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Theorem idqtop 20292
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 5566 . . . . . . 7  |-  `' (  _I  |`  X )  =  (  _I  |`  X )
21imaeq1i 5246 . . . . . 6  |-  ( `' (  _I  |`  X )
" x )  =  ( (  _I  |`  X )
" x )
3 resiima 5263 . . . . . . 7  |-  ( x 
C_  X  ->  (
(  _I  |`  X )
" x )  =  x )
43adantl 464 . . . . . 6  |-  ( ( J  e.  (TopOn `  X )  /\  x  C_  X )  ->  (
(  _I  |`  X )
" x )  =  x )
52, 4syl5eq 2435 . . . . 5  |-  ( ( J  e.  (TopOn `  X )  /\  x  C_  X )  ->  ( `' (  _I  |`  X )
" x )  =  x )
65eleq1d 2451 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  x  C_  X )  ->  (
( `' (  _I  |`  X ) " x
)  e.  J  <->  x  e.  J ) )
76pm5.32da 639 . . 3  |-  ( J  e.  (TopOn `  X
)  ->  ( (
x  C_  X  /\  ( `' (  _I  |`  X )
" x )  e.  J )  <->  ( x  C_  X  /\  x  e.  J ) ) )
8 f1oi 5759 . . . . 5  |-  (  _I  |`  X ) : X -1-1-onto-> X
9 f1ofo 5731 . . . . 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 20289 . . . 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 666 . . 3  |-  ( J  e.  (TopOn `  X
)  ->  ( x  e.  ( J qTop  (  _I  |`  X ) )  <->  ( x  C_  X  /\  ( `' (  _I  |`  X )
" x )  e.  J ) ) )
13 toponss 19515 . . . . 5  |-  ( ( J  e.  (TopOn `  X )  /\  x  e.  J )  ->  x  C_  X )
1413ex 432 . . . 4  |-  ( J  e.  (TopOn `  X
)  ->  ( x  e.  J  ->  x  C_  X ) )
1514pm4.71rd 633 . . 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 2379 1  |-  ( J  e.  (TopOn `  X
)  ->  ( J qTop  (  _I  |`  X ) )  =  J )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    = wceq 1399    e. wcel 1826    C_ wss 3389    _I cid 4704   `'ccnv 4912    |` cres 4915   "cima 4916   -onto->wfo 5494   -1-1-onto->wf1o 5495   ` cfv 5496  (class class class)co 6196   qTop cqtop 14910  TopOnctopon 19480
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-rep 4478  ax-sep 4488  ax-nul 4496  ax-pow 4543  ax-pr 4601  ax-un 6491
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-ral 2737  df-rex 2738  df-reu 2739  df-rab 2741  df-v 3036  df-sbc 3253  df-csb 3349  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-nul 3712  df-if 3858  df-pw 3929  df-sn 3945  df-pr 3947  df-op 3951  df-uni 4164  df-iun 4245  df-br 4368  df-opab 4426  df-mpt 4427  df-id 4709  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5460  df-fun 5498  df-fn 5499  df-f 5500  df-f1 5501  df-fo 5502  df-f1o 5503  df-fv 5504  df-ov 6199  df-oprab 6200  df-mpt2 6201  df-qtop 14914  df-topon 19487
This theorem is referenced by: (None)
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