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Theorem utopsnneiplem 20478
Description: The neighborhoods of a point  P for the topology induced by an uniform space  U. (Contributed by Thierry Arnoux, 11-Jan-2018.)
Hypotheses
Ref Expression
utoptop.1  |-  J  =  (unifTop `  U )
utopsnneip.1  |-  K  =  { a  e.  ~P X  |  A. p  e.  a  a  e.  ( N `  p ) }
utopsnneip.2  |-  N  =  ( p  e.  X  |->  ran  ( v  e.  U  |->  ( v " { p } ) ) )
Assertion
Ref Expression
utopsnneiplem  |-  ( ( U  e.  (UnifOn `  X )  /\  P  e.  X )  ->  (
( nei `  J
) `  { P } )  =  ran  ( v  e.  U  |->  ( v " { P } ) ) )
Distinct variable groups:    p, a, K    N, a, p    v, p, P    v, a, U, p    X, a, p, v
Allowed substitution hints:    P( a)    J( v, p, a)    K( v)    N( v)

Proof of Theorem utopsnneiplem
Dummy variables  b 
q  u  w are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 utoptop.1 . . . . . . . 8  |-  J  =  (unifTop `  U )
2 utopval 20463 . . . . . . . 8  |-  ( U  e.  (UnifOn `  X
)  ->  (unifTop `  U
)  =  { a  e.  ~P X  |  A. p  e.  a  E. w  e.  U  ( w " {
p } )  C_  a } )
31, 2syl5eq 2513 . . . . . . 7  |-  ( U  e.  (UnifOn `  X
)  ->  J  =  { a  e.  ~P X  |  A. p  e.  a  E. w  e.  U  ( w " { p } ) 
C_  a } )
4 simpll 753 . . . . . . . . . . 11  |-  ( ( ( U  e.  (UnifOn `  X )  /\  a  e.  ~P X )  /\  p  e.  a )  ->  U  e.  (UnifOn `  X ) )
5 simpr 461 . . . . . . . . . . . . 13  |-  ( ( U  e.  (UnifOn `  X )  /\  a  e.  ~P X )  -> 
a  e.  ~P X
)
65elpwid 4013 . . . . . . . . . . . 12  |-  ( ( U  e.  (UnifOn `  X )  /\  a  e.  ~P X )  -> 
a  C_  X )
76sselda 3497 . . . . . . . . . . 11  |-  ( ( ( U  e.  (UnifOn `  X )  /\  a  e.  ~P X )  /\  p  e.  a )  ->  p  e.  X )
8 simpr 461 . . . . . . . . . . . . . 14  |-  ( ( U  e.  (UnifOn `  X )  /\  p  e.  X )  ->  p  e.  X )
9 mptexg 6121 . . . . . . . . . . . . . . . 16  |-  ( U  e.  (UnifOn `  X
)  ->  ( v  e.  U  |->  ( v
" { p }
) )  e.  _V )
10 rnexg 6706 . . . . . . . . . . . . . . . 16  |-  ( ( v  e.  U  |->  ( v " { p } ) )  e. 
_V  ->  ran  ( v  e.  U  |->  ( v
" { p }
) )  e.  _V )
119, 10syl 16 . . . . . . . . . . . . . . 15  |-  ( U  e.  (UnifOn `  X
)  ->  ran  ( v  e.  U  |->  ( v
" { p }
) )  e.  _V )
1211adantr 465 . . . . . . . . . . . . . 14  |-  ( ( U  e.  (UnifOn `  X )  /\  p  e.  X )  ->  ran  ( v  e.  U  |->  ( v " {
p } ) )  e.  _V )
13 utopsnneip.2 . . . . . . . . . . . . . . 15  |-  N  =  ( p  e.  X  |->  ran  ( v  e.  U  |->  ( v " { p } ) ) )
1413fvmpt2 5948 . . . . . . . . . . . . . 14  |-  ( ( p  e.  X  /\  ran  ( v  e.  U  |->  ( v " {
p } ) )  e.  _V )  -> 
( N `  p
)  =  ran  (
v  e.  U  |->  ( v " { p } ) ) )
158, 12, 14syl2anc 661 . . . . . . . . . . . . 13  |-  ( ( U  e.  (UnifOn `  X )  /\  p  e.  X )  ->  ( N `  p )  =  ran  ( v  e.  U  |->  ( v " { p } ) ) )
1615eleq2d 2530 . . . . . . . . . . . 12  |-  ( ( U  e.  (UnifOn `  X )  /\  p  e.  X )  ->  (
a  e.  ( N `
 p )  <->  a  e.  ran  ( v  e.  U  |->  ( v " {
p } ) ) ) )
17 vex 3109 . . . . . . . . . . . . 13  |-  a  e. 
