( TopOpen ` CCfld ) |`t ( A i^i b ) ) |
31 |
29
|
cnfldtop |
|- ( TopOpen ` CCfld ) e. Top |
32 |
|
unicntop |
|- CC = U. ( TopOpen ` CCfld ) |
33 |
32
|
restid |
|- ( ( TopOpen ` CCfld ) e. Top -> ( ( TopOpen ` CCfld ) |`t CC ) = ( TopOpen ` CCfld ) ) |
34 |
31 33
|
ax-mp |
|- ( ( TopOpen ` CCfld ) |`t CC ) = ( TopOpen ` CCfld ) |
35 |
34
|
eqcomi |
|- ( TopOpen ` CCfld ) = ( ( TopOpen ` CCfld ) |`t CC ) |
36 |
29 30 35
|
cncfcn |
|- ( ( ( A i^i b ) C_ CC /\ CC C_ CC ) -> ( ( A i^i b ) -cn-> CC ) = ( ( ( TopOpen ` CCfld ) |`t ( A i^i b ) ) Cn ( TopOpen ` CCfld ) ) ) |
37 |
27 28 36
|
syl2anc |
|- ( ph -> ( ( A i^i b ) -cn-> CC ) = ( ( ( TopOpen ` CCfld ) |`t ( A i^i b ) ) Cn ( TopOpen ` CCfld ) ) ) |
38 |
37
|
eqcomd |
|- ( ph -> ( ( ( TopOpen ` CCfld ) |`t ( A i^i b ) ) Cn ( TopOpen ` CCfld ) ) = ( ( A i^i b ) -cn-> CC ) ) |
39 |
24 38
|
eleq12d |
|- ( ph -> ( ( F |` ( A i^i b ) ) e. ( ( ( TopOpen ` CCfld ) |`t ( A i^i b ) ) Cn ( TopOpen ` CCfld ) ) <-> ( F |` b ) e. ( ( A i^i b ) -cn-> CC ) ) ) |
40 |
39
|
adantr |
|- ( ( ph /\ b e. B ) -> ( ( F |` ( A i^i b ) ) e. ( ( ( TopOpen ` CCfld ) |`t ( A i^i b ) ) Cn ( TopOpen ` CCfld ) ) <-> ( F |` b ) e. ( ( A i^i b ) -cn-> CC ) ) ) |
41 |
5 40
|
mpbird |
|- ( ( ph /\ b e. B ) -> ( F |` ( A i^i b ) ) e. ( ( ( TopOpen ` CCfld ) |`t ( A i^i b ) ) Cn ( TopOpen ` CCfld ) ) ) |
42 |
41
|
3adant3 |
|- ( ( ph /\ b e. B /\ x e. ( A i^i b ) ) -> ( F |` ( A i^i b ) ) e. ( ( ( TopOpen ` CCfld ) |`t ( A i^i b ) ) Cn ( TopOpen ` CCfld ) ) ) |
43 |
29
|
cnfldtopon |
|- ( TopOpen ` CCfld ) e. ( TopOn ` CC ) |
44 |
43
|
a1i |
|- ( ph -> ( TopOpen ` CCfld ) e. ( TopOn ` CC ) ) |
45 |
|
resttopon |
|- ( ( ( TopOpen ` CCfld ) e. ( TopOn ` CC ) /\ ( A i^i b ) C_ CC ) -> ( ( TopOpen ` CCfld ) |`t ( A i^i b ) ) e. ( TopOn ` ( A i^i b ) ) ) |
46 |
44 27 45
|
syl2anc |
|- ( ph -> ( ( TopOpen ` CCfld ) |`t ( A i^i b ) ) e. ( TopOn ` ( A i^i b ) ) ) |
47 |
46
|
3ad2ant1 |
|- ( ( ph /\ b e. B /\ x e. ( A i^i b ) ) -> ( ( TopOpen ` CCfld ) |`t ( A i^i b ) ) e. ( TopOn ` ( A i^i b ) ) ) |
