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Theorem idqtop 17397
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 5322 . . . . . . 7  |-  `' (  _I  |`  X )  =  (  _I  |`  X )
21imaeq1i 5009 . . . . . 6  |-  ( `' (  _I  |`  X )
" x )  =  ( (  _I  |`  X )
" x )
3 resiima 5029 . . . . . . 7  |-  ( x 
C_  X  ->  (
(  _I  |`  X )
" x )  =  x )
43adantl 452 . . . . . 6  |-  ( ( J  e.  (TopOn `  X )  /\  x  C_  X )  ->  (
(  _I  |`  X )
" x )  =  x )
52, 4syl5eq 2327 . . . . 5  |-  ( ( J  e.  (TopOn `  X )  /\  x  C_  X )  ->  ( `' (  _I  |`  X )
" x )  =  x )
65eleq1d 2349 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  x  C_  X )  ->  (
( `' (  _I  |`  X ) " x
)  e.  J  <->  x  e.  J ) )
76pm5.32da 622 . . 3  |-  ( J  e.  (TopOn `  X
)  ->  ( (
x  C_  X  /\  ( `' (  _I  |`  X )
" x )  e.  J )  <->  ( x  C_  X  /\  x  e.  J ) ) )
8 f1oi 5511 . . . . 5  |-  (  _I  |`  X ) : X -1-1-onto-> X
9 f1ofo 5479 . . . . 5  |-  ( (  _I  |`  X ) : X -1-1-onto-> X  ->  (  _I  |`  X ) : X -onto-> X )
108, 9mp1i 11 . . . 4  |-  ( J  e.  (TopOn `  X
)  ->  (  _I  |`  X ) : X -onto-> X )
11 elqtop3 17394 . . . 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 649 . . 3  |-  ( J  e.  (TopOn `  X
)  ->  ( x  e.  ( J qTop  (  _I  |`  X ) )  <->  ( x  C_  X  /\  ( `' (  _I  |`  X )
" x )  e.  J ) ) )
13 toponss 16667 . . . . 5  |-  ( ( J  e.  (TopOn `  X )  /\  x  e.  J )  ->  x  C_  X )
1413ex 423 . . . 4  |-  ( J  e.  (TopOn `  X
)  ->  ( x  e.  J  ->  x  C_  X ) )
1514pm4.71rd 616 . . 3  |-  ( J  e.  (TopOn `  X
)  ->  ( x  e.  J  <->  ( x  C_  X  /\  x  e.  J
) ) )
167, 12, 153bitr4d 276 . 2  |-  ( J  e.  (TopOn `  X
)  ->  ( x  e.  ( J qTop  (  _I  |`  X ) )  <->  x  e.  J ) )
1716eqrdv 2281 1  |-  ( J  e.  (TopOn `  X
)  ->  ( J qTop  (  _I  |`  X ) )  =  J )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684    C_ wss 3152    _I cid 4304   `'ccnv 4688    |` cres 4691   "cima 4692   -onto->wfo 5253   -1-1-onto->wf1o 5254   ` cfv 5255  (class class class)co 5858   qTop cqtop 13406  TopOnctopon 16632
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-qtop 13410  df-topon 16639
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