_V
18 eqid 2460 . . . . . . . . . . . . . 14  |-  ( v  e.  U  |->  ( v
" { p }
) )  =  ( v  e.  U  |->  ( v " { p } ) )
1918elrnmpt 5240 . . . . . . . . . . . . 13  |-  ( a  e.  _V  ->  (
a  e.  ran  (
v  e.  U  |->  ( v " { p } ) )  <->  E. v  e.  U  a  =  ( v " {
p } ) ) )
2017, 19ax-mp 5 . . . . . . . . . . . 12  |-  ( a  e.  ran  ( v  e.  U  |->  ( v
" { p }
) )  <->  E. v  e.  U  a  =  ( v " {
p } ) )
2116, 20syl6bb 261 . . . . . . . . . . 11  |-  ( ( U  e.  (UnifOn `  X )  /\  p  e.  X )  ->  (
a  e.  ( N `
 p )  <->  E. v  e.  U  a  =  ( v " {
p } ) ) )
224, 7, 21syl2anc 661 . . . . . . . . . 10  |-  ( ( ( U  e.  (UnifOn `  X )  /\  a  e.  ~P X )  /\  p  e.  a )  ->  ( a  e.  ( N `  p )  <->  E. v  e.  U  a  =  ( v " { p } ) ) )
23 nfv 1678 . . . . . . . . . . . . 13  |-  F/ v ( ( U  e.  (UnifOn `  X )  /\  a  e.  ~P X )  /\  p  e.  a )
24 nfre1 2918 . . . . . . . . . . . . 13  |-  F/ v E. v  e.  U  a  =  ( v " { p } )
2523, 24nfan 1870 . . . . . . . . . . . 12  |-  F/ v ( ( ( U  e.  (UnifOn `  X
)  /\  a  e.  ~P X )  /\  p  e.  a )  /\  E. v  e.  U  a  =  ( v " { p } ) )
26 simplr 754 . . . . . . . . . . . . 13  |-  ( ( ( ( ( ( U  e.  (UnifOn `  X )  /\  a  e.  ~P X )  /\  p  e.  a )  /\  E. v  e.  U  a  =  ( v " { p } ) )  /\  v  e.  U )  /\  a  =  ( v " { p } ) )  ->  v  e.  U )
27 eqimss2 3550 . . . . . . . . . . . . . 14  |-  ( a  =  ( v " { p } )  ->  ( v " { p } ) 
C_  a )
2827adantl 466 . . . . . . . . . . . . 13  |-  ( ( ( ( ( ( U  e.  (UnifOn `  X )  /\  a  e.  ~P X )  /\  p  e.  a )  /\  E. v  e.  U  a  =  ( v " { p } ) )  /\  v  e.  U )  /\  a  =  ( v " { p } ) )  ->  ( v " { p } ) 
C_  a )
29 imaeq1 5323 . . . . . . . . . . . . . . 15  |-  ( w  =  v  ->  (
w " { p } )  =  ( v " { p } ) )
3029sseq1d 3524 . . . . . . . . . . . . . 14  |-  ( w  =  v  ->  (
( w " {
p } )  C_  a 
<->  ( v " {
p } )  C_  a ) )
3130rspcev 3207 . . . . . . . . . . . . 13  |-  ( ( v  e.  U  /\  ( v " {
p } )  C_  a )  ->  E. w  e.  U  ( w " { p } ) 
C_  a )
3226, 28, 31syl2anc 661 . . . . . . . . . . . 12  |-  ( ( ( ( ( ( U  e.  (UnifOn `  X )  /\  a  e.  ~P X )  /\  p  e.  a )  /\  E. v  e.  U  a  =  ( v " { p } ) )  /\  v  e.  U )  /\  a  =  ( v " { p } ) )  ->  E. w  e.  U  ( w " { p } ) 
C_  a )
33 simpr 461 . . . . . . . . . . . 12  |-  ( ( ( ( U  e.  (UnifOn `  X )  /\  a  e.  ~P X )  /\  p  e.  a )  /\  E. v  e.  U  a  =  ( v " { p } ) )  ->  E. v  e.  U  a  =  ( v " {
p } ) )
