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Theorem ustuqtoplem 20477
Description: Lemma for ustuqtop 20484 (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 4039 . . . . . . . . . 10  |-  ( ( p  =  q  /\  v  e.  U )  ->  { p }  =  { q } )
43imaeq2d 5335 . . . . . . . . 9  |-  ( ( p  =  q  /\  v  e.  U )  ->  ( v " {
p } )  =  ( v " {
q } ) )
54mpteq2dva 4533 . . . . . . . 8  |-  ( p  =  q  ->  (
v  e.  U  |->  ( v " { p } ) )  =  ( v  e.  U  |->  ( v " {
q } ) ) )
65rneqd 5228 . . . . . . 7  |-  ( p  =  q  ->  ran  ( v  e.  U  |->  ( v " {
p } ) )  =  ran  ( v  e.  U  |->  ( v
" { q } ) ) )
76cbvmptv 4538 . . . . . 6  |-  ( p  e.  X  |->  ran  (
v  e.  U  |->  ( v " { p } ) ) )  =  ( q  e.  X  |->  ran  ( v  e.  U  |->  ( v
" { q } ) ) )
81, 7eqtri 2496 . . . . 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 1003 . . . . . . . . 9  |-  ( ( U  e.  (UnifOn `  X )  /\  ( P  e.  X  /\  q  =  P  /\  v  e.  U )
)  ->  q  =  P )
1110sneqd 4039 . . . . . . . 8  |-  ( ( U  e.  (UnifOn `  X )  /\  ( P  e.  X  /\  q  =  P  /\  v  e.  U )
)  ->  { q }  =  { P } )
1211imaeq2d 5335 . . . . . . 7  |-  ( ( U  e.  (UnifOn `  X )  /\  ( P  e.  X  /\  q  =  P  /\  v  e.  U )
)  ->  ( v " { q } )  =  ( v " { P } ) )
13123anassrs 1218 . . . . . 6  |-  ( ( ( ( U  e.  (UnifOn `  X )  /\  P  e.  X
)  /\  q  =  P )  /\  v  e.  U )  ->  (
v " { q } )  =  ( v " { P } ) )
1413mpteq2dva 4533 . . . . 5  |-  ( ( ( U  e.  (UnifOn `  X )  /\  P  e.  X )  /\  q  =  P )  ->  (
v  e.  U  |->  ( v " { q } ) )  =  ( v  e.  U  |->  ( v " { P } ) ) )
1514rneqd 5228 . . . 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 6128 . . . . . 6  |-  ( U  e.  (UnifOn `  X
)  ->  ( v  e.  U  |->  ( v
" { P }
) )  e.  _V )
18 rnexg 6713 . . . . . 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 5953 . . 3  |-  ( ( U  e.  (UnifOn `  X )  /\  P  e.  X )  ->  ( N `  P )  =  ran  ( v  e.  U  |->  ( v " { P } ) ) )
2221eleq2d 2537 . 2  |-  ( ( U  e.  (UnifOn `  X )  /\  P  e.  X )  ->  ( A  e.  ( N `  P )  <->  A  e.  ran  ( v  e.  U  |->  ( v " { P } ) ) ) )
23 imaeq1 5330 . . . 4  |-  ( v  =  w  ->  (
v " { P } )  =  ( w " { P } ) )
2423cbvmptv 4538 . . 3  |-  ( v  e.  U  |->  ( v
" { P }
) )  =  ( w  e.  U  |->  ( w " { P } ) )
2524elrnmpt 5247 . 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 973    = wceq 1379    e. wcel 1767   E.wrex 2815   _Vcvv 3113   {csn 4027    |-> cmpt 4505   ran crn 5000   "cima 5002   ` cfv 5586  UnifOncust 20437
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pr 4686  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594
This theorem is referenced by:  ustuqtop1  20479  ustuqtop2  20480  ustuqtop3  20481  ustuqtop4  20482  ustuqtop5  20483  utopsnneiplem  20485
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