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Theorem ustuqtoplem 21265
Description: Lemma for ustuqtop 21272. (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 463 . . . . . . . . . . 11  |-  ( ( p  =  q  /\  v  e.  U )  ->  p  =  q )
32sneqd 3948 . . . . . . . . . 10  |-  ( ( p  =  q  /\  v  e.  U )  ->  { p }  =  { q } )
43imaeq2d 5146 . . . . . . . . 9  |-  ( ( p  =  q  /\  v  e.  U )  ->  ( v " {
p } )  =  ( v " {
q } ) )
54mpteq2dva 4461 . . . . . . . 8  |-  ( p  =  q  ->  (
v  e.  U  |->  ( v " { p } ) )  =  ( v  e.  U  |->  ( v " {
q } ) ) )
65rneqd 5040 . . . . . . 7  |-  ( p  =  q  ->  ran  ( v  e.  U  |->  ( v " {
p } ) )  =  ran  ( v  e.  U  |->  ( v
" { q } ) ) )
76cbvmptv 4467 . . . . . 6  |-  ( p  e.  X  |->  ran  (
v  e.  U  |->  ( v " { p } ) ) )  =  ( q  e.  X  |->  ran  ( v  e.  U  |->  ( v
" { q } ) ) )
81, 7eqtri 2474 . . . . 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 1016 . . . . . . . . 9  |-  ( ( U  e.  (UnifOn `  X )  /\  ( P  e.  X  /\  q  =  P  /\  v  e.  U )
)  ->  q  =  P )
1110sneqd 3948 . . . . . . . 8  |-  ( ( U  e.  (UnifOn `  X )  /\  ( P  e.  X  /\  q  =  P  /\  v  e.  U )
)  ->  { q }  =  { P } )
1211imaeq2d 5146 . . . . . . 7  |-  ( ( U  e.  (UnifOn `  X )  /\  ( P  e.  X  /\  q  =  P  /\  v  e.  U )
)  ->  ( v " { q } )  =  ( v " { P } ) )
13123anassrs 1235 . . . . . 6  |-  ( ( ( ( U  e.  (UnifOn `  X )  /\  P  e.  X
)  /\  q  =  P )  /\  v  e.  U )  ->  (
v " { q } )  =  ( v " { P } ) )
1413mpteq2dva 4461 . . . . 5  |-  ( ( ( U  e.  (UnifOn `  X )  /\  P  e.  X )  /\  q  =  P )  ->  (
v  e.  U  |->  ( v " { q } ) )  =  ( v  e.  U  |->  ( v " { P } ) ) )
1514rneqd 5040 . . . 4  |-  ( ( ( U  e.  (UnifOn `  X )  /\  P  e.  X )  /\  q  =  P )  ->  ran  ( v  e.  U  |->  ( v " {
q } ) )  =  ran  ( v  e.  U  |->  ( v
" { P }
) ) )
16 simpr 467 . . . 4  |-  ( ( U  e.  (UnifOn `  X )  /\  P  e.  X )  ->  P  e.  X )
17 mptexg 6121 . . . . . 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 17 . . . . 5  |-  ( U  e.  (UnifOn `  X
)  ->  ran  ( v  e.  U  |->  ( v
" { P }
) )  e.  _V )
2019adantr 471 . . . 4  |-  ( ( U  e.  (UnifOn `  X )  /\  P  e.  X )  ->  ran  ( v  e.  U  |->  ( v " { P } ) )  e. 
_V )
219, 15, 16, 20fvmptd 5938 . . 3  |-  ( ( U  e.  (UnifOn `  X )  /\  P  e.  X )  ->  ( N `  P )  =  ran  ( v  e.  U  |->  ( v " { P } ) ) )
2221eleq2d 2515 . 2  |-  ( ( U  e.  (UnifOn `  X )  /\  P  e.  X )  ->  ( A  e.  ( N `  P )  <->  A  e.  ran  ( v  e.  U  |->  ( v " { P } ) ) ) )
23 imaeq1 5141 . . . 4  |-  ( v  =  w  ->  (
v " { P } )  =  ( w " { P } ) )
2423cbvmptv 4467 . . 3  |-  ( v  e.  U  |->  ( v
" { P }
) )  =  ( w  e.  U  |->  ( w " { P } ) )
2524elrnmpt 5059 . 2  |-  ( A  e.  V  ->  ( A  e.  ran  ( v  e.  U  |->  ( v
" { P }
) )  <->  E. w  e.  U  A  =  ( w " { P } ) ) )
2622, 25sylan9bb 711 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 189    /\ wa 375    /\ w3a 986    = wceq 1448    e. wcel 1891   E.wrex 2738   _Vcvv 3013   {csn 3936    |-> cmpt 4433   ran crn 4813   "cima 4815   ` cfv 5561  UnifOncust 21225
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1673  ax-4 1686  ax-5 1762  ax-6 1809  ax-7 1855  ax-8 1893  ax-9 1900  ax-10 1919  ax-11 1924  ax-12 1937  ax-13 2092  ax-ext 2432  ax-rep 4487  ax-sep 4497  ax-nul 4506  ax-pr 4612  ax-un 6571
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3an 988  df-tru 1451  df-ex 1668  df-nf 1672  df-sb 1802  df-eu 2304  df-mo 2305  df-clab 2439  df-cleq 2445  df-clel 2448  df-nfc 2582  df-ne 2624  df-ral 2742  df-rex 2743  df-reu 2744  df-rab 2746  df-v 3015  df-sbc 3236  df-csb 3332  df-dif 3375  df-un 3377  df-in 3379  df-ss 3386  df-nul 3700  df-if 3850  df-sn 3937  df-pr 3939  df-op 3943  df-uni 4169  df-iun 4250  df-br 4375  df-opab 4434  df-mpt 4435  df-id 4727  df-xp 4818  df-rel 4819  df-cnv 4820  df-co 4821  df-dm 4822  df-rn 4823  df-res 4824  df-ima 4825  df-iota 5525  df-fun 5563  df-fn 5564  df-f 5565  df-f1 5566  df-fo 5567  df-f1o 5568  df-fv 5569
This theorem is referenced by:  ustuqtop1  21267  ustuqtop2  21268  ustuqtop3  21269  ustuqtop4  21270  ustuqtop5  21271  utopsnneiplem  21273
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