3425, 32, 33r19.29af 2994 . . . . . . . . . . 11  |-  ( ( ( ( U  e.  (UnifOn `  X )  /\  a  e.  ~P X )  /\  p  e.  a )  /\  E. v  e.  U  a  =  ( v " { p } ) )  ->  E. w  e.  U  ( w " { p } ) 
C_  a )
35 nfv 1678 . . . . . . . . . . . . 13  |-  F/ w
( ( U  e.  (UnifOn `  X )  /\  a  e.  ~P X )  /\  p  e.  a )
36 nfre1 2918 . . . . . . . . . . . . 13  |-  F/ w E. w  e.  U  ( w " {
p } )  C_  a
3735, 36nfan 1870 . . . . . . . . . . . 12  |-  F/ w
( ( ( U  e.  (UnifOn `  X
)  /\  a  e.  ~P X )  /\  p  e.  a )  /\  E. w  e.  U  (
w " { p } )  C_  a
)
384ad2antrr 725 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ( U  e.  (UnifOn `  X
)  /\  a  e.  ~P X )  /\  p  e.  a )  /\  w  e.  U )  /\  (
w " { p } )  C_  a
)  ->  U  e.  (UnifOn `  X ) )
397ad2antrr 725 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ( U  e.  (UnifOn `  X
)  /\  a  e.  ~P X )  /\  p  e.  a )  /\  w  e.  U )  /\  (
w " { p } )  C_  a
)  ->  p  e.  X )
4038, 39jca 532 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( U  e.  (UnifOn `  X
)  /\  a  e.  ~P X )  /\  p  e.  a )  /\  w  e.  U )  /\  (
w " { p } )  C_  a
)  ->  ( U  e.  (UnifOn `  X )  /\  p  e.  X
) )
41 simpr 461 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( U  e.  (UnifOn `  X
)  /\  a  e.  ~P X )  /\  p  e.  a )  /\  w  e.  U )  /\  (
w " { p } )  C_  a
)  ->  ( w " { p } ) 
C_  a )
426ad3antrrr 729 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( U  e.  (UnifOn `  X
)  /\  a  e.  ~P X )  /\  p  e.  a )  /\  w  e.  U )  /\  (
w " { p } )  C_  a
)  ->  a  C_  X )
43 simplr 754 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ( U  e.  (UnifOn `  X
)  /\  a  e.  ~P X )  /\  p  e.  a )  /\  w  e.  U )  /\  (
w " { p } )  C_  a
)  ->  w  e.  U )
44 eqid 2460 . . . . . . . . . . . . . . . . . . 19  |-  ( w
" { p }
)  =  ( w
" { p }
)
45 imaeq1 5323 . . . . . . . . . . . . . . . . . . . . 21  |-  ( u  =  w  ->  (
u " { p } )  =  ( w " { p } ) )
4645eqeq2d 2474 . . . . . . . . . . . . . . . . . . . 20  |-  ( u  =  w  ->  (
( w " {
p } )  =  ( u " {
p } )  <->  ( w " { p } )  =  ( w " { p } ) ) )
4746rspcev 3207 . . . . . . . . . . . . . . . . . . 19  |-  ( ( w  e.  U  /\  ( w " {
p } )  =  ( w " {
p } ) )  ->  E. u  e.  U  ( w " {
p } )  =  ( u " {
p } ) )
4844, 47mpan2 671 . . . . . . . . . . . . . . . . . 18  |-  ( w  e.  U  ->  E. u  e.  U  ( w " { p } )  =  ( u " { p } ) )
4948adantl 466 . . . . . . . . . . . . . . . . 17  |-  ( ( ( U  e.  (UnifOn `  X )  /\  p  e.  X )  /\  w  e.  U )  ->  E. u  e.  U  ( w " { p } )  =  ( u " { p } ) )
50 vex 3109 . . . . . . . . . . . . . . . . . . . 20  |-  w  e. 