48 |
43
|
a1i |
|- ( ( ph /\ b e. B /\ x e. ( A i^i b ) ) -> ( TopOpen ` CCfld ) e. ( TopOn ` CC ) ) |
49 |
|
cncnp |
|- ( ( ( ( TopOpen ` CCfld ) |`t ( A i^i b ) ) e. ( TopOn ` ( A i^i b ) ) /\ ( TopOpen ` CCfld ) e. ( TopOn ` CC ) ) -> ( ( F |` ( A i^i b ) ) e. ( ( ( TopOpen ` CCfld ) |`t ( A i^i b ) ) Cn ( TopOpen ` CCfld ) ) <-> ( ( F |` ( A i^i b ) ) : ( A i^i b ) --> CC /\ A. x e. ( A i^i b ) ( F |` ( A i^i b ) ) e. ( ( ( ( TopOpen ` CCfld ) |`t ( A i^i b ) ) CnP ( TopOpen ` CCfld ) ) ` x ) ) ) ) |
50 |
47 48 49
|
syl2anc |
|- ( ( ph /\ b e. B /\ x e. ( A i^i b ) ) -> ( ( F |` ( A i^i b ) ) e. ( ( ( TopOpen ` CCfld ) |`t ( A i^i b ) ) Cn ( TopOpen ` CCfld ) ) <-> ( ( F |` ( A i^i b ) ) : ( A i^i b ) --> CC /\ A. x e. ( A i^i b ) ( F |` ( A i^i b ) ) e. ( ( ( ( TopOpen ` CCfld ) |`t ( A i^i b ) ) CnP ( TopOpen ` CCfld ) ) ` x ) ) ) ) |
51 |
42 50
|
mpbid |
|- ( ( ph /\ b e. B /\ x e. ( A i^i b ) ) -> ( ( F |` ( A i^i b ) ) : ( A i^i b ) --> CC /\ A. x e. ( A i^i b ) ( F |` ( A i^i b ) ) e. ( ( ( ( TopOpen ` CCfld ) |`t ( A i^i b ) ) CnP ( TopOpen ` CCfld ) ) ` x ) ) ) |
52 |
51
|
simprd |
|- ( ( ph /\ b e. B /\ x e. ( A i^i b ) ) -> A. x e. ( A i^i b ) ( F |` ( A i^i b ) ) e. ( ( ( ( TopOpen ` CCfld ) |`t ( A i^i b ) ) CnP ( TopOpen ` CCfld ) ) ` x ) ) |
53 |
|
simp3 |
|- ( ( ph /\ b e. B /\ x e. ( A i^i b ) ) -> x e. ( A i^i b ) ) |
54 |
|
rspa |
|- ( ( A. x e. ( A i^i b ) ( F |` ( A i^i b ) ) e. ( ( ( ( TopOpen ` CCfld ) |`t ( A i^i b ) ) CnP ( TopOpen ` CCfld ) ) ` x ) /\ x e. ( A i^i b ) ) -> ( F |` ( A i^i b ) ) e. ( ( ( ( TopOpen ` CCfld ) |`t ( A i^i b ) ) CnP ( TopOpen ` CCfld ) ) ` x ) ) |
55 |
52 53 54
|
syl2anc |
|- ( ( ph /\ b e. B /\ x e. ( A i^i b ) ) -> ( F |` ( A i^i b ) ) e. ( ( ( ( TopOpen ` CCfld ) |`t ( A i^i b ) ) CnP ( TopOpen ` CCfld ) ) ` x ) ) |
56 |
31
|
a1i |
|- ( ph -> ( TopOpen ` CCfld ) e. Top ) |
57 |
|
cnex |
|- CC e. _V |
58 |
57
|
ssex |
|- ( A C_ CC -> A e. _V ) |
59 |
1 58
|
syl |
