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

Theorem qtopres 19962
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 5306 . . . . . . 7  |-  ( ( F  |`  X ) " X )  =  ( F " X )
21pweqi 4014 . . . . . 6  |-  ~P (
( F  |`  X )
" X )  =  ~P ( F " X )
3 rabeq 3107 . . . . . 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 5305 . . . . . . . . . . 11  |-  ( ( F  |`  X )  |`  X )  =  ( F  |`  X )
65cnveqi 5177 . . . . . . . . . 10  |-  `' ( ( F  |`  X )  |`  X )  =  `' ( F  |`  X )
76imaeq1i 5334 . . . . . . . . 9  |-  ( `' ( ( F  |`  X )  |`  X )
" s )  =  ( `' ( F  |`  X ) " s
)
8 cnvresima 5496 . . . . . . . . 9  |-  ( `' ( ( F  |`  X )  |`  X )
" s )  =  ( ( `' ( F  |`  X ) " s )  i^i 
X )
9 cnvresima 5496 . . . . . . . . 9  |-  ( `' ( F  |`  X )
" s )  =  ( ( `' F " s )  i^i  X
)
107, 8, 93eqtr3i 2504 . . . . . . . 8  |-  ( ( `' ( F  |`  X ) " s
)  i^i  X )  =  ( ( `' F " s )  i^i  X )
1110eleq1i 2544 . . . . . . 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 3102 . . . . 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 2497 . . . 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 19959 . . . 4  |-  ( ( J  e.  _V  /\  F  e.  V )  ->  ( J qTop  F )  =  { s  e. 
~P ( F " X )  |  ( ( `' F "
s )  i^i  X
)  e.  J }
)
17 resexg 5316 . . . . 5  |-  ( F  e.  V  ->  ( F  |`  X )  e. 
_V )
1815qtopval 19959 . . . . 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 474 . . . 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 2534 . . 3  |-  ( ( J  e.  _V  /\  F  e.  V )  ->  ( J qTop  F )  =  ( J qTop  ( F  |`  X ) ) )
2120expcom 435 . 2  |-  ( F  e.  V  ->  ( J  e.  _V  ->  ( J qTop  F )  =  ( J qTop  ( F  |`  X ) ) ) )
22 df-qtop 14762 . . . . 5  |- qTop  =  ( j  e.  _V , 
f  e.  _V  |->  { s  e.  ~P (
f " U. j
)  |  ( ( `' f " s
)  i^i  U. j
)  e.  j } )
2322reldmmpt2 6397 . . . 4  |-  Rel  dom qTop
2423ovprc1 6312 . . 3  |-  ( -.  J  e.  _V  ->  ( J qTop  F )  =  (/) )
2523ovprc1 6312 . . 3  |-  ( -.  J  e.  _V  ->  ( J qTop  ( F  |`  X ) )  =  (/) )
2624, 25eqtr4d 2511 . 2  |-  ( -.  J  e.  _V  ->  ( J qTop  F )  =  ( J qTop  ( F  |`  X ) ) )
2721, 26pm2.61d1 159 1  |-  ( F  e.  V  ->  ( J qTop  F )  =  ( J qTop  ( F  |`  X ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767   {crab 2818   _Vcvv 3113    i^i cin 3475   (/)c0 3785   ~Pcpw 4010   U.cuni 4245   `'ccnv 4998    |` cres 5001   "cima 5002  (class class class)co 6284   qTop cqtop 14758
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fv 5596  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-qtop 14762
This theorem is referenced by:  qtoptop2  19963
  Copyright terms: Public domain W3C validator