_V
51 imaexg 6711 . . . . . . . . . . . . . . . . . . . 20  |-  ( w  e.  _V  ->  (
w " { p } )  e.  _V )
5250, 51ax-mp 5 . . . . . . . . . . . . . . . . . . 19  |-  ( w
" { p }
)  e.  _V
5313ustuqtoplem 20470 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( U  e.  (UnifOn `  X )  /\  p  e.  X )  /\  (
w " { p } )  e.  _V )  ->  ( ( w
" { p }
)  e.  ( N `
 p )  <->  E. u  e.  U  ( w " { p } )  =  ( u " { p } ) ) )
5452, 53mpan2 671 . . . . . . . . . . . . . . . . . 18  |-  ( ( U  e.  (UnifOn `  X )  /\  p  e.  X )  ->  (
( w " {
p } )  e.  ( N `  p
)  <->  E. u  e.  U  ( w " {
p } )  =  ( u " {
p } ) ) )
5554adantr 465 . . . . . . . . . . . . . . . . 17  |-  ( ( ( U  e.  (UnifOn `  X )  /\  p  e.  X )  /\  w  e.  U )  ->  (
( w " {
p } )  e.  ( N `  p
)  <->  E. u  e.  U  ( w " {
p } )  =  ( u " {
p } ) ) )
5649, 55mpbird 232 . . . . . . . . . . . . . . . 16  |-  ( ( ( U  e.  (UnifOn `  X )  /\  p  e.  X )  /\  w  e.  U )  ->  (
w " { p } )  e.  ( N `  p ) )
5738, 39, 43, 56syl21anc 1222 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( U  e.  (UnifOn `  X
)  /\  a  e.  ~P X )  /\  p  e.  a )  /\  w  e.  U )  /\  (
w " { p } )  C_  a
)  ->  ( w " { p } )  e.  ( N `  p ) )
58 sseq1 3518 . . . . . . . . . . . . . . . . . . . 20  |-  ( b  =  ( w " { p } )  ->  ( b  C_  a 
<->  ( w " {
p } )  C_  a ) )
59583anbi2d 1299 . . . . . . . . . . . . . . . . . . 19  |-  ( b  =  ( w " { p } )  ->  ( ( ( U  e.  (UnifOn `  X )  /\  p  e.  X )  /\  b  C_  a  /\  a  C_  X )  <->  ( ( U  e.  (UnifOn `  X
)  /\  p  e.  X )  /\  (
w " { p } )  C_  a  /\  a  C_  X ) ) )
60 eleq1 2532 . . . . . . . . . . . . . . . . . . 19  |-  ( b  =  ( w " { p } )  ->  ( b  e.  ( N `  p
)  <->  ( w " { p } )  e.  ( N `  p ) ) )
6159, 60anbi12d 710 . . . . . . . . . . . . . . . . . 18  |-  ( b  =  ( w " { p } )  ->  ( ( ( ( U  e.  (UnifOn `  X )  /\  p  e.  X )  /\  b  C_  a  /\  a  C_  X )  /\  b  e.  ( N `  p
) )  <->  ( (
( U  e.  (UnifOn `  X )  /\  p  e.  X )  /\  (
w " { p } )  C_  a  /\  a  C_  X )  /\  ( w " { p } )  e.  ( N `  p ) ) ) )
6261imbi1d 317 . . . . . . . . . . . . . . . . 17  |-  ( b  =  ( w " { p } )  ->  ( ( ( ( ( U  e.  (UnifOn `  X )  /\  p  e.  X