|- ( ph -> A e. _V ) |
60 |
|
restabs |
|- ( ( ( TopOpen ` CCfld ) e. Top /\ ( A i^i b ) C_ A /\ A e. _V ) -> ( ( ( TopOpen ` CCfld ) |`t A ) |`t ( A i^i b ) ) = ( ( TopOpen ` CCfld ) |`t ( A i^i b ) ) ) |
61 |
56 26 59 60
|
syl3anc |
|- ( ph -> ( ( ( TopOpen ` CCfld ) |`t A ) |`t ( A i^i b ) ) = ( ( TopOpen ` CCfld ) |`t ( A i^i b ) ) ) |
62 |
61
|
eqcomd |
|- ( ph -> ( ( TopOpen ` CCfld ) |`t ( A i^i b ) ) = ( ( ( TopOpen ` CCfld ) |`t A ) |`t ( A i^i b ) ) ) |
63 |
62
|
oveq1d |
|- ( ph -> ( ( ( TopOpen ` CCfld ) |`t ( A i^i b ) ) CnP ( TopOpen ` CCfld ) ) = ( ( ( ( TopOpen ` CCfld ) |`t A ) |`t ( A i^i b ) ) CnP ( TopOpen ` CCfld ) ) ) |
64 |
63
|
fveq1d |
|- ( ph -> ( ( ( ( TopOpen ` CCfld ) |`t ( A i^i b ) ) CnP ( TopOpen ` CCfld ) ) ` x ) = ( ( ( ( ( TopOpen ` CCfld ) |`t A ) |`t ( A i^i b ) ) CnP ( TopOpen ` CCfld ) ) ` x ) ) |
65 |
64
|
3ad2ant1 |
|- ( ( ph /\ b e. B /\ x e. ( A i^i b ) ) -> ( ( ( ( TopOpen ` CCfld ) |`t ( A i^i b ) ) CnP ( TopOpen ` CCfld ) ) ` x ) = ( ( ( ( ( TopOpen ` CCfld ) |`t A ) |`t ( A i^i b ) ) CnP ( TopOpen ` CCfld ) ) ` x ) ) |
66 |
55 65
|
eleqtrd |
|- ( ( ph /\ b e. B /\ x e. ( A i^i b ) ) -> ( F |` ( A i^i b ) ) e. ( ( ( ( ( TopOpen ` CCfld ) |`t A ) |`t ( A i^i b ) ) CnP ( TopOpen ` CCfld ) ) ` x ) ) |
67 |
|
resttop |
|- ( ( ( TopOpen ` CCfld ) e. Top /\ A e. _V ) -> ( ( TopOpen ` CCfld ) |`t A ) e. Top ) |
68 |
56 59 67
|
syl2anc |
|- ( ph -> ( ( TopOpen ` CCfld ) |`t A ) e. Top ) |
69 |
68
|
3ad2ant1 |
|- ( ( ph /\ b e. B /\ x e. ( A i^i b ) ) -> ( ( TopOpen ` CCfld ) |`t A ) e. Top ) |
70 |
32
|
restuni |
|- ( ( ( TopOpen ` CCfld ) e. Top /\ A C_ CC ) -> A = U. ( ( TopOpen ` CCfld ) |`t A ) ) |
71 |
56 1 70
|
syl2anc |
|- ( ph -> A = U. ( ( TopOpen ` CCfld ) |`t A ) ) |
72 |
26 71
|
sseqtrd |
|- ( ph -> ( A i^i b ) C_ U. ( ( TopOpen ` CCfld ) |`t A ) ) |
73 |
72
|
3ad2ant1 |
|- ( ( ph /\ b e. B /\ x e. ( A i^i b ) ) -> ( A i^i b ) C_ U. ( ( TopOpen ` CCfld ) |`t A ) ) |
74 |
4
|
3adant3 |
