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

Theorem idqtop 19942
Description: The quotient topology induced by the identity function is the original topology. (Contributed by Mario Carneiro, 30-Aug-2015.)
Assertion
Ref Expression
idqtop  |-  ( J  e.  (TopOn `  X
)  ->  ( J qTop  (  _I  |`  X ) )  =  J )

Proof of Theorem idqtop
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 cnvresid 5656 . . . . . . 7  |-  `' (  _I  |`  X )  =  (  _I  |`  X )
21imaeq1i 5332 . . . . . 6  |-  ( `' (  _I  |`  X )
" x )  =  ( (  _I  |`  X )
" x )
3 resiima 5349 . . . . . . 7  |-  ( x 
C_  X  ->  (
(  _I  |`  X )
" x )  =  x )
43adantl 466 . . . . . 6  |-  ( ( J  e.  (TopOn `  X )  /\  x  C_  X )  ->  (
(  _I  |`  X )
" x )  =  x )
52, 4syl5eq 2520 . . . . 5  |-  ( ( J  e.  (TopOn `  X )  /\  x  C_  X )  ->  ( `' (  _I  |`  X )
" x )  =  x )
65eleq1d 2536 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  x  C_  X )  ->  (
( `' (  _I  |`  X ) " x
)  e.  J  <->  x  e.  J ) )
76pm5.32da 641 . . 3  |-  ( J  e.  (TopOn `  X
)  ->  ( (
x  C_  X  /\  ( `' (  _I  |`  X )
" x )  e.  J )  <->  ( x  C_  X  /\  x  e.  J ) ) )
8 f1oi 5849 . . . . 5  |-  (  _I  |`  X ) : X -1-1-onto-> X
9 f1ofo 5821 . . . . 5  |-  ( (  _I  |`  X ) : X -1-1-onto-> X  ->  (  _I  |`  X ) : X -onto-> X )
108, 9mp1i 12 . . . 4  |-  ( J  e.  (TopOn `  X
)  ->  (  _I  |`  X ) : X -onto-> X )
11 elqtop3 19939 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  (  _I  |`  X ) : X -onto-> X )  ->  (
x  e.  ( J qTop  (  _I  |`  X ) )  <->  ( x  C_  X  /\  ( `' (  _I  |`  X ) " x )  e.  J ) ) )
1210, 11mpdan 668 . . 3  |-  ( J  e.  (TopOn `  X
)  ->  ( x  e.  ( J qTop  (  _I  |`  X ) )  <->  ( x  C_  X  /\  ( `' (  _I  |`  X )
" x )  e.  J ) ) )
13 toponss 19197 . . . . 5  |-  ( ( J  e.  (TopOn `  X )  /\  x  e.  J )  ->  x  C_  X )
1413ex 434 . . . 4  |-  ( J  e.  (TopOn `  X
)  ->  ( x  e.  J  ->  x  C_  X ) )
1514pm4.71rd 635 . . 3  |-  ( J  e.  (TopOn `  X
)  ->  ( x  e.  J  <->  ( x  C_  X  /\  x  e.  J
) ) )
167, 12, 153bitr4d 285 . 2  |-  ( J  e.  (TopOn `  X
)  ->  ( x  e.  ( J qTop  (  _I  |`  X ) )  <->  x  e.  J ) )
1716eqrdv 2464 1  |-  ( J  e.  (TopOn `  X
)  ->  ( J qTop  (  _I  |`  X ) )  =  J )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767    C_ wss 3476    _I cid 4790   `'ccnv 4998    |` cres 5001   "cima 5002   -onto->wfo 5584   -1-1-onto->wf1o 5585   ` cfv 5586  (class class class)co 6282   qTop cqtop 14754  TopOnctopon 19162
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-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574
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-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  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-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  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 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-qtop 14758  df-topon 19169
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator