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Theorem utopsnneiplem 21254
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 21239 . . . . . . . 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 2476 . . . . . . 7  |-  ( U  e.  (UnifOn `  X
)  ->  J  =  { a  e.  ~P X  |  A. p  e.  a  E. w  e.  U  ( w " { p } ) 
C_  a } )
4 simpll 759 . . . . . . . . . . 11  |-  ( ( ( U  e.  (UnifOn `  X )  /\  a  e.  ~P X )  /\  p  e.  a )  ->  U  e.  (UnifOn `  X ) )
5 simpr 463 . . . . . . . . . . . . 13  |-  ( ( U  e.  (UnifOn `  X )  /\  a  e.  ~P X )  -> 
a  e.  ~P X
)
65elpwid 3990 . . . . . . . . . . . 12  |-  ( ( U  e.  (UnifOn `  X )  /\  a  e.  ~P X )  -> 
a  C_  X )
76sselda 3465 . . . . . . . . . . 11  |-  ( ( ( U  e.  (UnifOn `  X )  /\  a  e.  ~P X )  /\  p  e.  a )  ->  p  e.  X )
8 simpr 463 . . . . . . . . . . . . . 14  |-  ( ( U  e.  (UnifOn `  X )  /\  p  e.  X )  ->  p  e.  X )
9 mptexg 6148 . . . . . . . . . . . . . . . 16  |-  ( U  e.  (UnifOn `  X
)  ->  ( v  e.  U  |->  ( v
" { p }
) )  e.  _V )
10 rnexg 6737 . . . . . . . . . . . . . . . 16  |-  ( ( v  e.  U  |->  ( v " { p } ) )  e. 
_V  ->  ran  ( v  e.  U  |->  ( v
" { p }
) )  e.  _V )
119, 10syl 17 . . . . . . . . . . . . . . 15  |-  ( U  e.  (UnifOn `  X
)  ->  ran  ( v  e.  U  |->  ( v
" { p }
) )  e.  _V )
1211adantr 467 . . . . . . . . . . . . . 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 5971 . . . . . . . . . . . . . 14  |-  ( ( p  e.  X  /\  ran  ( v  e.  U  |->  ( v " {
p } ) )  e.  _V )  -> 
( N `  p
)  =  ran  (
v  e.  U  |->  ( v " { p } ) ) )
158, 12, 14syl2anc 666 . . . . . . . . . . . . 13  |-  ( ( U  e.  (UnifOn `  X )  /\  p  e.  X )  ->  ( N `  p )  =  ran  ( v  e.  U  |->  ( v " { p } ) ) )
1615eleq2d 2493 . . . . . . . . . . . 12  |-  ( ( U  e.  (UnifOn `  X )  /\  p  e.  X )  ->  (
a  e.  ( N `
 p )  <->  a  e.  ran  ( v  e.  U  |->  ( v " {
p } ) ) ) )
17 vex 3085 . . . . . . . . . . . . 13  |-  a  e. 