|- ( ( ph /\ b e. B /\ x e. ( A i^i b ) ) -> ( A i^i b ) e. ( ( TopOpen ` CCfld ) |`t A ) ) |
75 |
|
eqid |
|- U. ( ( TopOpen ` CCfld ) |`t A ) = U. ( ( TopOpen ` CCfld ) |`t A ) |
76 |
75
|
isopn3 |
|- ( ( ( ( TopOpen ` CCfld ) |`t A ) e. Top /\ ( A i^i b ) C_ U. ( ( TopOpen ` CCfld ) |`t A ) ) -> ( ( A i^i b ) e. ( ( TopOpen ` CCfld ) |`t A ) <-> ( ( int ` ( ( TopOpen ` CCfld ) |`t A ) ) ` ( A i^i b ) ) = ( A i^i b ) ) ) |
77 |
69 73 76
|
syl2anc |
|- ( ( ph /\ b e. B /\ x e. ( A i^i b ) ) -> ( ( A i^i b ) e. ( ( TopOpen ` CCfld ) |`t A ) <-> ( ( int ` ( ( TopOpen ` CCfld ) |`t A ) ) ` ( A i^i b ) ) = ( A i^i b ) ) ) |
78 |
74 77
|
mpbid |
|- ( ( ph /\ b e. B /\ x e. ( A i^i b ) ) -> ( ( int ` ( ( TopOpen ` CCfld ) |`t A ) ) ` ( A i^i b ) ) = ( A i^i b ) ) |
79 |
78
|
eqcomd |
|- ( ( ph /\ b e. B /\ x e. ( A i^i b ) ) -> ( A i^i b ) = ( ( int ` ( ( TopOpen ` CCfld ) |`t A ) ) ` ( A i^i b ) ) ) |
80 |
53 79
|
eleqtrd |
|- ( ( ph /\ b e. B /\ x e. ( A i^i b ) ) -> x e. ( ( int ` ( ( TopOpen ` CCfld ) |`t A ) ) ` ( A i^i b ) ) ) |
81 |
71
|
feq2d |
|- ( ph -> ( F : A --> CC <-> F : U. ( ( TopOpen ` CCfld ) |`t A ) --> CC ) ) |
82 |
2 81
|
mpbid |
|- ( ph -> F : U. ( ( TopOpen ` CCfld ) |`t A ) --> CC ) |
83 |
82
|
3ad2ant1 |
|- ( ( ph /\ b e. B /\ x e. ( A i^i b ) ) -> F : U. ( ( TopOpen ` CCfld ) |`t A ) --> CC ) |
84 |
75 32
|
cnprest |
|- ( ( ( ( ( TopOpen ` CCfld ) |`t A ) e. Top /\ ( A i^i b ) C_ U. ( ( TopOpen ` CCfld ) |`t A ) ) /\ ( x e. ( ( int ` ( ( TopOpen ` CCfld ) |`t A ) ) ` ( A i^i b ) ) /\ F : U. ( ( TopOpen ` CCfld ) |`t A ) --> CC ) ) -> ( F e. ( ( ( ( TopOpen ` CCfld ) |`t A ) CnP ( TopOpen ` CCfld ) ) ` x ) <-> ( F |` ( A i^i b ) ) e. ( ( ( ( ( TopOpen ` CCfld ) |`t A ) |`t ( A i^i b ) ) CnP ( TopOpen ` CCfld ) ) ` x ) ) ) |
85 |
69 73 80 83 84
|
syl22anc |
|- ( ( ph /\ b e. B /\ x e. ( A i^i b ) ) -> ( F e. ( ( ( ( TopOpen ` CCfld ) |`t A ) CnP ( TopOpen ` CCfld ) ) ` x ) <-> ( F |` ( A i^i b ) ) e. ( ( ( ( ( TopOpen ` CCfld ) |`t A ) |`t ( A i^i b ) ) CnP ( TopOpen ` CCfld ) ) ` x ) ) ) |
