MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  qtopval Unicode version

Theorem qtopval 17688
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 2932 . 2  |-  ( J  e.  V  ->  J  e.  _V )
2 elex 2932 . 2  |-  ( F  e.  W  ->  F  e.  _V )
3 imaexg 5184 . . . . 5  |-  ( F  e.  _V  ->  ( F " X )  e. 
_V )
4 pwexg 4351 . . . . 5  |-  ( ( F " X )  e.  _V  ->  ~P ( F " X )  e.  _V )
5 rabexg 4321 . . . . 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 3994 . . . . . . . 8  |-  ( ( j  =  J  /\  f  =  F )  ->  U. j  =  U. J )
11 qtopval.1 . . . . . . . 8  |-  X  = 
U. J
1210, 11syl6eqr 2462 . . . . . . 7  |-  ( ( j  =  J  /\  f  =  F )  ->  U. j  =  X )
138, 12imaeq12d 5171 . . . . . 6  |-  ( ( j  =  J  /\  f  =  F )  ->  ( f " U. j )  =  ( F " X ) )
1413pweqd 3772 . . . . 5  |-  ( ( j  =  J  /\  f  =  F )  ->  ~P ( f " U. j )  =  ~P ( F " X ) )
158cnveqd 5015 . . . . . . . 8  |-  ( ( j  =  J  /\  f  =  F )  ->  `' f  =  `' F )
1615imaeq1d 5169 . . . . . . 7  |-  ( ( j  =  J  /\  f  =  F )  ->  ( `' f "
s )  =  ( `' F " s ) )
1716, 12ineq12d 3511 . . . . . 6  |-  ( ( j  =  J  /\  f  =  F )  ->  ( ( `' f
" s )  i^i  U. j )  =  ( ( `' F "
s )  i^i  X
) )
1817, 9eleq12d 2480 . . . . 5  |-  ( ( j  =  J  /\  f  =  F )  ->  ( ( ( `' f " s )  i^i  U. j )  e.  j  <->  ( ( `' F " s )  i^i  X )  e.  J ) )
1914, 18rabeqbidv 2919 . . . 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 13696 . . . 4  |- qTop  =  ( j  e.  _V , 
f  e.  _V  |->  { s  e.  ~P (
f " U. j
)  |  ( ( `' f " s
)  i^i  U. j
)  e.  j } )
2119, 20ovmpt2ga 6170 . . 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 2678   _Vcvv 2924    i^i cin 3287   ~Pcpw 3767   U.cuni 3983   `'ccnv 4844   "cima 4848  (class class class)co 6048   qTop cqtop 13692
This theorem is referenced by:  qtopval2  17689  qtopres  17691  imastopn  17713
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 2393  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668
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 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-ral 2679  df-rex 2680  df-rab 2683  df-v 2926  df-sbc 3130  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-op 3791  df-uni 3984  df-br 4181  df-opab 4235  df-id 4466  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fv 5429  df-ov 6051  df-oprab 6052  df-mpt2 6053  df-qtop 13696
  Copyright terms: Public domain W3C validator