_V
18 eqid 2423 . . . . . . . . . . . . . 14  |-  ( v  e.  U  |->  ( v
" { p }
) )  =  ( v  e.  U  |->  ( v " { p } ) )
1918elrnmpt 5098 . . . . . . . . . . . . 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 265 . . . . . . . . . . 11  |-  ( ( U  e.  (UnifOn `  X )  /\  p  e.  X )  ->  (
a  e.  ( N `
 p )  <->  E. v  e.  U  a  =  ( v " {
p } ) ) )
224, 7, 21syl2anc 666 . . . . . . . . . 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 1752 . . . . . . . . . . . . 13  |-  F/ v ( ( U  e.  (UnifOn `  X )  /\  a  e.  ~P X )  /\  p  e.  a )
24 nfre1 2887 . . . . . . . . . . . . 13  |-  F/ v E. v  e.  U  a  =  ( v " { p } )
2523, 24nfan 1985 . . . . . . . . . . . 12  |-  F/ v ( ( ( U  e.  (UnifOn `  X
)  /\  a  e.  ~P X )  /\  p  e.  a )  /\  E. v  e.  U  a  =  ( v " { p } ) )
26 simplr 761 . . . . . . . . . . . . 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 3518 . . . . . . . . . . . . . 14  |-  ( a  =  ( v " { p } )  ->  ( v " { p } ) 
C_  a )
2827adantl 468 . . . . . . . . . . . . 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 5180 . . . . . . . . . . . . . . 15  |-  ( w  =  v  ->  (
w " { p } )  =  ( v " { p } ) )
3029sseq1d 3492 . . . . . . . . . . . . . 14  |-  ( w  =  v  ->  (
( w " {
p } )  C_  a 
<->  ( v " {
p } )  C_  a ) )
3130rspcev 3183 . . . . . . . . . . . . 13  |-  ( ( v  e.  U  /\  ( v " {
p } )  C_  a )  ->  E. w  e.  U  ( w " { p } ) 
C_  a )
3226, 28, 31syl2anc 666 . . . . . . . . . . . 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 463 . . . . . . . . . . . 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 2969 . . . . . . . . . . 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 )
354ad2antrr 731 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( U  e.  (UnifOn `  X
)  /\  a  e.  ~P X )  /\  p  e.  a )  /\  w  e.  U )  /\  (
w " { p } )  C_  a
)  ->  U  e.  (UnifOn `  X ) )
367ad2antrr 731 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( U  e.  (UnifOn `  X
)  /\  a  e.  ~P X )  /\  p  e.  a )  /\  w  e.  U )  /\  (
w " { p } )  C_  a
)  ->  p  e.  X )
3735, 36jca 535 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( 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
) )
38 simpr 463 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( U  e.  (UnifOn `  X
)  /\  a  e.  ~P X )  /\  p  e.  a )  /\  w  e.  U )  /\  (
w " { p } )  C_  a
)  ->  ( w " { p } ) 
C_  a )
396ad3antrrr 735 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( U  e.  (UnifOn `  X
)  /\  a  e.  ~P X )  /\  p  e.  a )  /\  w  e.  U )  /\  (
w " { p } )  C_  a
)  ->  a  C_  X )
40 simplr 761 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( U  e.  (UnifOn `  X
)  /\  a  e.  ~P X )  /\  p  e.  a )  /\  w  e.  U )  /\  (
w " { p } )  C_  a
)  ->  w  e.  U )
41 eqid 2423 . . . . . . . . . . . . . . . . . 18  |-  ( w
" { p }
)  =  ( w
" { p }
)
42 imaeq1 5180 . . . . . . . . . . . . . . . . . . . 20  |-  ( u  =  w  ->  (
u " { p } )  =  ( w " { p } ) )
4342eqeq2d 2437 . . . . . . . . . . . . . . . . . . 19  |-  ( u  =  w  ->  (
( w " {
p } )  =  ( u " {
p } )  <->  ( w " { p } )  =  ( w " { p } ) ) )
4443rspcev 3183 . . . . . . . . . . . . . . . . . 18  |-  ( ( w  e.  U  /\  ( w " {
p } )  =  ( w " {
p } ) )  ->  E. u  e.  U  ( w " {
p } )  =  ( u " {
p } ) )
4541, 44mpan2 676 . . . . . . . . . . . . . . . . 17  |-  ( w  e.  U  ->  E. u  e.  U  ( w " { p } )  =  ( u " { p } ) )
4645adantl 468 . . . . . . . . . . . . . . . 16  |-  ( ( ( U  e.  (UnifOn `  X )  /\  p  e.  X )  /\  w  e.  U )  ->  E. u  e.  U  ( w " { p } )  =  ( u " { p } ) )
47 vex 3085 . . . . . . . . . . . . . . . . . . 19  |-  w  e. 