)  /\  b  C_  a  /\  a  C_  X
)  /\  b  e.  ( N `  p ) )  ->  a  e.  ( N `  p ) )  <->  ( ( ( ( U  e.  (UnifOn `  X )  /\  p  e.  X )  /\  (
w " { p } )  C_  a  /\  a  C_  X )  /\  ( w " { p } )  e.  ( N `  p ) )  -> 
a  e.  ( N `
 p ) ) ) )
6313ustuqtop1 20472 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( U  e.  (UnifOn `  X )  /\  p  e.  X
)  /\  b  C_  a  /\  a  C_  X
)  /\  b  e.  ( N `  p ) )  ->  a  e.  ( N `  p ) )
6462, 63vtoclg 3164 . . . . . . . . . . . . . . . 16  |-  ( ( w " { p } )  e.  _V  ->  ( ( ( ( U  e.  (UnifOn `  X )  /\  p  e.  X )  /\  (
w " { p } )  C_  a  /\  a  C_  X )  /\  ( w " { p } )  e.  ( N `  p ) )  -> 
a  e.  ( N `
 p ) ) )
6550, 51, 64mp2b 10 . . . . . . . . . . . . . . 15  |-  ( ( ( ( U  e.  (UnifOn `  X )  /\  p  e.  X
)  /\  ( w " { p } ) 
C_  a  /\  a  C_  X )  /\  (
w " { p } )  e.  ( N `  p ) )  ->  a  e.  ( N `  p ) )
6640, 41, 42, 57, 65syl31anc 1226 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( U  e.  (UnifOn `  X
)  /\  a  e.  ~P X )  /\  p  e.  a )  /\  w  e.  U )  /\  (
w " { p } )  C_  a
)  ->  a  e.  ( N `  p ) )
6740, 21syl 16 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( U  e.  (UnifOn `  X
)  /\  a  e.  ~P X )  /\  p  e.  a )  /\  w  e.  U )  /\  (
w " { p } )  C_  a
)  ->  ( a  e.  ( N `  p
)  <->  E. v  e.  U  a  =  ( v " { p } ) ) )
6866, 67mpbid 210 . . . . . . . . . . . . 13  |-  ( ( ( ( ( U  e.  (UnifOn `  X
)  /\  a  e.  ~P X )  /\  p  e.  a )  /\  w  e.  U )  /\  (
w " { p } )  C_  a
)  ->  E. v  e.  U  a  =  ( v " {
p } ) )
6968adantllr 718 . . . . . . . . . . . 12  |-  ( ( ( ( ( ( U  e.  (UnifOn `  X )  /\  a  e.  ~P X )  /\  p  e.  a )  /\  E. w  e.  U  ( w " {
p } )  C_  a )  /\  w  e.  U )  /\  (
w " { p } )  C_  a
)  ->  E. v  e.  U  a  =  ( v " {
p } ) )
70 simpr 461 . . . . . . . . . . . 12  |-  ( ( ( ( U  e.  (UnifOn `  X )  /\  a  e.  ~P X )  /\  p  e.  a )  /\  E. w  e.  U  (
w " { p } )  C_  a
)  ->  E. w  e.  U  ( w " { p } ) 
C_  a )
7137, 69, 70r19.29af 2994 . . . . . . . . . . 11  |-  ( ( ( ( U  e.  (UnifOn `  X )  /\  a  e.  ~P X )  /\  p  e.  a )  /\  E. w  e.  U  (
w " { p } )  C_  a
