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Theorem qtopres 20790
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 5143 . . . . . . 7  |-  ( ( F  |`  X ) " X )  =  ( F " X )
21pweqi 3946 . . . . . 6  |-  ~P (
( F  |`  X )
" X )  =  ~P ( F " X )
3 rabeq 3024 . . . . . 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 5142 . . . . . . . . . . 11  |-  ( ( F  |`  X )  |`  X )  =  ( F  |`  X )
65cnveqi 5014 . . . . . . . . . 10  |-  `' ( ( F  |`  X )  |`  X )  =  `' ( F  |`  X )
76imaeq1i 5171 . . . . . . . . 9  |-  ( `' ( ( F  |`  X )  |`  X )
" s )  =  ( `' ( F  |`  X ) " s
)
8 cnvresima 5331 . . . . . . . . 9  |-  ( `' ( ( F  |`  X )  |`  X )
" s )  =  ( ( `' ( F  |`  X ) " s )  i^i 
X )
9 cnvresima 5331 . . . . . . . . 9  |-  ( `' ( F  |`  X )
" s )  =  ( ( `' F " s )  i^i  X
)
107, 8, 93eqtr3i 2501 . . . . . . . 8  |-  ( ( `' ( F  |`  X ) " s
)  i^i  X )  =  ( ( `' F " s )  i^i  X )
1110eleq1i 2540 . . . . . . 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 3019 . . . . 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 2494 . . . 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 20787 . . . 4  |-  ( ( J  e.  _V  /\  F  e.  V )  ->  ( J qTop  F )  =  { s  e. 
~P ( F " X )  |  ( ( `' F "
s )  i^i  X
)  e.  J }
)
17 resexg 5153 . . . . 5  |-  ( F  e.  V  ->  ( F  |`  X )  e. 
_V )
1815qtopval 20787 . . . . 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 482 . . . 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 2531 . . 3  |-  ( ( J  e.  _V  /\  F  e.  V )  ->  ( J qTop  F )  =  ( J qTop  ( F  |`  X ) ) )
2120expcom 442 . 2  |-  ( F  e.  V  ->  ( J  e.  _V  ->  ( J qTop  F )  =  ( J qTop  ( F  |`  X ) ) ) )
22 df-qtop 15484 . . . . 5  |- qTop  =  ( j  e.  _V , 
f  e.  _V  |->  { s  e.  ~P (
f " U. j
)  |  ( ( `' f " s
)  i^i  U. j
)  e.  j } )
2322reldmmpt2 6426 . . . 4  |-  Rel  dom qTop
2423ovprc1 6339 . . 3  |-  ( -.  J  e.  _V  ->  ( J qTop  F )  =  (/) )
2523ovprc1 6339 . . 3  |-  ( -.  J  e.  _V  ->  ( J qTop  ( F  |`  X ) )  =  (/) )
2624, 25eqtr4d 2508 . 2  |-  ( -.  J  e.  _V  ->  ( J qTop  F )  =  ( J qTop  ( F  |`  X ) ) )
2721, 26pm2.61d1 164 1  |-  ( F  e.  V  ->  ( J qTop  F )  =  ( J qTop  ( F  |`  X ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 189    /\ wa 376    = wceq 1452    e. wcel 1904   {crab 2760   _Vcvv 3031    i^i cin 3389   (/)c0 3722   ~Pcpw 3942   U.cuni 4190   `'ccnv 4838    |` cres 4841   "cima 4842  (class class class)co 6308   qTop cqtop 15479
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-sbc 3256  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-br 4396  df-opab 4455  df-id 4754  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5553  df-fun 5591  df-fv 5597  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-qtop 15484
This theorem is referenced by:  qtoptop2  20791
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