_V
48 imaexg 6742 . . . . . . . . . . . . . . . . . . 19  |-  ( w  e.  _V  ->  (
w " { p } )  e.  _V )
4947, 48ax-mp 5 . . . . . . . . . . . . . . . . . 18  |-  ( w
" { p }
)  e.  _V
5013ustuqtoplem 21246 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( U  e.  (UnifOn `  X )  /\  p  e.  X )  /\  (
w " { p } )  e.  _V )  ->  ( ( w
" { p }
)  e.  ( N `
 p )  <->  E. u  e.  U  ( w " { p } )  =  ( u " { p } ) ) )
5149, 50mpan2 676 . . . . . . . . . . . . . . . . 17  |-  ( ( U  e.  (UnifOn `  X )  /\  p  e.  X )  ->  (
( w " {
p } )  e.  ( N `  p
)  <->  E. u  e.  U  ( w " {
p } )  =  ( u " {
p } ) ) )
5251adantr 467 . . . . . . . . . . . . . . . 16  |-  ( ( ( U  e.  (UnifOn `  X )  /\  p  e.  X )  /\  w  e.  U )  ->  (
( w " {
p } )  e.  ( N `  p
)  <->  E. u  e.  U  ( w " {
p } )  =  ( u " {
p } ) ) )
5346, 52mpbird 236 . . . . . . . . . . . . . . 15  |-  ( ( ( U  e.  (UnifOn `  X )  /\  p  e.  X )  /\  w  e.  U )  ->  (
w " { p } )  e.  ( N `  p ) )
5435, 36, 40, 53syl21anc 1264 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( U  e.  (UnifOn `  X
)  /\  a  e.  ~P X )  /\  p  e.  a )  /\  w  e.  U )  /\  (
w " { p } )  C_  a
)  ->  ( w " { p } )  e.  ( N `  p ) )
55 sseq1 3486 . . . . . . . . . . . . . . . . . . 19  |-  ( b  =  ( w " { p } )  ->  ( b  C_  a 
<->  ( w " {
p } )  C_  a ) )
56553anbi2d 1341 . . . . . . . . . . . . . . . . . 18  |-  ( 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 ) ) )
57 eleq1 2495 . . . . . . . . . . . . . . . . . 18  |-  ( b  =  ( w " { p } )  ->  ( b  e.  ( N `  p
)  <->  ( w " { p } )  e.  ( N `  p ) ) )
5856, 57anbi12d 716 . . . . . . . . . . . . . . . . 17  |-  ( 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 ) ) ) )
5958imbi1d 319 . . . . . . . . . . . . . . . 16  |-  ( 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 ) ) ) )
6013ustuqtop1 21248 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( U  e.  (UnifOn `  X )  /\  p  e.  X
)  /\  b  C_  a  /\  a  C_  X
)  /\  b  e.  ( N `  p ) )  ->  a  e.  ( N `  p ) )
6159, 60vtoclg 3140 . . . . . . . . . . . . . . 15  |-  ( ( 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 ) ) )
6247, 48, 61mp2b 10 . . . . . . . . . . . . . 14  |-  ( ( ( ( U  e.  (UnifOn `  X )  /\  p  e.  X
)  /\  ( w " { p } ) 
C_  a  /\  a  C_  X )  /\  (
w " { p } )  e.  ( N `  p ) )  ->  a  e.  ( N `  p ) )
6337, 38, 39, 54, 62syl31anc 1268 . . . . . . . . . . . . 13  |-  ( ( ( ( ( U  e.  (UnifOn `  X
)  /\  a  e.  ~P X )  /\  p  e.  a )  /\  w  e.  U )  /\  (
w " { p } )  C_  a
)  ->  a  e.  ( N `  p ) )
6437, 21syl 17 . . . . . . . . . . . . 13  |-  ( ( ( ( ( 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 } ) ) )
6563, 64mpbid 214 . . . . . . . . . . . 12  |-  ( ( ( ( ( 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 } ) )
6665r19.29an 2970 . . . . . . . . . . 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 } ) )
