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

Theorem idqtop 17738
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 5523 . . . . . . 7  |-  `' (  _I  |`  X )  =  (  _I  |`  X )
21imaeq1i 5200 . . . . . 6  |-  ( `' (  _I  |`  X )
" x )  =  ( (  _I  |`  X )
" x )
3 resiima 5220 . . . . . . 7  |-  ( x 
C_  X  ->  (
(  _I  |`  X )
" x )  =  x )
43adantl 453 . . . . . 6  |-  ( ( J  e.  (TopOn `  X )  /\  x  C_  X )  ->  (
(  _I  |`  X )
" x )  =  x )
52, 4syl5eq 2480 . . . . 5  |-  ( ( J  e.  (TopOn `  X )  /\  x  C_  X )  ->  ( `' (  _I  |`  X )
" x )  =  x )
65eleq1d 2502 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  x  C_  X )  ->  (
( `' (  _I  |`  X ) " x
)  e.  J  <->  x  e.  J ) )
76pm5.32da 623 . . 3  |-  ( J  e.  (TopOn `  X
)  ->  ( (
x  C_  X  /\  ( `' (  _I  |`  X )
" x )  e.  J )  <->  ( x  C_  X  /\  x  e.  J ) ) )
8 f1oi 5713 . . . . 5  |-  (  _I  |`  X ) : X -1-1-onto-> X
9 f1ofo 5681 . . . . 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 17735 . . . 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 650 . . 3  |-  ( J  e.  (TopOn `  X
)  ->  ( x  e.  ( J qTop  (  _I  |`  X ) )  <->  ( x  C_  X  /\  ( `' (  _I  |`  X )
" x )  e.  J ) ) )
13 toponss 16994 . . . . 5  |-  ( ( J  e.  (TopOn `  X )  /\  x  e.  J )  ->  x  C_  X )
1413ex 424 . . . 4  |-  ( J  e.  (TopOn `  X
)  ->  ( x  e.  J  ->  x  C_  X ) )
1514pm4.71rd 617 . . 3  |-  ( J  e.  (TopOn `  X
)  ->  ( x  e.  J  <->  ( x  C_  X  /\  x  e.  J
) ) )
167, 12, 153bitr4d 277 . 2  |-  ( J  e.  (TopOn `  X
)  ->  ( x  e.  ( J qTop  (  _I  |`  X ) )  <->  x  e.  J ) )
1716eqrdv 2434 1  |-  ( J  e.  (TopOn `  X
)  ->  ( J qTop  (  _I  |`  X ) )  =  J )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725    C_ wss 3320    _I cid 4493   `'ccnv 4877    |` cres 4880   "cima 4881   -onto->wfo 5452   -1-1-onto->wf1o 5453   ` cfv 5454  (class class class)co 6081   qTop cqtop 13729  TopOnctopon 16959
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 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
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 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-qtop 13733  df-topon 16966
  Copyright terms: Public domain W3C validator