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Theorem qtopres 20644
Description: The quotient topology is unaffected by restriction to the base set. This property makes it slightly more convenient to use, since we don't have to require that  F be a function with domain  X. (Contributed by Mario Carneiro, 23-Mar-2015.)
Hypothesis
Ref Expression
qtopval.1  |-  X  = 
U. J
Assertion
Ref Expression
qtopres  |-  ( F  e.  V  ->  ( J qTop  F )  =  ( J qTop  ( F  |`  X ) ) )

Proof of Theorem qtopres
Dummy variables  s 
f  j are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 resima 5157 . . . . . . 7  |-  ( ( F  |`  X ) " X )  =  ( F " X )
21pweqi 3989 . . . . . 6  |-  ~P (
( F  |`  X )
" X )  =  ~P ( F " X )
3 rabeq 3081 . . . . . 6  |-  ( ~P ( ( F  |`  X ) " X
)  =  ~P ( F " X )  ->  { s  e.  ~P ( ( F  |`  X ) " X
)  |  ( ( `' ( F  |`  X ) " s
)  i^i  X )  e.  J }  =  {
s  e.  ~P ( F " X )  |  ( ( `' ( F  |`  X ) " s )  i^i 
X )  e.  J } )
42, 3ax-mp 5 . . . . 5  |-  { s  e.  ~P ( ( F  |`  X ) " X )  |  ( ( `' ( F  |`  X ) " s
)  i^i  X )  e.  J }  =  {
s  e.  ~P ( F " X )  |  ( ( `' ( F  |`  X ) " s )  i^i 
X )  e.  J }
5 residm 5156 . . . . . . . . . . 11  |-  ( ( F  |`  X )  |`  X )  =  ( F  |`  X )
65cnveqi 5029 . . . . . . . . . 10  |-  `' ( ( F  |`  X )  |`  X )  =  `' ( F  |`  X )
76imaeq1i 5185 . . . . . . . . 9  |-  ( `' ( ( F  |`  X )  |`  X )
" s )  =  ( `' ( F  |`  X ) " s
)
8 cnvresima 5344 . . . . . . . . 9  |-  ( `' ( ( F  |`  X )  |`  X )
" s )  =  ( ( `' ( F  |`  X ) " s )  i^i 
X )
9 cnvresima 5344 . . . . . . . . 9  |-  ( `' ( F  |`  X )
" s )  =  ( ( `' F " s )  i^i  X
)
107, 8, 93eqtr3i 2466 . . . . . . . 8  |-  ( ( `' ( F  |`  X ) " s
)  i^i  X )  =  ( ( `' F " s )  i^i  X )
1110eleq1i 2506 . . . . . . 7  |-  ( ( ( `' ( F  |`  X ) " s
)  i^i  X )  e.  J  <->  ( ( `' F " s )  i^i  X )  e.  J )
1211a1i 11 . . . . . 6  |-  ( s  e.  ~P ( F
" X )  -> 
( ( ( `' ( F  |`  X )
" s )  i^i 
X )  e.  J  <->  ( ( `' F "
s )  i^i  X
)  e.  J ) )
1312rabbiia 3076 . . . . 5  |-  { s  e.  ~P ( F
" X )  |  ( ( `' ( F  |`  X ) " s )  i^i 
X )  e.  J }  =  { s  e.  ~P ( F " X )  |  ( ( `' F "
s )  i^i  X
)  e.  J }
144, 13eqtr2i 2459 . . . 4  |-  { s  e.  ~P ( F
" X )  |  ( ( `' F " s )  i^i  X
)  e.  J }  =  { s  e.  ~P ( ( F  |`  X ) " X
)  |  ( ( `' ( F  |`  X ) " s
)  i^i  X )  e.  J }
15 qtopval.1 . . . . 5  |-  X  = 
U. J
1615qtopval 20641 . . . 4  |-  ( ( J  e.  _V  /\  F  e.  V )  ->  ( J qTop  F )  =  { s  e. 
~P ( F " X )  |  ( ( `' F "
s )  i^i  X
)  e.  J }
)
17 resexg 5167 . . . . 5  |-  ( F  e.  V  ->  ( F  |`  X )  e. 
_V )
1815qtopval 20641 . . . . 5  |-  ( ( J  e.  _V  /\  ( F  |`  X )  e.  _V )  -> 
( J qTop  ( F  |`  X ) )  =  { s  e.  ~P ( ( F  |`  X ) " X
)  |  ( ( `' ( F  |`  X ) " s
)  i^i  X )  e.  J } )
1917, 18sylan2 476 . . . 4  |-  ( ( J  e.  _V  /\  F  e.  V )  ->  ( J qTop  ( F  |`  X ) )  =  { s  e.  ~P ( ( F  |`  X ) " X
)  |  ( ( `' ( F  |`  X ) " s
)  i^i  X )  e.  J } )
2014, 16, 193eqtr4a 2496 . . 3  |-  ( ( J  e.  _V  /\  F  e.  V )  ->  ( J qTop  F )  =  ( J qTop  ( F  |`  X ) ) )
2120expcom 436 . 2  |-  ( F  e.  V  ->  ( J  e.  _V  ->  ( J qTop  F )  =  ( J qTop  ( F  |`  X ) ) ) )
22 df-qtop 15364 . . . . 5  |- qTop  =  ( j  e.  _V , 
f  e.  _V  |->  { s  e.  ~P (
f " U. j
)  |  ( ( `' f " s
)  i^i  U. j
)  e.  j } )
2322reldmmpt2 6421 . . . 4  |-  Rel  dom qTop
2423ovprc1 6336 . . 3  |-  ( -.  J  e.  _V  ->  ( J qTop  F )  =  (/) )
2523ovprc1 6336 . . 3  |-  ( -.  J  e.  _V  ->  ( J qTop  ( F  |`  X ) )  =  (/) )
2624, 25eqtr4d 2473 . 2  |-  ( -.  J  e.  _V  ->  ( J qTop  F )  =  ( J qTop  ( F  |`  X ) ) )
2721, 26pm2.61d1 162 1  |-  ( F  e.  V  ->  ( J qTop  F )  =  ( J qTop  ( F  |`  X ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 187    /\ wa 370    = wceq 1437    e. wcel 1870   {crab 2786   _Vcvv 3087    i^i cin 3441   (/)c0 3767   ~Pcpw 3985   U.cuni 4222   `'ccnv 4853    |` cres 4856   "cima 4857  (class class class)co 6305   qTop cqtop 15360
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-rab 2791  df-v 3089  df-sbc 3306  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-br 4427  df-opab 4485  df-id 4769  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-iota 5565  df-fun 5603  df-fv 5609  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-qtop 15364
This theorem is referenced by:  qtoptop2  20645
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