MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ustuqtoplem Structured version   Unicode version

Theorem ustuqtoplem 21036
Description: Lemma for ustuqtop 21043 (Contributed by Thierry Arnoux, 11-Jan-2018.)
Hypothesis
Ref Expression
utopustuq.1  |-  N  =  ( p  e.  X  |->  ran  ( v  e.  U  |->  ( v " { p } ) ) )
Assertion
Ref Expression
ustuqtoplem  |-  ( ( ( U  e.  (UnifOn `  X )  /\  P  e.  X )  /\  A  e.  V )  ->  ( A  e.  ( N `  P )  <->  E. w  e.  U  A  =  ( w " { P } ) ) )
Distinct variable groups:    w, A    w, v, P    v, p, w, U    X, p, v
Allowed substitution hints:    A( v, p)    P( p)    N( w, v, p)    V( w, v, p)    X( w)

Proof of Theorem ustuqtoplem
Dummy variable  q is distinct from all other variables.
StepHypRef Expression
1 utopustuq.1 . . . . . 6  |-  N  =  ( p  e.  X  |->  ran  ( v  e.  U  |->  ( v " { p } ) ) )
2 simpl 457 . . . . . . . . . . 11  |-  ( ( p  =  q  /\  v  e.  U )  ->  p  =  q )
32sneqd 3986 . . . . . . . . . 10  |-  ( ( p  =  q  /\  v  e.  U )  ->  { p }  =  { q } )
43imaeq2d 5159 . . . . . . . . 9  |-  ( ( p  =  q  /\  v  e.  U )  ->  ( v " {
p } )  =  ( v " {
q } ) )
54mpteq2dva 4483 . . . . . . . 8  |-  ( p  =  q  ->  (
v  e.  U  |->  ( v " { p } ) )  =  ( v  e.  U  |->  ( v " {
q } ) ) )
65rneqd 5053 . . . . . . 7  |-  ( p  =  q  ->  ran  ( v  e.  U  |->  ( v " {
p } ) )  =  ran  ( v  e.  U  |->  ( v
" { q } ) ) )
76cbvmptv 4489 . . . . . 6  |-  ( p  e.  X  |->  ran  (
v  e.  U  |->  ( v " { p } ) ) )  =  ( q  e.  X  |->  ran  ( v  e.  U  |->  ( v
" { q } ) ) )
81, 7eqtri 2433 . . . . 5  |-  N  =  ( q  e.  X  |->  ran  ( v  e.  U  |->  ( v " { q } ) ) )
98a1i 11 . . . 4  |-  ( ( U  e.  (UnifOn `  X )  /\  P  e.  X )  ->  N  =  ( q  e.  X  |->  ran  ( v  e.  U  |->  ( v
" { q } ) ) ) )
10 simpr2 1006 . . . . . . . . 9  |-  ( ( U  e.  (UnifOn `  X )  /\  ( P  e.  X  /\  q  =  P  /\  v  e.  U )
)  ->  q  =  P )
1110sneqd 3986 . . . . . . . 8  |-  ( ( U  e.  (UnifOn `  X )  /\  ( P  e.  X  /\  q  =  P  /\  v  e.  U )
)  ->  { q }  =  { P } )
1211imaeq2d 5159 . . . . . . 7  |-  ( ( U  e.  (UnifOn `  X )  /\  ( P  e.  X  /\  q  =  P  /\  v  e.  U )
)  ->  ( v " { q } )  =  ( v " { P } ) )
13123anassrs 1222 . . . . . 6  |-  ( ( ( ( U  e.  (UnifOn `  X )  /\  P  e.  X
)  /\  q  =  P )  /\  v  e.  U )  ->  (
v " { q } )  =  ( v " { P } ) )
1413mpteq2dva 4483 . . . . 5  |-  ( ( ( U  e.  (UnifOn `  X )  /\  P  e.  X )  /\  q  =  P )  ->  (
v  e.  U  |->  ( v " { q } ) )  =  ( v  e.  U  |->  ( v " { P } ) ) )
1514rneqd 5053 . . . 4  |-  ( ( ( U  e.  (UnifOn `  X )  /\  P  e.  X )  /\  q  =  P )  ->  ran  ( v  e.  U  |->  ( v " {
q } ) )  =  ran  ( v  e.  U  |->  ( v
" { P }
) ) )
16 simpr 461 . . . 4  |-  ( ( U  e.  (UnifOn `  X )  /\  P  e.  X )  ->  P  e.  X )
17 mptexg 6125 . . . . . 6  |-  ( U  e.  (UnifOn `  X
)  ->  ( v  e.  U  |->  ( v
" { P }
) )  e.  _V )
18 rnexg 6718 . . . . . 6  |-  ( ( v  e.  U  |->  ( v " { P } ) )  e. 
