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Theorem ustuqtoplem 21252
Description: Lemma for ustuqtop 21259 (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 458 . . . . . . . . . . 11  |-  ( ( p  =  q  /\  v  e.  U )  ->  p  =  q )
32sneqd 4010 . . . . . . . . . 10  |-  ( ( p  =  q  /\  v  e.  U )  ->  { p }  =  { q } )
43imaeq2d 5187 . . . . . . . . 9  |-  ( ( p  =  q  /\  v  e.  U )  ->  ( v " {
p } )  =  ( v " {
q } ) )
54mpteq2dva 4510 . . . . . . . 8  |-  ( p  =  q  ->  (
v  e.  U  |->  ( v " { p } ) )  =  ( v  e.  U  |->  ( v " {
q } ) ) )
65rneqd 5081 . . . . . . 7  |-  ( p  =  q  ->  ran  ( v  e.  U  |->  ( v " {
p } ) )  =  ran  ( v  e.  U  |->  ( v
" { q } ) ) )
76cbvmptv 4516 . . . . . 6  |-  ( p  e.  X  |->  ran  (
v  e.  U  |->  ( v " { p } ) ) )  =  ( q  e.  X  |->  ran  ( v  e.  U  |->  ( v
" { q } ) ) )
81, 7eqtri 2451 . . . . 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 1012 . . . . . . . . 9  |-  ( ( U  e.  (UnifOn `  X )  /\  ( P  e.  X  /\  q  =  P  /\  v  e.  U )
)  ->  q  =  P )
1110sneqd 4010 . . . . . . . 8  |-  ( ( U  e.  (UnifOn `  X )  /\  ( P  e.  X  /\  q  =  P  /\  v  e.  U )
)  ->  { q }  =  { P } )
1211imaeq2d 5187 . . . . . . 7  |-  ( ( U  e.  (UnifOn `  X )  /\  ( P  e.  X  /\  q  =  P  /\  v  e.  U )
)  ->  ( v " { q } )  =  ( v " { P } ) )
13123anassrs 1228 . . . . . 6  |-  ( ( ( ( U  e.  (UnifOn `  X )  /\  P  e.  X
)  /\  q  =  P )  /\  v  e.  U )  ->  (
v " { q } )  =  ( v " { P } ) )
1413mpteq2dva 4510 . . . . 5  |-  ( ( ( U  e.  (UnifOn `  X )  /\  P  e.  X )  /\  q  =  P )  ->  (
v  e.  U  |->  ( v " { q } ) )  =  ( v  e.  U  |->  ( v " { P } ) ) )
1514rneqd 5081 . . . 4  |-  ( ( ( U  e.  (UnifOn `  X )  /\  P  e.  X )  /\  q  =  P )  ->  ran  ( v  e.  U  |->  ( v " {
q } ) )  =  ran  ( v  e.  U  |->  ( v
" { P }
) ) )
16 simpr 462 . . . 4  |-  ( ( U  e.  (UnifOn `  X )  /\  P  e.  X )  ->  P  e.  X )
17 mptexg 6150 . . . . . 6  |-  ( U  e.  (UnifOn `  X
)  ->  ( v  e.  U  |->  ( v
" { P }
) )  e.  _V )
18 rnexg 6739 . . . . . 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 466 . . . 4  |-  ( ( U  e.  (UnifOn `  X )  /\  P  e.  X )  ->  ran  ( v  e.  U  |->  ( v " { P } ) )  e. 
_V )
219, 15, 16, 20fvmptd 5970 . . 3  |-  ( ( U  e.  (UnifOn `  X )  /\  P  e.  X )  ->  ( N `  P )  =  ran  ( v  e.  U  |->  ( v " { P } ) ) )
2221eleq2d 2492 . 2  |-  ( ( U  e.  (UnifOn `  X )  /\  P  e.  X )  ->  ( A  e.  ( N `  P )  <->  A  e.  ran  ( v  e.  U  |->  ( v " { P } ) ) ) )
23 imaeq1 5182 . . . 4  |-  ( v  =  w  ->  (
v " { P } )  =  ( w " { P } ) )
2423cbvmptv 4516 . . 3  |-  ( v  e.  U  |->  ( v
" { P }
) )  =  ( w  e.  U  |->  ( w " { P } ) )
2524elrnmpt 5100 . 2  |-  ( A  e.  V  ->  ( A  e.  ran  ( v  e.  U  |->  ( v
" { P }
) )  <->  E. w  e.  U  A  =  ( w " { P } ) ) )
2622, 25sylan9bb 704 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 187    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1872   E.wrex 2772   _Vcvv 3080   {csn 3998    |-> cmpt 4482   ran crn 4854   "cima 4856   ` cfv 5601  UnifOncust 21212
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-rep 4536  ax-sep 4546  ax-nul 4555  ax-pr 4660  ax-un 6597
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2273  df-mo 2274  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-ral 2776  df-rex 2777  df-reu 2778  df-rab 2780  df-v 3082  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-if 3912  df-sn 3999  df-pr 4001  df-op 4005  df-uni 4220  df-iun 4301  df-br 4424  df-opab 4483  df-mpt 4484  df-id 4768  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609
This theorem is referenced by:  ustuqtop1  21254  ustuqtop2  21255  ustuqtop3  21256  ustuqtop4  21257  ustuqtop5  21258  utopsnneiplem  21260
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