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Theorem ustuqtoplem 19814
Description: Lemma for ustuqtop 19821 (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 3889 . . . . . . . . . 10  |-  ( ( p  =  q  /\  v  e.  U )  ->  { p }  =  { q } )
43imaeq2d 5169 . . . . . . . . 9  |-  ( ( p  =  q  /\  v  e.  U )  ->  ( v " {
p } )  =  ( v " {
q } ) )
54mpteq2dva 4378 . . . . . . . 8  |-  ( p  =  q  ->  (
v  e.  U  |->  ( v " { p } ) )  =  ( v  e.  U  |->  ( v " {
q } ) ) )
65rneqd 5067 . . . . . . 7  |-  ( p  =  q  ->  ran  ( v  e.  U  |->  ( v " {
p } ) )  =  ran  ( v  e.  U  |->  ( v
" { q } ) ) )
76cbvmptv 4383 . . . . . 6  |-  ( p  e.  X  |->  ran  (
v  e.  U  |->  ( v " { p } ) ) )  =  ( q  e.  X  |->  ran  ( v  e.  U  |->  ( v
" { q } ) ) )
81, 7eqtri 2463 . . . . 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 995 . . . . . . . . 9  |-  ( ( U  e.  (UnifOn `  X )  /\  ( P  e.  X  /\  q  =  P  /\  v  e.  U )
)  ->  q  =  P )
1110sneqd 3889 . . . . . . . 8  |-  ( ( U  e.  (UnifOn `  X )  /\  ( P  e.  X  /\  q  =  P  /\  v  e.  U )
)  ->  { q }  =  { P } )
1211imaeq2d 5169 . . . . . . 7  |-  ( ( U  e.  (UnifOn `  X )  /\  ( P  e.  X  /\  q  =  P  /\  v  e.  U )
)  ->  ( v " { q } )  =  ( v " { P } ) )
13123anassrs 1209 . . . . . 6  |-  ( ( ( ( U  e.  (UnifOn `  X )  /\  P  e.  X
)  /\  q  =  P )  /\  v  e.  U )  ->  (
v " { q } )  =  ( v " { P } ) )
1413mpteq2dva 4378 . . . . 5  |-  ( ( ( U  e.  (UnifOn `  X )  /\  P  e.  X )  /\  q  =  P )  ->  (
v  e.  U  |->  ( v " { q } ) )  =  ( v  e.  U  |->  ( v " { P } ) ) )
1514rneqd 5067 . . . 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 5947 . . . . . 6  |-  ( U  e.  (UnifOn `  X
)  ->  ( v  e.  U  |->  ( v
" { P }
) )  e.  _V )
18 rnexg 6510 . . . . . 6  |-  ( ( v  e.  U  |->  ( v " { P } ) )  e. 
_V  ->  ran  ( v  e.  U  |->  ( v
" { P }
) )  e.  _V )
1917, 18syl 16 . . . . 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 5779 . . 3  |-  ( ( U  e.  (UnifOn `  X )  /\  P  e.  X )  ->  ( N `  P )  =  ran  ( v  e.  U  |->  ( v " { P } ) ) )
2221eleq2d 2510 . 2  |-  ( ( U  e.  (UnifOn `  X )  /\  P  e.  X )  ->  ( A  e.  ( N `  P )  <->  A  e.  ran  ( v  e.  U  |->  ( v " { P } ) ) ) )
23 imaeq1 5164 . . . 4  |-  ( v  =  w  ->  (
v " { P } )  =  ( w " { P } ) )
2423cbvmptv 4383 . . 3  |-  ( v  e.  U  |->  ( v
" { P }
) )  =  ( w  e.  U  |->  ( w " { P } ) )
2524elrnmpt 5086 . 2  |-  ( A  e.  V  ->  ( A  e.  ran  ( v  e.  U  |->  ( v
" { P }
) )  <->  E. w  e.  U  A  =  ( w " { P } ) ) )
2622, 25sylan9bb 699 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 184    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756   E.wrex 2716   _Vcvv 2972   {csn 3877    e. cmpt 4350   ran crn 4841   "cima 4843   ` cfv 5418  UnifOncust 19774
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4403  ax-sep 4413  ax-nul 4421  ax-pr 4531  ax-un 6372
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-reu 2722  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-sn 3878  df-pr 3880  df-op 3884  df-uni 4092  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-id 4636  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426
This theorem is referenced by:  ustuqtop1  19816  ustuqtop2  19817  ustuqtop3  19818  ustuqtop4  19819  ustuqtop5  19820  utopsnneiplem  19822
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