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Theorem utopsnneiplem 21262
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 21247 . . . . . . . 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 2497 . . . . . . 7  |-  ( U  e.  (UnifOn `  X
)  ->  J  =  { a  e.  ~P X  |  A. p  e.  a  E. w  e.  U  ( w " { p } ) 
C_  a } )
4 simpll 760 . . . . . . . . . . 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 3961 . . . . . . . . . . . 12  |-  ( ( U  e.  (UnifOn `  X )  /\  a  e.  ~P X )  -> 
a  C_  X )
76sselda 3432 . . . . . . . . . . 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 6135 . . . . . . . . . . . . . . . 16  |-  ( U  e.  (UnifOn `  X
)  ->  ( v  e.  U  |->  ( v
" { p }
) )  e.  _V )
10 rnexg 6725 . . . . . . . . . . . . . . . 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 5957 . . . . . . . . . . . . . 14  |-  ( ( p  e.  X  /\  ran  ( v  e.  U  |->  ( v " {
p } ) )  e.  _V )  -> 
( N `  p
)  =  ran  (
v  e.  U  |->  ( v " { p } ) ) )
158, 12, 14syl2anc 667 . . . . . . . . . . . . 13  |-  ( ( U  e.  (UnifOn `  X )  /\  p  e.  X )  ->  ( N `  p )  =  ran  ( v  e.  U  |->  ( v " { p } ) ) )
1615eleq2d 2514 . . . . . . . . . . . 12  |-  ( ( U  e.  (UnifOn `  X )  /\  p  e.  X )  ->  (
a  e.  ( N `
 p )  <->  a  e.  ran  ( v  e.  U  |->  ( v " {
p } ) ) ) )
17 vex 3048 . . . . . . . . . . . . 13  |-  a  e. 
_V
18 eqid 2451 . . . . . . . . . . . . . 14  |-  ( v  e.  U  |->  ( v
" { p }
) )  =  ( v  e.  U  |->  ( v " { p } ) )
1918elrnmpt 5081 . . . . . . . . . . . . 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 667 . . . . . . . . . 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 1761 . . . . . . . . . . . . 13  |-  F/ v ( ( U  e.  (UnifOn `  X )  /\  a  e.  ~P X )  /\  p  e.  a )
24 nfre1 2848 . . . . . . . . . . . . 13  |-  F/ v E. v  e.  U  a  =  ( v " { p } )
2523, 24nfan 2011 . . . . . . . . . . . 12  |-  F/ v ( ( ( U  e.  (UnifOn `  X
)  /\  a  e.  ~P X )  /\  p  e.  a )  /\  E. v  e.  U  a  =  ( v " { p } ) )
26 simplr 762 . . . . . . . . . . . . 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 3485 . . . . . . . . . . . . . 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 5163 . . . . . . . . . . . . . . 15  |-  ( w  =  v  ->  (
w " { p } )  =  ( v " { p } ) )
3029sseq1d 3459 . . . . . . . . . . . . . 14  |-  ( w  =  v  ->  (
( w " {
p } )  C_  a 
<->  ( v " {
p } )  C_  a ) )
3130rspcev 3150 . . . . . . . . . . . . 13  |-  ( ( v  e.  U  /\  ( v " {
p } )  C_  a )  ->  E. w  e.  U  ( w " { p } ) 
C_  a )
3226, 28, 31syl2anc 667 . . . . . . . . . . . 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 2930 . . . . . . . . . . 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 732 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( U  e.  (UnifOn `  X
)  /\  a  e.  ~P X )  /\  p  e.  a )  /\  w  e.  U )  /\  (
w " { p } )  C_  a
)  ->  U  e.  (UnifOn `  X ) )
367ad2antrr 732 . . . . . . . . . . . . . . 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 736 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( U  e.  (UnifOn `  X
)  /\  a  e.  ~P X )  /\  p  e.  a )  /\  w  e.  U )  /\  (
w " { p } )  C_  a
)  ->  a  C_  X )
40 simplr 762 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( U  e.  (UnifOn `  X
)  /\  a  e.  ~P X )  /\  p  e.  a )  /\  w  e.  U )  /\  (
w " { p } )  C_  a
)  ->  w  e.  U )
41 eqid 2451 . . . . . . . . . . . . . . . . . 18  |-  ( w
" { p }
)  =  ( w
" { p }
)
42 imaeq1 5163 . . . . . . . . . . . . . . . . . . . 20  |-  ( u  =  w  ->  (
u " { p } )  =  ( w " { p } ) )
4342eqeq2d 2461 . . . . . . . . . . . . . . . . . . 19  |-  ( u  =  w  ->  (
( w " {
p } )  =  ( u " {
p } )  <->  ( w " { p } )  =  ( w " { p } ) ) )
4443rspcev 3150 . . . . . . . . . . . . . . . . . 18  |-  ( ( w  e.  U  /\  ( w " {
p } )  =  ( w " {
p } ) )  ->  E. u  e.  U  ( w " {
p } )  =  ( u " {
p } ) )
4541, 44mpan2 677 . . . . . . . . . . . . . . . . 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 3048 . . . . . . . . . . . . . . . . . . 19  |-  w  e. 
