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Theorem qtopres 20706
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 5136 . . . . . . 7  |-  ( ( F  |`  X ) " X )  =  ( F " X )
21pweqi 3954 . . . . . 6  |-  ~P (
( F  |`  X )
" X )  =  ~P ( F " X )
3 rabeq 3037 . . . . . 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 5135 . . . . . . . . . . 11  |-  ( ( F  |`  X )  |`  X )  =  ( F  |`  X )
65cnveqi 5008 . . . . . . . . . 10  |-  `' ( ( F  |`  X )  |`  X )  =  `' ( F  |`  X )
76imaeq1i 5164 . . . . . . . . 9  |-  ( `' ( ( F  |`  X )  |`  X )
" s )  =  ( `' ( F  |`  X ) " s
)
8 cnvresima 5323 . . . . . . . . 9  |-  ( `' ( ( F  |`  X )  |`  X )
" s )  =  ( ( `' ( F  |`  X ) " s )  i^i 
X )
9 cnvresima 5323 . . . . . . . . 9  |-  ( `' ( F  |`  X )
" s )  =  ( ( `' F " s )  i^i  X
)
107, 8, 93eqtr3i 2480 . . . . . . . 8  |-  ( ( `' ( F  |`  X ) " s
)  i^i  X )  =  ( ( `' F " s )  i^i  X )
1110eleq1i 2519 . . . . . . 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 3032 . . . . 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 2473 . . . 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 20703 . . . 4  |-  ( ( J  e.  _V  /\  F  e.  V )  ->  ( J qTop  F )  =  { s  e. 
~P ( F " X )  |  ( ( `' F "
s )  i^i  X
)  e.  J }
)
17 resexg 5146 . . . . 5  |-  ( F  e.  V  ->  ( F  |`  X )  e. 
_V )
1815qtopval 20703 . . . . 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 477 . . . 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 2510 . . 3  |-  ( ( J  e.  _V  /\  F  e.  V )  ->  ( J qTop  F )  =  ( J qTop  ( F  |`  X ) ) )
2120expcom 437 . 2  |-  ( F  e.  V  ->  ( J  e.  _V  ->  ( J qTop  F )  =  ( J qTop  ( F  |`  X ) ) ) )
22 df-qtop 15399 . . . . 5  |- qTop  =  ( j  e.  _V , 
f  e.  _V  |->  { s  e.  ~P (
f " U. j
)  |  ( ( `' f " s
)  i^i  U. j
)  e.  j } )
2322reldmmpt2 6404 . . . 4  |-  Rel  dom qTop
2423ovprc1 6319 . . 3  |-  ( -.  J  e.  _V  ->  ( J qTop  F )  =  (/) )
2523ovprc1 6319 . . 3  |-  ( -.  J  e.  _V  ->  ( J qTop  ( F  |`  X ) )  =  (/) )
2624, 25eqtr4d 2487 . 2  |-  ( -.  J  e.  _V  ->  ( J qTop  F )  =  ( J qTop  ( F  |`  X ) ) )
2721, 26pm2.61d1 163 1  |-  ( F  e.  V  ->  ( J qTop  F )  =  ( J qTop  ( F  |`  X ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 188    /\ wa 371    = wceq 1443    e. wcel 1886   {crab 2740   _Vcvv 3044    i^i cin 3402   (/)c0 3730   ~Pcpw 3950   U.cuni 4197   `'ccnv 4832    |` cres 4835   "cima 4836  (class class class)co 6288   qTop cqtop 15394
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-8 1888  ax-9 1895  ax-10 1914  ax-11 1919  ax-12 1932  ax-13 2090  ax-ext 2430  ax-sep 4524  ax-nul 4533  ax-pow 4580  ax-pr 4638  ax-un 6580
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 986  df-tru 1446  df-ex 1663  df-nf 1667  df-sb 1797  df-eu 2302  df-mo 2303  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2580  df-ne 2623  df-ral 2741  df-rex 2742  df-rab 2745  df-v 3046  df-sbc 3267  df-dif 3406  df-un 3408  df-in 3410  df-ss 3417  df-nul 3731  df-if 3881  df-pw 3952  df-sn 3968  df-pr 3970  df-op 3974  df-uni 4198  df-br 4402  df-opab 4461  df-id 4748  df-xp 4839  df-rel 4840  df-cnv 4841  df-co 4842  df-dm 4843  df-rn 4844  df-res 4845  df-ima 4846  df-iota 5545  df-fun 5583  df-fv 5589  df-ov 6291  df-oprab 6292  df-mpt2 6293  df-qtop 15399
This theorem is referenced by:  qtoptop2  20707
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