)  ->  E. v  e.  U  a  =  ( v " {
p } ) )
7234, 71impbida 829 . . . . . . . . . 10  |-  ( ( ( U  e.  (UnifOn `  X )  /\  a  e.  ~P X )  /\  p  e.  a )  ->  ( E. v  e.  U  a  =  ( v " { p } )  <->  E. w  e.  U  ( w " { p } ) 
C_  a ) )
7322, 72bitrd 253 . . . . . . . . 9  |-  ( ( ( U  e.  (UnifOn `  X )  /\  a  e.  ~P X )  /\  p  e.  a )  ->  ( a  e.  ( N `  p )  <->  E. w  e.  U  ( w " {
p } )  C_  a ) )
7473ralbidva 2893 . . . . . . . 8  |-  ( ( U  e.  (UnifOn `  X )  /\  a  e.  ~P X )  -> 
( A. p  e.  a  a  e.  ( N `  p )  <->  A. p  e.  a  E. w  e.  U  ( w " {
p } )  C_  a ) )
7574rabbidva 3097 . . . . . . 7  |-  ( U  e.  (UnifOn `  X
)  ->  { a  e.  ~P X  |  A. p  e.  a  a  e.  ( N `  p
) }  =  {
a  e.  ~P X  |  A. p  e.  a  E. w  e.  U  ( w " {
p } )  C_  a } )
763, 75eqtr4d 2504 . . . . . 6  |-  ( U  e.  (UnifOn `  X
)  ->  J  =  { a  e.  ~P X  |  A. p  e.  a  a  e.  ( N `  p ) } )
77 utopsnneip.1 . . . . . 6  |-  K  =  { a  e.  ~P X  |  A. p  e.  a  a  e.  ( N `  p ) }
7876, 77syl6eqr 2519 . . . . 5  |-  ( U  e.  (UnifOn `  X
)  ->  J  =  K )
7978fveq2d 5861 . . . 4  |-  ( U  e.  (UnifOn `  X
)  ->  ( nei `  J )  =  ( nei `  K ) )
8079fveq1d 5859 . . 3  |-  ( U  e.  (UnifOn `  X
)  ->  ( ( nei `  J ) `  { P } )  =  ( ( nei `  K
) `  { P } ) )
8180adantr 465 . 2  |-  ( ( U  e.  (UnifOn `  X )  /\  P  e.  X )  ->  (
( nei `  J
) `  { P } )  =  ( ( nei `  K
) `  { P } ) )
8213ustuqtop0 20471 . . . . 5  |-  ( U  e.  (UnifOn `  X
)  ->  N : X
--> ~P ~P X )
8313ustuqtop1 20472 . . . . 5  |-  ( ( ( ( U  e.  (UnifOn `  X )  /\  p  e.  X
)  /\  a  C_  b  /\  b  C_  X
)  /\  a  e.  ( N `  p ) )  ->  b  e.  ( N `  p ) )
8413ustuqtop2 20473 . . . . 5  |-  ( ( U  e.  (UnifOn `  X )  /\  p  e.  X )  ->  ( fi `  ( N `  p ) )  C_  ( N `  p ) )
8513ustuqtop3 20474 . . . . 5  |-  ( ( ( U  e.  (UnifOn `  X )  /\  p  e.  X )  /\  a  e.  ( N `  p
) )  ->  p  e.  a )
8613ustuqtop4 20475 . . . . 5  |-  ( ( ( U  e.  (UnifOn `  X )  /\  p  e.  X )  /\  a  e.  ( N `  p
) )  ->  E. b  e.  ( N `  p
) A. q  e.  b  a  e.  ( N `  q ) )
8713ustuqtop5 20476 . . . . 5  |-  ( ( U  e.  (UnifOn `  X )  /\  p  e.  X )  ->  X  e.  ( N `  p
) )