6734, 66impbida 841 . . . . . . . . . 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 ) )
6822, 67bitrd 257 . . . . . . . . 9  |-  ( ( ( U  e.  (UnifOn `  X )  /\  a  e.  ~P X )  /\  p  e.  a )  ->  ( a  e.  ( N `  p )  <->  E. w  e.  U  ( w " {
p } )  C_  a ) )
6968ralbidva 2862 . . . . . . . 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 ) )
7069rabbidva 3072 . . . . . . 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 } )
713, 70eqtr4d 2467 . . . . . 6  |-  ( U  e.  (UnifOn `  X
)  ->  J  =  { a  e.  ~P X  |  A. p  e.  a  a  e.  ( N `  p ) } )
72 utopsnneip.1 . . . . . 6  |-  K  =  { a  e.  ~P X  |  A. p  e.  a  a  e.  ( N `  p ) }
7371, 72syl6eqr 2482 . . . . 5  |-  ( U  e.  (UnifOn `  X
)  ->  J  =  K )
7473fveq2d 5883 . . . 4  |-  ( U  e.  (UnifOn `  X
)  ->  ( nei `  J )  =  ( nei `  K ) )
7574fveq1d 5881 . . 3  |-  ( U  e.  (UnifOn `  X
)  ->  ( ( nei `  J ) `  { P } )  =  ( ( nei `  K
) `  { P } ) )
7675adantr 467 . 2  |-  ( ( U  e.  (UnifOn `  X )  /\  P  e.  X )  ->  (
( nei `  J
) `  { P } )  =  ( ( nei `  K
) `  { P } ) )
7713ustuqtop0 21247 . . . . 5  |-  ( U  e.  (UnifOn `  X
)  ->  N : X
--> ~P ~P X )
7813ustuqtop1 21248 . . . . 5  |-  ( ( ( ( U  e.  (UnifOn `  X )  /\  p  e.  X
)  /\  a  C_  b  /\  b  C_  X
)  /\  a  e.  ( N `  p ) )  ->  b  e.  ( N `  p ) )
7913ustuqtop2 21249 . . . . 5  |-  ( ( U  e.  (UnifOn `  X )  /\  p  e.  X )  ->  ( fi `  ( N `  p ) )  C_  ( N `  p ) )
8013ustuqtop3 21250 . . . . 5  |-  ( ( ( U  e.  (UnifOn `  X )  /\  p  e.  X )  /\  a  e.  ( N `  p
) )  ->  p  e.  a )
8113ustuqtop4 21251 . . . . 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 ) )
8213ustuqtop5 21252 . . . . 5  |-  ( ( U  e.  (UnifOn `  X )  /\  p  e.  X )  ->  X  e.  ( N `  p
) )
8372, 77, 78, 79, 80, 81, 82neiptopnei 20140 . . . 4  |-  ( U  e.  (UnifOn `  X
)  ->  N  =  ( p  e.  X  |->  ( ( nei `  K
) `  { p } ) ) )
8483adantr 467 . . 3  |-  ( ( U  e.  (UnifOn `  X )  /\  P  e.  X )  ->  N  =  ( p  e.  X  |->  ( ( nei `  K ) `  {
p } ) ) )
85 simpr 463 . . . . 5  |-  ( ( ( U  e.  (UnifOn `  X )  /\  P  e.  X )  /\  p  =  P )  ->  p  =  P )
8685sneqd 4009 . . . 4  |-  ( ( ( U  e.  (UnifOn `  X )  /\  P  e.  X )  /\  p  =  P )  ->  { p }  =  { P } )
8786fveq2d 5883 . . 3  |-  ( ( ( U  e.  (UnifOn `  X )  /\  P  e.  X )  /\  p  =  P )  ->  (
( nei `  K
) `  { p } )  =  ( ( nei `  K
) `  { P } ) )
88 simpr 463 . . 3  |-  ( ( U  e.  (UnifOn `  X )  /\  P  e.  X )  ->  P  e.  X )
89 fvex 5889 . . . 4  |-  ( ( nei `  K ) `
 { P }
)  e.  _V
9089a1i 11 . . 3  |-  ( ( U  e.  (UnifOn `  X )  /\  P  e.  X )  ->  (
( nei `  K
) `  { P } )  e.  _V )
9184, 87, 88, 90fvmptd 5968 . 2  |-  ( ( U  e.  (UnifOn `  X )  /\  P  e.  X )  ->  ( N `  P )  =  ( ( nei `  K ) `  { P } ) )
92 mptexg 6148 . . . . 5  |-  ( U  e.  (UnifOn `  X
)  ->  ( v  e.  U  |->  ( v
" { P }
) )  e.  _V )
93 rnexg 6737 . . . . 5  |-  ( ( v  e.  U  |->  ( v " { P } ) )  e. 