86 |
66 85
|
mpbird |
|- ( ( ph /\ b e. B /\ x e. ( A i^i b ) ) -> F e. ( ( ( ( TopOpen ` CCfld ) |`t A ) CnP ( TopOpen ` CCfld ) ) ` x ) ) |
87 |
9 10 14 86
|
syl3anc |
|- ( ( ( ph /\ x e. A ) /\ b e. B /\ x e. b ) -> F e. ( ( ( ( TopOpen ` CCfld ) |`t A ) CnP ( TopOpen ` CCfld ) ) ` x ) ) |
88 |
87
|
rexlimdv3a |
|- ( ( ph /\ x e. A ) -> ( E. b e. B x e. b -> F e. ( ( ( ( TopOpen ` CCfld ) |`t A ) CnP ( TopOpen ` CCfld ) ) ` x ) ) ) |
89 |
8 88
|
mpd |
|- ( ( ph /\ x e. A ) -> F e. ( ( ( ( TopOpen ` CCfld ) |`t A ) CnP ( TopOpen ` CCfld ) ) ` x ) ) |
90 |
89
|
ralrimiva |
|- ( ph -> A. x e. A F e. ( ( ( ( TopOpen ` CCfld ) |`t A ) CnP ( TopOpen ` CCfld ) ) ` x ) ) |
91 |
|
resttopon |
|- ( ( ( TopOpen ` CCfld ) e. ( TopOn ` CC ) /\ A C_ CC ) -> ( ( TopOpen ` CCfld ) |`t A ) e. ( TopOn ` A ) ) |
92 |
44 1 91
|
syl2anc |
|- ( ph -> ( ( TopOpen ` CCfld ) |`t A ) e. ( TopOn ` A ) ) |
93 |
|
cncnp |
|- ( ( ( ( TopOpen ` CCfld ) |`t A ) e. ( TopOn ` A ) /\ ( TopOpen ` CCfld ) e. ( TopOn ` CC ) ) -> ( F e. ( ( ( TopOpen ` CCfld ) |`t A ) Cn ( TopOpen ` CCfld ) ) <-> ( F : A --> CC /\ A. x e. A F e. ( ( ( ( TopOpen ` CCfld ) |`t A ) CnP ( TopOpen ` CCfld ) ) ` x ) ) ) ) |
94 |
92 44 93
|
syl2anc |
|- ( ph -> ( F e. ( ( ( TopOpen ` CCfld ) |`t A ) Cn ( TopOpen ` CCfld ) ) <-> ( F : A --> CC /\ A. x e. A F e. ( ( ( ( TopOpen ` CCfld ) |`t A ) CnP ( TopOpen ` CCfld ) ) ` x ) ) ) ) |
95 |
2 90 94
|
mpbir2and |
|- ( ph -> F e. ( ( ( TopOpen ` CCfld ) |`t A ) Cn ( TopOpen ` CCfld ) ) ) |
96 |
|
eqid |
|- ( ( TopOpen ` CCfld ) |`t A ) = ( ( TopOpen ` CCfld ) |`t A ) |
97 |
29 96 35
|
cncfcn |
|- ( ( A C_ CC /\ CC C_ CC ) -> ( A -cn-> CC ) = ( ( ( TopOpen ` CCfld ) |`t A ) Cn ( TopOpen ` CCfld ) ) ) |
98 |
1 28 97
|
syl2anc |
|- ( ph -> ( A -cn-> CC ) = ( ( ( TopOpen ` CCfld ) |`t A ) Cn ( TopOpen ` CCfld ) ) ) |
99 |
98
|
eqcomd |
|- ( ph -> ( ( ( TopOpen ` CCfld ) |`t A ) Cn ( TopOpen ` CCfld ) ) = ( A -cn-> CC ) ) |
100 |
95 99
|
eleqtrd |
|- ( ph -> F e. ( A -cn-> CC ) ) |