8877, 82, 83, 84, 85, 86, 87neiptopnei 19392 . . . 4  |-  ( U  e.  (UnifOn `  X
)  ->  N  =  ( p  e.  X  |->  ( ( nei `  K
) `  { p } ) ) )
8988adantr 465 . . 3  |-  ( ( U  e.  (UnifOn `  X )  /\  P  e.  X )  ->  N  =  ( p  e.  X  |->  ( ( nei `  K ) `  {
p } ) ) )
90 simpr 461 . . . . 5  |-  ( ( ( U  e.  (UnifOn `  X )  /\  P  e.  X )  /\  p  =  P )  ->  p  =  P )
9190sneqd 4032 . . . 4  |-  ( ( ( U  e.  (UnifOn `  X )  /\  P  e.  X )  /\  p  =  P )  ->  { p }  =  { P } )
9291fveq2d 5861 . . 3  |-  ( ( ( U  e.  (UnifOn `  X )  /\  P  e.  X )  /\  p  =  P )  ->  (
( nei `  K
) `  { p } )  =  ( ( nei `  K
) `  { P } ) )
93 simpr 461 . . 3  |-  ( ( U  e.  (UnifOn `  X )  /\  P  e.  X )  ->  P  e.  X )
94 fvex 5867 . . . 4  |-  ( ( nei `  K ) `
 { P }
)  e.  _V
9594a1i 11 . . 3  |-  ( ( U  e.  (UnifOn `  X )  /\  P  e.  X )  ->  (
( nei `  K
) `  { P } )  e.  _V )
9689, 92, 93, 95fvmptd 5946 . 2  |-  ( ( U  e.  (UnifOn `  X )  /\  P  e.  X )  ->  ( N `  P )  =  ( ( nei `  K ) `  { P } ) )
97 mptexg 6121 . . . . 5  |-  ( U  e.  (UnifOn `  X
)  ->  ( v  e.  U  |->  ( v
" { P }
) )  e.  _V )
98 rnexg 6706 . . . . 5  |-  ( ( v  e.  U  |->  ( v " { P } ) )  e. 
_V  ->  ran  ( v  e.  U  |->  ( v
" { P }
) )  e.  _V )
9997, 98syl 16 . . . 4  |-  ( U  e.  (UnifOn `  X
)  ->  ran  ( v  e.  U  |->  ( v
" { P }
) )  e.  _V )
10099adantr 465 . . 3  |-  ( ( U  e.  (UnifOn `  X )  /\  P  e.  X )  ->  ran  ( v  e.  U  |->  ( v " { P } ) )  e. 
_V )
10113a1i 11 . . . 4  |-  ( ( P  e.  X  /\  ran  ( v  e.  U  |->  ( v " { P } ) )  e. 
_V )  ->  N  =  ( p  e.  X  |->  ran  ( v  e.  U  |->  ( v
" { p }
) ) ) )
102 nfv 1678 . . . . . . . 8  |-  F/ v  P  e.  X
103 nfmpt1 4529 . . . . . . . . . 10  |-  F/_ v
( v  e.  U  |->  ( v " { P } ) )
104103nfrn 5236 . . . . . . . . 9  |-  F/_ v ran  ( v  e.  U  |->  ( v " { P } ) )
105104nfel1 2638 . . . . . . . 8  |-  F/ v ran  ( v  e.  U  |->  ( v " { P } ) )  e.  _V
106102, 105nfan 1870 . . . . . . 7  |-  F/ v ( P  e.  X  /\  ran  ( v  e.  U  |->  ( v " { P } ) )  e.  _V )
107 nfv 1678 . . . . . . 7  |-  F/ v  p  =  P
108106, 107nfan 1870 . . . . . 6  |-  F/ v ( ( P  e.  X  /\  ran  (
v  e.  U  |->  ( v " { P } ) )  e. 