_V  ->  ran  ( v  e.  U  |->  ( v
" { P }
) )  e.  _V )
9492, 93syl 17 . . . 4  |-  ( U  e.  (UnifOn `  X
)  ->  ran  ( v  e.  U  |->  ( v
" { P }
) )  e.  _V )
9594adantr 467 . . 3  |-  ( ( U  e.  (UnifOn `  X )  /\  P  e.  X )  ->  ran  ( v  e.  U  |->  ( v " { P } ) )  e. 
_V )
9613a1i 11 . . . 4  |-  ( ( P  e.  X  /\  ran  ( v  e.  U  |->  ( v " { P } ) )  e. 
_V )  ->  N  =  ( p  e.  X  |->  ran  ( v  e.  U  |->  ( v
" { p }
) ) ) )
97 nfv 1752 . . . . . . . 8  |-  F/ v  P  e.  X
98 nfmpt1 4511 . . . . . . . . . 10  |-  F/_ v
( v  e.  U  |->  ( v " { P } ) )
9998nfrn 5094 . . . . . . . . 9  |-  F/_ v ran  ( v  e.  U  |->  ( v " { P } ) )
10099nfel1 2601 . . . . . . . 8  |-  F/ v ran  ( v  e.  U  |->  ( v " { P } ) )  e.  _V
10197, 100nfan 1985 . . . . . . 7  |-  F/ v ( P  e.  X  /\  ran  ( v  e.  U  |->  ( v " { P } ) )  e.  _V )
102 nfv 1752 . . . . . . 7  |-  F/ v  p  =  P
103101, 102nfan 1985 . . . . . 6  |-  F/ v ( ( P  e.  X  /\  ran  (
v  e.  U  |->  ( v " { P } ) )  e. 
_V )  /\  p  =  P )
104 simpr2 1013 . . . . . . . . 9  |-  ( ( P  e.  X  /\  ( ran  ( v  e.  U  |->  ( v " { P } ) )  e.  _V  /\  p  =  P  /\  v  e.  U ) )  ->  p  =  P )
105104sneqd 4009 . . . . . . . 8  |-  ( ( P  e.  X  /\  ( ran  ( v  e.  U  |->  ( v " { P } ) )  e.  _V  /\  p  =  P  /\  v  e.  U ) )  ->  { p }  =  { P } )
106105imaeq2d 5185 . . . . . . 7  |-  ( ( P  e.  X  /\  ( ran  ( v  e.  U  |->  ( v " { P } ) )  e.  _V  /\  p  =  P  /\  v  e.  U ) )  -> 
( v " {
p } )  =  ( v " { P } ) )
1071063anassrs 1229 . . . . . 6  |-  ( ( ( ( P  e.  X  /\  ran  (
v  e.  U  |->  ( v " { P } ) )  e. 
