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Theorem qtopval 19271
Description: Value of the quotient topology function. (Contributed by Mario Carneiro, 23-Mar-2015.)
Hypothesis
Ref Expression
qtopval.1  |-  X  = 
U. J
Assertion
Ref Expression
qtopval  |-  ( ( J  e.  V  /\  F  e.  W )  ->  ( J qTop  F )  =  { s  e. 
~P ( F " X )  |  ( ( `' F "
s )  i^i  X
)  e.  J }
)
Distinct variable groups:    F, s    J, s    V, s    X, s
Allowed substitution hint:    W( s)

Proof of Theorem qtopval
Dummy variables  f 
j are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elex 2984 . 2  |-  ( J  e.  V  ->  J  e.  _V )
2 elex 2984 . 2  |-  ( F  e.  W  ->  F  e.  _V )
3 imaexg 6518 . . . . 5  |-  ( F  e.  _V  ->  ( F " X )  e. 
_V )
4 pwexg 4479 . . . . 5  |-  ( ( F " X )  e.  _V  ->  ~P ( F " X )  e.  _V )
5 rabexg 4445 . . . . 5  |-  ( ~P ( F " X
)  e.  _V  ->  { s  e.  ~P ( F " X )  |  ( ( `' F " s )  i^i  X
)  e.  J }  e.  _V )
63, 4, 53syl 20 . . . 4  |-  ( F  e.  _V  ->  { s  e.  ~P ( F
" X )  |  ( ( `' F " s )  i^i  X
)  e.  J }  e.  _V )
76adantl 466 . . 3  |-  ( ( J  e.  _V  /\  F  e.  _V )  ->  { s  e.  ~P ( F " X )  |  ( ( `' F " s )  i^i  X )  e.  J }  e.  _V )
8 simpr 461 . . . . . . 7  |-  ( ( j  =  J  /\  f  =  F )  ->  f  =  F )
9 simpl 457 . . . . . . . . 9  |-  ( ( j  =  J  /\  f  =  F )  ->  j  =  J )
109unieqd 4104 . . . . . . . 8  |-  ( ( j  =  J  /\  f  =  F )  ->  U. j  =  U. J )
11 qtopval.1 . . . . . . . 8  |-  X  = 
U. J
1210, 11syl6eqr 2493 . . . . . . 7  |-  ( ( j  =  J  /\  f  =  F )  ->  U. j  =  X )
138, 12imaeq12d 5173 . . . . . 6  |-  ( ( j  =  J  /\  f  =  F )  ->  ( f " U. j )  =  ( F " X ) )
1413pweqd 3868 . . . . 5  |-  ( ( j  =  J  /\  f  =  F )  ->  ~P ( f " U. j )  =  ~P ( F " X ) )
158cnveqd 5018 . . . . . . . 8  |-  ( ( j  =  J  /\  f  =  F )  ->  `' f  =  `' F )
1615imaeq1d 5171 . . . . . . 7  |-  ( ( j  =  J  /\  f  =  F )  ->  ( `' f "
s )  =  ( `' F " s ) )
1716, 12ineq12d 3556 . . . . . 6  |-  ( ( j  =  J  /\  f  =  F )  ->  ( ( `' f
" s )  i^i  U. j )  =  ( ( `' F "
s )  i^i  X
) )
1817, 9eleq12d 2511 . . . . 5  |-  ( ( j  =  J  /\  f  =  F )  ->  ( ( ( `' f " s )  i^i  U. j )  e.  j  <->  ( ( `' F " s )  i^i  X )  e.  J ) )
1914, 18rabeqbidv 2970 . . . 4  |-  ( ( j  =  J  /\  f  =  F )  ->  { s  e.  ~P ( f " U. j )  |  ( ( `' f "
s )  i^i  U. j )  e.  j }  =  { s  e.  ~P ( F
" X )  |  ( ( `' F " s )  i^i  X
)  e.  J }
)
20 df-qtop 14448 . . . 4  |- qTop  =  ( j  e.  _V , 
f  e.  _V  |->  { s  e.  ~P (
f " U. j
)  |  ( ( `' f " s
)  i^i  U. j
)  e.  j } )
2119, 20ovmpt2ga 6223 . . 3  |-  ( ( J  e.  _V  /\  F  e.  _V  /\  {
s  e.  ~P ( F " X )  |  ( ( `' F " s )  i^i  X
)  e.  J }  e.  _V )  ->  ( J qTop  F )  =  {
s  e.  ~P ( F " X )  |  ( ( `' F " s )  i^i  X
)  e.  J }
)
227, 21mpd3an3 1315 . 2  |-  ( ( J  e.  _V  /\  F  e.  _V )  ->  ( J qTop  F )  =  { s  e. 
~P ( F " X )  |  ( ( `' F "
s )  i^i  X
)  e.  J }
)
231, 2, 22syl2an 477 1  |-  ( ( J  e.  V  /\  F  e.  W )  ->  ( J qTop  F )  =  { s  e. 
~P ( F " X )  |  ( ( `' F "
s )  i^i  X
)  e.  J }
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756   {crab 2722   _Vcvv 2975    i^i cin 3330   ~Pcpw 3863   U.cuni 4094   `'ccnv 4842   "cima 4846  (class class class)co 6094   qTop cqtop 14444
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-sep 4416  ax-nul 4424  ax-pow 4473  ax-pr 4534  ax-un 6375
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 2571  df-ne 2611  df-ral 2723  df-rex 2724  df-rab 2727  df-v 2977  df-sbc 3190  df-dif 3334  df-un 3336  df-in 3338  df-ss 3345  df-nul 3641  df-if 3795  df-pw 3865  df-sn 3881  df-pr 3883  df-op 3887  df-uni 4095  df-br 4296  df-opab 4354  df-id 4639  df-xp 4849  df-rel 4850  df-cnv 4851  df-co 4852  df-dm 4853  df-rn 4854  df-res 4855  df-ima 4856  df-iota 5384  df-fun 5423  df-fv 5429  df-ov 6097  df-oprab 6098  df-mpt2 6099  df-qtop 14448
This theorem is referenced by:  qtopval2  19272  qtopres  19274  imastopn  19296
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