_V )  /\  p  =  P )
109 simpr2 998 . . . . . . . . 9  |-  ( ( P  e.  X  /\  ( ran  ( v  e.  U  |->  ( v " { P } ) )  e.  _V  /\  p  =  P  /\  v  e.  U ) )  ->  p  =  P )
110109sneqd 4032 . . . . . . . 8  |-  ( ( P  e.  X  /\  ( ran  ( v  e.  U  |->  ( v " { P } ) )  e.  _V  /\  p  =  P  /\  v  e.  U ) )  ->  { p }  =  { P } )
111110imaeq2d 5328 . . . . . . 7  |-  ( ( P  e.  X  /\  ( ran  ( v  e.  U  |->  ( v " { P } ) )  e.  _V  /\  p  =  P  /\  v  e.  U ) )  -> 
( v " {
p } )  =  ( v " { P } ) )
1121113anassrs 1213 . . . . . 6  |-  ( ( ( ( P  e.  X  /\  ran  (
v  e.  U  |->  ( v " { P } ) )  e. 
_V )  /\  p  =  P )  /\  v  e.  U )  ->  (
v " { p } )  =  ( v " { P } ) )
113108, 112mpteq2da 4525 . . . . 5  |-  ( ( ( P  e.  X  /\  ran  ( v  e.  U  |->  ( v " { P } ) )  e.  _V )  /\  p  =  P )  ->  ( v  e.  U  |->  ( v " {
p } ) )  =  ( v  e.  U  |->  ( v " { P } ) ) )
114113rneqd 5221 . . . 4  |-  ( ( ( P  e.  X  /\  ran  ( v  e.  U  |->  ( v " { P } ) )  e.  _V )  /\  p  =  P )  ->  ran  ( v  e.  U  |->  ( v " { p } ) )  =  ran  (
v  e.  U  |->  ( v " { P } ) ) )
115 simpl 457 . . . 4  |-  ( ( P  e.  X  /\  ran  ( v  e.  U  |->  ( v " { P } ) )  e. 
_V )  ->  P  e.  X )
116 simpr 461 . . . 4  |-  ( ( P  e.  X  /\  ran  ( v  e.  U  |->  ( v " { P } ) )  e. 
_V )  ->  ran  ( v  e.  U  |->  ( v " { P } ) )  e. 
_V )
117101, 114, 115, 116fvmptd 5946 . . 3  |-  ( ( P  e.  X  /\  ran  ( v  e.  U  |->  ( v " { P } ) )  e. 
_V )  ->  ( N `  P )  =  ran  ( v  e.  U  |->  ( v " { P } ) ) )
11893, 100, 117syl2anc 661 . 2  |-  ( ( U  e.  (UnifOn `  X )  /\  P  e.  X )  ->  ( N `  P )  =  ran  ( v  e.  U  |->  ( v " { P } ) ) )
11981, 96, 1183eqtr2d 2507 1  |-  ( ( U  e.  (UnifOn `  X )  /\  P  e.  X )  ->  (
( nei `  J
) `  { P } )  =  ran  ( v  e.  U  |->  ( v " { P } ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 968    = wceq 1374    e. wcel 1762   A.wral 2807   E.wrex 2808   {crab 2811   _Vcvv 3106    C_ wss 3469   ~Pcpw 4003   {csn 4020    |-> cmpt 4498   ran crn 4993   "cima 4995   ` cfv 5579   neicnei 19357  UnifOncust 20430  unifTopcutop 20461
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-int 4276  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-we 4833  df-ord 4874  df-on 4875  df-lim 4876  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-om 6672  df-recs 7032  df-rdg 7066  df-1o 7120  df-oadd 7124  df-er 7301  df-en 7507  df-fin 7510  df-fi 7860  df-top 19159  df-nei 19358  df-ust 20431  df-utop 20462
This theorem is referenced by:  utopsnneip  20479
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