_V )  /\  p  =  P )  /\  v  e.  U )  ->  (
v " { p } )  =  ( v " { P } ) )
108103, 107mpteq2da 4507 . . . . 5  |-  ( ( ( P  e.  X  /\  ran  ( v  e.  U  |->  ( v " { P } ) )  e.  _V )  /\  p  =  P )  ->  ( v  e.  U  |->  ( v " {
p } ) )  =  ( v  e.  U  |->  ( v " { P } ) ) )
109108rneqd 5079 . . . 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 } ) ) )
110 simpl 459 . . . 4  |-  ( ( P  e.  X  /\  ran  ( v  e.  U  |->  ( v " { P } ) )  e. 
_V )  ->  P  e.  X )
111 simpr 463 . . . 4  |-  ( ( P  e.  X  /\  ran  ( v  e.  U  |->  ( v " { P } ) )  e. 
_V )  ->  ran  ( v  e.  U  |->  ( v " { P } ) )  e. 
_V )
11296, 109, 110, 111fvmptd 5968 . . 3  |-  ( ( P  e.  X  /\  ran  ( v  e.  U  |->  ( v " { P } ) )  e. 
_V )  ->  ( N `  P )  =  ran  ( v  e.  U  |->  ( v " { P } ) ) )
11388, 95, 112syl2anc 666 . 2  |-  ( ( U  e.  (UnifOn `  X )  /\  P  e.  X )  ->  ( N `  P )  =  ran  ( v  e.  U  |->  ( v " { P } ) ) )
11476, 91, 1133eqtr2d 2470 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 188    /\ wa 371    /\ w3a 983    = wceq 1438    e. wcel 1869   A.wral 2776   E.wrex 2777   {crab 2780   _Vcvv 3082    C_ wss 3437   ~Pcpw 3980   {csn 3997    |-> cmpt 4480   ran crn 4852   "cima 4854   ` cfv 5599   neicnei 20105  UnifOncust 21206  unifTopcutop 21237
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1666  ax-4 1679  ax-5 1749  ax-6 1795  ax-7 1840  ax-8 1871  ax-9 1873  ax-10 1888  ax-11 1893  ax-12 1906  ax-13 2054  ax-ext 2401  ax-rep 4534  ax-sep 4544  ax-nul 4553  ax-pow 4600  ax-pr 4658  ax-un 6595
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 984  df-3an 985  df-tru 1441  df-ex 1661  df-nf 1665  df-sb 1788  df-eu 2270  df-mo 2271  df-clab 2409  df-cleq 2415  df-clel 2418  df-nfc 2573  df-ne 2621  df-ral 2781  df-rex 2782  df-reu 2783  df-rab 2785  df-v 3084  df-sbc 3301  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-pss 3453  df-nul 3763  df-if 3911  df-pw 3982  df-sn 3998  df-pr 4000  df-tp 4002  df-op 4004  df-uni 4218  df-int 4254  df-iun 4299  df-br 4422  df-opab 4481  df-mpt 4482  df-tr 4517  df-eprel 4762  df-id 4766  df-po 4772  df-so 4773  df-fr 4810  df-we 4812  df-xp 4857  df-rel 4858  df-cnv 4859  df-co 4860  df-dm 4861  df-rn 4862  df-res 4863  df-ima 4864  df-pred 5397  df-ord 5443  df-on 5444  df-lim 5445  df-suc 5446  df-iota 5563  df-fun 5601  df-fn 5602  df-f 5603  df-f1 5604  df-fo 5605  df-f1o 5606  df-fv 5607  df-ov 6306  df-oprab 6307  df-mpt2 6308  df-om 6705  df-wrecs 7034  df-recs 7096  df-rdg 7134  df-1o 7188  df-oadd 7192  df-er 7369  df-en 7576  df-fin 7579  df-fi 7929  df-top 19913  df-nei 20106  df-ust 21207  df-utop 21238
This theorem is referenced by:  utopsnneip  21255
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