_V
48 imaexg 6730 . . . . . . . . . . . . . . . . . . 19  |-  ( w  e.  _V  ->  (
w " { p } )  e.  _V )
4947, 48ax-mp 5 . . . . . . . . . . . . . . . . . 18  |-  ( w
" { p }
)  e.  _V
5013ustuqtoplem 21254 . . . . . . . . . . . . . . . . . 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 677 . . . . . . . . . . . . . . . . 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 1267 . . . . . . . . . . . . . 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 3453 . . . . . . . . . . . . . . . . . . 19  |-  ( b  =  ( w " { p } )  ->  ( b  C_  a 
<->  ( w " {
p } )  C_  a ) )
56553anbi2d 1344 . . . . . . . . . . . . . . . . . 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 2517 . . . . . . . . . . . . . . . . . 18  |-  ( b  =  ( w " { p } )  ->  ( b  e.  ( N `  p
)  <->  ( w " { p } )  e.  ( N `  p ) ) )
5856, 57anbi12d 717 . . . . . . . . . . . . . . . . 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 21256 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( U  e.  (UnifOn `  X )  /\  p  e.  X
)  /\  b  C_  a  /\  a  C_  X
)  /\  b  e.  ( N `  p ) )  ->  a  e.  ( N `  p ) )
6159, 60vtoclg 3107 . . . . . . . . . . . . . . 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 1271 . . . . . . . . . . . . 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 2931 . . . . . . . . . . 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 843 . . . . . . . . . 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 2824 . . . . . . . 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 3035 . . . . . . 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 2488 . . . . . 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 2503 . . . . 5  |-  ( U  e.  (UnifOn `  X
)  ->  J  =  K )
7473fveq2d 5869 . . . 4  |-  ( U  e.  (UnifOn `  X
)  ->  ( nei `  J )  =  ( nei `  K ) )
7574fveq1d 5867 . . 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 21255 . . . . 5  |-  ( U  e.  (UnifOn `  X
)  ->  N : X
--> ~P ~P X )
7813ustuqtop1 21256 . . . . 5  |-  ( ( ( ( U  e.  (UnifOn `  X )  /\  p  e.  X
)  /\  a  C_  b  /\  b  C_  X
)  /\  a  e.  ( N `  p ) )  ->  b  e.  ( N `  p ) )
7913ustuqtop2 21257 . . . . 5  |-  ( ( U  e.  (UnifOn `  X )  /\  p  e.  X )  ->  ( fi `  ( N `  p ) )  C_  ( N `  p ) )
8013ustuqtop3 21258 . . . . 5  |-  ( ( ( U  e.  (UnifOn `  X )  /\  p  e.  X )  /\  a  e.  ( N `  p
) )  ->  p  e.  a )
8113ustuqtop4 21259 . . . . 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 21260 . . . . 5  |-  ( ( U  e.  (UnifOn `  X )  /\  p  e.  X )  ->  X  e.  ( N `  p
) )
8372, 77, 78, 79, 80, 81, 82neiptopnei 20148 . . . 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 3980 . . . 4  |-  ( ( ( U  e.  (UnifOn `  X )  /\  P  e.  X )  /\  p  =  P )  ->  { p }  =  { P } )
8786fveq2d 5869 . . 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 5875 . . . 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 5954 . 2  |-  ( ( U  e.  (UnifOn `  X )  /\  P  e.  X )  ->  ( N `  P )  =  ( ( nei `  K ) `  { P } ) )
