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Theorem qtopval 17680
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 2924 . 2  |-  ( J  e.  V  ->  J  e.  _V )
2 elex 2924 . 2  |-  ( F  e.  W  ->  F  e.  _V )
3 imaexg 5176 . . . . 5  |-  ( F  e.  _V  ->  ( F " X )  e. 
_V )
4 pwexg 4343 . . . . 5  |-  ( ( F " X )  e.  _V  ->  ~P ( F " X )  e.  _V )
5 rabexg 4313 . . . . 5  |-  ( ~P ( F " X
)  e.  _V  ->  { s  e.  ~P ( F " X )  |  ( ( `' F " s )  i^i  X
)  e.  J }  e.  _V )
63, 4, 53syl 19 . . . 4  |-  ( F  e.  _V  ->  { s  e.  ~P ( F
" X )  |  ( ( `' F " s )  i^i  X
)  e.  J }  e.  _V )
76adantl 453 . . 3  |-  ( ( J  e.  _V  /\  F  e.  _V )  ->  { s  e.  ~P ( F " X )  |  ( ( `' F " s )  i^i  X )  e.  J }  e.  _V )
8 simpr 448 . . . . . . 7  |-  ( ( j  =  J  /\  f  =  F )  ->  f  =  F )
9 simpl 444 . . . . . . . . 9  |-  ( ( j  =  J  /\  f  =  F )  ->  j  =  J )
109unieqd 3986 . . . . . . . 8  |-  ( ( j  =  J  /\  f  =  F )  ->  U. j  =  U. J )
11 qtopval.1 . . . . . . . 8  |-  X  = 
U. J
1210, 11syl6eqr 2454 . . . . . . 7  |-  ( ( j  =  J  /\  f  =  F )  ->  U. j  =  X )
138, 12imaeq12d 5163 . . . . . 6  |-  ( ( j  =  J  /\  f  =  F )  ->  ( f " U. j )  =  ( F " X ) )
1413pweqd 3764 . . . . 5  |-  ( ( j  =  J  /\  f  =  F )  ->  ~P ( f " U. j )  =  ~P ( F " X ) )
158cnveqd 5007 . . . . . . . 8  |-  ( ( j  =  J  /\  f  =  F )  ->  `' f  =  `' F )
1615imaeq1d 5161 . . . . . . 7  |-  ( ( j  =  J  /\  f  =  F )  ->  ( `' f "
s )  =  ( `' F " s ) )
1716, 12ineq12d 3503 . . . . . 6  |-  ( ( j  =  J  /\  f  =  F )  ->  ( ( `' f
" s )  i^i  U. j )  =  ( ( `' F "
s )  i^i  X
) )
1817, 9eleq12d 2472 . . . . 5  |-  ( ( j  =  J  /\  f  =  F )  ->  ( ( ( `' f " s )  i^i  U. j )  e.  j  <->  ( ( `' F " s )  i^i  X )  e.  J ) )
1914, 18rabeqbidv 2911 . . . 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 13688 . . . 4  |- qTop  =  ( j  e.  _V , 
f  e.  _V  |->  { s  e.  ~P (
f " U. j
)  |  ( ( `' f " s
)  i^i  U. j
)  e.  j } )
2119, 20ovmpt2ga 6162 . . 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 1280 . 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 464 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 set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1721   {crab 2670   _Vcvv 2916    i^i cin 3279   ~Pcpw 3759   U.cuni 3975   `'ccnv 4836   "cima 4840  (class class class)co 6040   qTop cqtop 13684
This theorem is referenced by:  qtopval2  17681  qtopres  17683  imastopn  17705
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-sbc 3122  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-br 4173  df-opab 4227  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-qtop 13688
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