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Theorem qtopval 19112
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 2973 . 2  |-  ( J  e.  V  ->  J  e.  _V )
2 elex 2973 . 2  |-  ( F  e.  W  ->  F  e.  _V )
3 imaexg 6506 . . . . 5  |-  ( F  e.  _V  ->  ( F " X )  e. 
_V )
4 pwexg 4466 . . . . 5  |-  ( ( F " X )  e.  _V  ->  ~P ( F " X )  e.  _V )
5 rabexg 4432 . . . . 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 463 . . 3  |-  ( ( J  e.  _V  /\  F  e.  _V )  ->  { s  e.  ~P ( F " X )  |  ( ( `' F " s )  i^i  X )  e.  J }  e.  _V )
8 simpr 458 . . . . . . 7  |-  ( ( j  =  J  /\  f  =  F )  ->  f  =  F )
9 simpl 454 . . . . . . . . 9  |-  ( ( j  =  J  /\  f  =  F )  ->  j  =  J )
109unieqd 4091 . . . . . . . 8  |-  ( ( j  =  J  /\  f  =  F )  ->  U. j  =  U. J )
11 qtopval.1 . . . . . . . 8  |-  X  = 
U. J
1210, 11syl6eqr 2485 . . . . . . 7  |-  ( ( j  =  J  /\  f  =  F )  ->  U. j  =  X )
138, 12imaeq12d 5160 . . . . . 6  |-  ( ( j  =  J  /\  f  =  F )  ->  ( f " U. j )  =  ( F " X ) )
1413pweqd 3855 . . . . 5  |-  ( ( j  =  J  /\  f  =  F )  ->  ~P ( f " U. j )  =  ~P ( F " X ) )
158cnveqd 5004 . . . . . . . 8  |-  ( ( j  =  J  /\  f  =  F )  ->  `' f  =  `' F )
1615imaeq1d 5158 . . . . . . 7  |-  ( ( j  =  J  /\  f  =  F )  ->  ( `' f "
s )  =  ( `' F " s ) )
1716, 12ineq12d 3543 . . . . . 6  |-  ( ( j  =  J  /\  f  =  F )  ->  ( ( `' f
" s )  i^i  U. j )  =  ( ( `' F "
s )  i^i  X
) )
1817, 9eleq12d 2503 . . . . 5  |-  ( ( j  =  J  /\  f  =  F )  ->  ( ( ( `' f " s )  i^i  U. j )  e.  j  <->  ( ( `' F " s )  i^i  X )  e.  J ) )
1914, 18rabeqbidv 2959 . . . 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 14430 . . . 4  |- qTop  =  ( j  e.  _V , 
f  e.  _V  |->  { s  e.  ~P (
f " U. j
)  |  ( ( `' f " s
)  i^i  U. j
)  e.  j } )
2119, 20ovmpt2ga 6211 . . 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 1310 . 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 474 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 1364    e. wcel 1757   {crab 2711   _Vcvv 2964    i^i cin 3317   ~Pcpw 3850   U.cuni 4081   `'ccnv 4828   "cima 4832  (class class class)co 6082   qTop cqtop 14426
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1671  ax-6 1709  ax-7 1729  ax-8 1759  ax-9 1761  ax-10 1776  ax-11 1781  ax-12 1793  ax-13 1944  ax-ext 2416  ax-sep 4403  ax-nul 4411  ax-pow 4460  ax-pr 4521  ax-un 6363
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1702  df-eu 2260  df-mo 2261  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2966  df-sbc 3178  df-dif 3321  df-un 3323  df-in 3325  df-ss 3332  df-nul 3628  df-if 3782  df-pw 3852  df-sn 3868  df-pr 3870  df-op 3874  df-uni 4082  df-br 4283  df-opab 4341  df-id 4625  df-xp 4835  df-rel 4836  df-cnv 4837  df-co 4838  df-dm 4839  df-rn 4840  df-res 4841  df-ima 4842  df-iota 5371  df-fun 5410  df-fv 5416  df-ov 6085  df-oprab 6086  df-mpt2 6087  df-qtop 14430
This theorem is referenced by:  qtopval2  19113  qtopres  19115  imastopn  19137
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