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Theorem qtopval 17386
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 2796 . 2  |-  ( J  e.  V  ->  J  e.  _V )
2 elex 2796 . 2  |-  ( F  e.  W  ->  F  e.  _V )
3 imaexg 5026 . . . . 5  |-  ( F  e.  _V  ->  ( F " X )  e. 
_V )
4 pwexg 4194 . . . . 5  |-  ( ( F " X )  e.  _V  ->  ~P ( F " X )  e.  _V )
5 rabexg 4164 . . . . 5  |-  ( ~P ( F " X
)  e.  _V  ->  { s  e.  ~P ( F " X )  |  ( ( `' F " s )  i^i  X
)  e.  J }  e.  _V )
63, 4, 53syl 18 . . . 4  |-  ( F  e.  _V  ->  { s  e.  ~P ( F
" X )  |  ( ( `' F " s )  i^i  X
)  e.  J }  e.  _V )
76adantl 452 . . 3  |-  ( ( J  e.  _V  /\  F  e.  _V )  ->  { s  e.  ~P ( F " X )  |  ( ( `' F " s )  i^i  X )  e.  J }  e.  _V )
8 simpr 447 . . . . . . . 8  |-  ( ( j  =  J  /\  f  =  F )  ->  f  =  F )
98imaeq1d 5011 . . . . . . 7  |-  ( ( j  =  J  /\  f  =  F )  ->  ( f " U. j )  =  ( F " U. j
) )
10 simpl 443 . . . . . . . . . 10  |-  ( ( j  =  J  /\  f  =  F )  ->  j  =  J )
1110unieqd 3838 . . . . . . . . 9  |-  ( ( j  =  J  /\  f  =  F )  ->  U. j  =  U. J )
12 qtopval.1 . . . . . . . . 9  |-  X  = 
U. J
1311, 12syl6eqr 2333 . . . . . . . 8  |-  ( ( j  =  J  /\  f  =  F )  ->  U. j  =  X )
1413imaeq2d 5012 . . . . . . 7  |-  ( ( j  =  J  /\  f  =  F )  ->  ( F " U. j )  =  ( F " X ) )
159, 14eqtrd 2315 . . . . . 6  |-  ( ( j  =  J  /\  f  =  F )  ->  ( f " U. j )  =  ( F " X ) )
1615pweqd 3630 . . . . 5  |-  ( ( j  =  J  /\  f  =  F )  ->  ~P ( f " U. j )  =  ~P ( F " X ) )
178cnveqd 4857 . . . . . . . 8  |-  ( ( j  =  J  /\  f  =  F )  ->  `' f  =  `' F )
1817imaeq1d 5011 . . . . . . 7  |-  ( ( j  =  J  /\  f  =  F )  ->  ( `' f "
s )  =  ( `' F " s ) )
1918, 13ineq12d 3371 . . . . . 6  |-  ( ( j  =  J  /\  f  =  F )  ->  ( ( `' f
" s )  i^i  U. j )  =  ( ( `' F "
s )  i^i  X
) )
2019, 10eleq12d 2351 . . . . 5  |-  ( ( j  =  J  /\  f  =  F )  ->  ( ( ( `' f " s )  i^i  U. j )  e.  j  <->  ( ( `' F " s )  i^i  X )  e.  J ) )
2116, 20rabeqbidv 2783 . . . 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 }
)
22 df-qtop 13410 . . . 4  |- qTop  =  ( j  e.  _V , 
f  e.  _V  |->  { s  e.  ~P (
f " U. j
)  |  ( ( `' f " s
)  i^i  U. j
)  e.  j } )
2321, 22ovmpt2ga 5977 . . 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 }
)
247, 23mpd3an3 1278 . 2  |-  ( ( J  e.  _V  /\  F  e.  _V )  ->  ( J qTop  F )  =  { s  e. 
~P ( F " X )  |  ( ( `' F "
s )  i^i  X
)  e.  J }
)
251, 2, 24syl2an 463 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 358    = wceq 1623    e. wcel 1684   {crab 2547   _Vcvv 2788    i^i cin 3151   ~Pcpw 3625   U.cuni 3827   `'ccnv 4688   "cima 4692  (class class class)co 5858   qTop cqtop 13406
This theorem is referenced by:  qtopval2  17387  qtopres  17389  imastopn  17411
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-qtop 13410
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