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Theorem qtopres 17720
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 5170 . . . . . . 7  |-  ( ( F  |`  X ) " X )  =  ( F " X )
21pweqi 3795 . . . . . 6  |-  ~P (
( F  |`  X )
" X )  =  ~P ( F " X )
3 rabeq 2942 . . . . . 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 8 . . . . 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 5169 . . . . . . . . . . 11  |-  ( ( F  |`  X )  |`  X )  =  ( F  |`  X )
65cnveqi 5039 . . . . . . . . . 10  |-  `' ( ( F  |`  X )  |`  X )  =  `' ( F  |`  X )
76imaeq1i 5192 . . . . . . . . 9  |-  ( `' ( ( F  |`  X )  |`  X )
" s )  =  ( `' ( F  |`  X ) " s
)
8 cnvresima 5351 . . . . . . . . 9  |-  ( `' ( ( F  |`  X )  |`  X )
" s )  =  ( ( `' ( F  |`  X ) " s )  i^i 
X )
9 cnvresima 5351 . . . . . . . . 9  |-  ( `' ( F  |`  X )
" s )  =  ( ( `' F " s )  i^i  X
)
107, 8, 93eqtr3i 2463 . . . . . . . 8  |-  ( ( `' ( F  |`  X ) " s
)  i^i  X )  =  ( ( `' F " s )  i^i  X )
1110eleq1i 2498 . . . . . . 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 2938 . . . . 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 2456 . . . 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 17717 . . . 4  |-  ( ( J  e.  _V  /\  F  e.  V )  ->  ( J qTop  F )  =  { s  e. 
~P ( F " X )  |  ( ( `' F "
s )  i^i  X
)  e.  J }
)
17 resexg 5177 . . . . 5  |-  ( F  e.  V  ->  ( F  |`  X )  e. 
_V )
1815qtopval 17717 . . . . 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 461 . . . 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 2493 . . 3  |-  ( ( J  e.  _V  /\  F  e.  V )  ->  ( J qTop  F )  =  ( J qTop  ( F  |`  X ) ) )
2120expcom 425 . 2  |-  ( F  e.  V  ->  ( J  e.  _V  ->  ( J qTop  F )  =  ( J qTop  ( F  |`  X ) ) ) )
22 df-qtop 13723 . . . . 5  |- qTop  =  ( j  e.  _V , 
f  e.  _V  |->  { s  e.  ~P (
f " U. j
)  |  ( ( `' f " s
)  i^i  U. j
)  e.  j } )
2322reldmmpt2 6173 . . . 4  |-  Rel  dom qTop
2423ovprc1 6101 . . 3  |-  ( -.  J  e.  _V  ->  ( J qTop  F )  =  (/) )
2523ovprc1 6101 . . 3  |-  ( -.  J  e.  _V  ->  ( J qTop  ( F  |`  X ) )  =  (/) )
2624, 25eqtr4d 2470 . 2  |-  ( -.  J  e.  _V  ->  ( J qTop  F )  =  ( J qTop  ( F  |`  X ) ) )
2721, 26pm2.61d1 153 1  |-  ( F  e.  V  ->  ( J qTop  F )  =  ( J qTop  ( F  |`  X ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725   {crab 2701   _Vcvv 2948    i^i cin 3311   (/)c0 3620   ~Pcpw 3791   U.cuni 4007   `'ccnv 4869    |` cres 4872   "cima 4873  (class class class)co 6073   qTop cqtop 13719
This theorem is referenced by:  qtoptop2  17721
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-qtop 13723
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