_V  ->  ran  ( v  e.  U  |->  ( v
" { P }
) )  e.  _V )
1917, 18syl 17 . . . . 5  |-  ( U  e.  (UnifOn `  X
)  ->  ran  ( v  e.  U  |->  ( v
" { P }
) )  e.  _V )
2019adantr 465 . . . 4  |-  ( ( U  e.  (UnifOn `  X )  /\  P  e.  X )  ->  ran  ( v  e.  U  |->  ( v " { P } ) )  e. 
_V )
219, 15, 16, 20fvmptd 5940 . . 3  |-  ( ( U  e.  (UnifOn `  X )  /\  P  e.  X )  ->  ( N `  P )  =  ran  ( v  e.  U  |->  ( v " { P } ) ) )
2221eleq2d 2474 . 2  |-  ( ( U  e.  (UnifOn `  X )  /\  P  e.  X )  ->  ( A  e.  ( N `  P )  <->  A  e.  ran  ( v  e.  U  |->  ( v " { P } ) ) ) )
23 imaeq1 5154 . . . 4  |-  ( v  =  w  ->  (
v " { P } )  =  ( w " { P } ) )
2423cbvmptv 4489 . . 3  |-  ( v  e.  U  |->  ( v
" { P }
) )  =  ( w  e.  U  |->  ( w " { P } ) )
2524elrnmpt 5072 . 2  |-  ( A  e.  V  ->  ( A  e.  ran  ( v  e.  U  |->  ( v
" { P }
) )  <->  E. w  e.  U  A  =  ( w " { P } ) ) )
2622, 25sylan9bb 700 1  |-  ( ( ( U  e.  (UnifOn `  X )  /\  P  e.  X )  /\  A  e.  V )  ->  ( A  e.  ( N `  P )  <->  E. w  e.  U  A  =  ( w " { P } ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 186    /\ wa 369    /\ w3a 976    = wceq 1407    e. wcel 1844   E.wrex 2757   _Vcvv 3061   {csn 3974    |-> cmpt 4455   ran crn 4826   "cima 4828   ` cfv 5571  UnifOncust 20996
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1641  ax-4 1654  ax-5 1727  ax-6 1773  ax-7 1816  ax-8 1846  ax-9 1848  ax-10 1863  ax-11 1868  ax-12 1880  ax-13 2028  ax-ext 2382  ax-rep 4509  ax-sep 4519  ax-nul 4527  ax-pr 4632  ax-un 6576
This theorem depends on definitions:  df-bi 187  df-or 370  df-an 371  df-3an 978  df-tru 1410  df-ex 1636  df-nf 1640  df-sb 1766  df-eu 2244  df-mo 2245  df-clab 2390  df-cleq 2396  df-clel 2399  df-nfc 2554  df-ne 2602  df-ral 2761  df-rex 2762  df-reu 2763  df-rab 2765  df-v 3063  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3741  df-if 3888  df-sn 3975  df-pr 3977  df-op 3981  df-uni 4194  df-iun 4275  df-br 4398  df-opab 4456  df-mpt 4457  df-id 4740  df-xp 4831  df-rel 4832  df-cnv 4833  df-co 4834  df-dm 4835  df-rn 4836  df-res 4837  df-ima 4838  df-iota 5535  df-fun 5573  df-fn 5574  df-f 5575  df-f1 5576  df-fo 5577  df-f1o 5578  df-fv 5579
This theorem is referenced by:  ustuqtop1  21038  ustuqtop2  21039  ustuqtop3  21040  ustuqtop4  21041  ustuqtop5  21042  utopsnneiplem  21044
  Copyright terms: Public domain W3C validator