92 mptexg 6135 . . . . 5  |-  ( U  e.  (UnifOn `  X
)  ->  ( v  e.  U  |->  ( v
" { P }
) )  e.  _V )
93 rnexg 6725 . . . . 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 1761 . . . . . . . 8  |-  F/ v  P  e.  X
98 nfmpt1 4492 . . . . . . . . . 10  |-  F/_ v
( v  e.  U  |->  ( v " { P } ) )
9998nfrn 5077 . . . . . . . . 9  |-  F/_ v ran  ( v  e.  U  |->  ( v " { P } ) )
10099nfel1 2606 . . . . . . . 8  |-  F/ v ran  ( v  e.  U  |->  ( v " { P } ) )  e.  _V
10197, 100nfan 2011 . . . . . . 7  |-  F/ v ( P  e.  X  /\  ran  ( v  e.  U  |->  ( v " { P } ) )  e.  _V )
102 nfv 1761 . . . . . . 7  |-  F/ v  p  =  P
103101, 102nfan 2011 . . . . . 6  |-  F/ v ( ( P  e.  X  /\  ran  (
v  e.  U  |->  ( v " { P } ) )  e. 
_V )  /\  p  =  P )
104 simpr2 1015 . . . . . . . . 9  |-  ( ( P  e.  X  /\  ( ran  ( v  e.  U  |->  ( v " { P } ) )  e.  _V  /\  p  =  P  /\  v  e.  U ) )  ->  p  =  P )
105104sneqd 3980 . . . . . . . 8  |-  ( ( P  e.  X  /\  ( ran  ( v  e.  U  |->  ( v " { P } ) )  e.  _V  /\  p  =  P  /\  v  e.  U ) )  ->  { p }  =  { P } )
106105imaeq2d 5168 . . . . . . 7  |-  ( ( P  e.  X  /\  ( ran  ( v  e.  U  |->  ( v " { P } ) )  e.  _V  /\  p  =  P  /\  v  e.  U ) )  -> 
( v " {
p } )  =  ( v " { P } ) )
1071063anassrs 1232 . . . . . 6  |-  ( ( ( ( P  e.  X  /\  ran  (
v  e.  U  |->  ( v " { P } ) )  e. 
_V )  /\  p  =  P )  /\  v  e.  U )  ->  (
v " { p } )  =  ( v " { P } ) )
108103, 107mpteq2da 4488 . . . . 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 5062 . . . 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 5954 . . 3  |-  ( ( P  e.  X  /\  ran  ( v  e.  U  |->  ( v " { P } ) )  e. 
_V )  ->  ( N `  P )  =  ran  ( v  e.  U  |->  ( v " { P } ) ) )
11388, 95, 112syl2anc 667 . 2  |-  ( ( U  e.  (UnifOn `  X )  /\  P  e.  X )  ->  ( N `  P )  =  ran  ( v  e.  U  |->  ( v " { P } ) ) )
11476, 91, 1133eqtr2d 2491 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 985    = wceq 1444    e. wcel 1887   A.wral 2737   E.wrex 2738   {crab 2741   _Vcvv 3045    C_ wss 3404   ~Pcpw 3951   {csn 3968    |-> cmpt 4461   ran crn 4835   "cima 4837   ` cfv 5582   neicnei 20113  UnifOncust 21214  unifTopcutop 21245
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-rep 4515  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-ral 2742  df-rex 2743  df-reu 2744  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-uni 4199  df-int 4235  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-tr 4498  df-eprel 4745  df-id 4749  df-po 4755  df-so 4756  df-fr 4793  df-we 4795  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-pred 5380  df-ord 5426  df-on 5427  df-lim 5428  df-suc 5429  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-om 6693  df-wrecs 7028  df-recs 7090  df-rdg 7128  df-1o 7182  df-oadd 7186  df-er 7363  df-en 7570  df-fin 7573  df-fi 7925  df-top 19921  df-nei 20114  df-ust 21215  df-utop 21246
This theorem is referenced by:  utopsnneip  21263
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