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Theorem qtopomap 17672
Description: If  F is a surjective continuous open map, then it is a quotient map. (An open map is a function that maps open sets to open sets.) (Contributed by Mario Carneiro, 24-Mar-2015.)
Hypotheses
Ref Expression
qtopomap.4  |-  ( ph  ->  K  e.  (TopOn `  Y ) )
qtopomap.5  |-  ( ph  ->  F  e.  ( J  Cn  K ) )
qtopomap.6  |-  ( ph  ->  ran  F  =  Y )
qtopomap.7  |-  ( (
ph  /\  x  e.  J )  ->  ( F " x )  e.  K )
Assertion
Ref Expression
qtopomap  |-  ( ph  ->  K  =  ( J qTop 
F ) )
Distinct variable groups:    x, F    x, J    x, K    ph, x    x, Y

Proof of Theorem qtopomap
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 qtopomap.5 . . 3  |-  ( ph  ->  F  e.  ( J  Cn  K ) )
2 qtopomap.4 . . 3  |-  ( ph  ->  K  e.  (TopOn `  Y ) )
3 qtopomap.6 . . 3  |-  ( ph  ->  ran  F  =  Y )
4 qtopss 17669 . . 3  |-  ( ( F  e.  ( J  Cn  K )  /\  K  e.  (TopOn `  Y
)  /\  ran  F  =  Y )  ->  K  C_  ( J qTop  F ) )
51, 2, 3, 4syl3anc 1184 . 2  |-  ( ph  ->  K  C_  ( J qTop  F ) )
6 cntop1 17227 . . . . . . 7  |-  ( F  e.  ( J  Cn  K )  ->  J  e.  Top )
71, 6syl 16 . . . . . 6  |-  ( ph  ->  J  e.  Top )
8 eqid 2388 . . . . . . 7  |-  U. J  =  U. J
98toptopon 16922 . . . . . 6  |-  ( J  e.  Top  <->  J  e.  (TopOn `  U. J ) )
107, 9sylib 189 . . . . 5  |-  ( ph  ->  J  e.  (TopOn `  U. J ) )
11 cnf2 17236 . . . . . . . 8  |-  ( ( J  e.  (TopOn `  U. J )  /\  K  e.  (TopOn `  Y )  /\  F  e.  ( J  Cn  K ) )  ->  F : U. J
--> Y )
1210, 2, 1, 11syl3anc 1184 . . . . . . 7  |-  ( ph  ->  F : U. J --> Y )
13 ffn 5532 . . . . . . 7  |-  ( F : U. J --> Y  ->  F  Fn  U. J )
1412, 13syl 16 . . . . . 6  |-  ( ph  ->  F  Fn  U. J
)
15 df-fo 5401 . . . . . 6  |-  ( F : U. J -onto-> Y  <->  ( F  Fn  U. J  /\  ran  F  =  Y ) )
1614, 3, 15sylanbrc 646 . . . . 5  |-  ( ph  ->  F : U. J -onto-> Y )
17 elqtop3 17657 . . . . 5  |-  ( ( J  e.  (TopOn `  U. J )  /\  F : U. J -onto-> Y )  ->  ( y  e.  ( J qTop  F )  <-> 
( y  C_  Y  /\  ( `' F "
y )  e.  J
) ) )
1810, 16, 17syl2anc 643 . . . 4  |-  ( ph  ->  ( y  e.  ( J qTop  F )  <->  ( y  C_  Y  /\  ( `' F " y )  e.  J ) ) )
19 foimacnv 5633 . . . . . . . 8  |-  ( ( F : U. J -onto-> Y  /\  y  C_  Y
)  ->  ( F " ( `' F "
y ) )  =  y )
2016, 19sylan 458 . . . . . . 7  |-  ( (
ph  /\  y  C_  Y )  ->  ( F " ( `' F " y ) )  =  y )
2120adantrr 698 . . . . . 6  |-  ( (
ph  /\  ( y  C_  Y  /\  ( `' F " y )  e.  J ) )  ->  ( F "
( `' F "
y ) )  =  y )
22 simprr 734 . . . . . . 7  |-  ( (
ph  /\  ( y  C_  Y  /\  ( `' F " y )  e.  J ) )  ->  ( `' F " y )  e.  J
)
23 qtopomap.7 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  J )  ->  ( F " x )  e.  K )
2423ralrimiva 2733 . . . . . . . 8  |-  ( ph  ->  A. x  e.  J  ( F " x )  e.  K )
2524adantr 452 . . . . . . 7  |-  ( (
ph  /\  ( y  C_  Y  /\  ( `' F " y )  e.  J ) )  ->  A. x  e.  J  ( F " x )  e.  K )
26 imaeq2 5140 . . . . . . . . 9  |-  ( x  =  ( `' F " y )  ->  ( F " x )  =  ( F " ( `' F " y ) ) )
2726eleq1d 2454 . . . . . . . 8  |-  ( x  =  ( `' F " y )  ->  (
( F " x
)  e.  K  <->  ( F " ( `' F "
y ) )  e.  K ) )
2827rspcv 2992 . . . . . . 7  |-  ( ( `' F " y )  e.  J  ->  ( A. x  e.  J  ( F " x )  e.  K  ->  ( F " ( `' F " y ) )  e.  K ) )
2922, 25, 28sylc 58 . . . . . 6  |-  ( (
ph  /\  ( y  C_  Y  /\  ( `' F " y )  e.  J ) )  ->  ( F "
( `' F "
y ) )  e.  K )
3021, 29eqeltrrd 2463 . . . . 5  |-  ( (
ph  /\  ( y  C_  Y  /\  ( `' F " y )  e.  J ) )  ->  y  e.  K
)
3130ex 424 . . . 4  |-  ( ph  ->  ( ( y  C_  Y  /\  ( `' F " y )  e.  J
)  ->  y  e.  K ) )
3218, 31sylbid 207 . . 3  |-  ( ph  ->  ( y  e.  ( J qTop  F )  -> 
y  e.  K ) )
3332ssrdv 3298 . 2  |-  ( ph  ->  ( J qTop  F ) 
C_  K )
345, 33eqssd 3309 1  |-  ( ph  ->  K  =  ( J qTop 
F ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1717   A.wral 2650    C_ wss 3264   U.cuni 3958   `'ccnv 4818   ran crn 4820   "cima 4822    Fn wfn 5390   -->wf 5391   -onto->wfo 5393   ` cfv 5395  (class class class)co 6021   qTop cqtop 13657   Topctop 16882  TopOnctopon 16883    Cn ccn 17211
This theorem is referenced by:  hmeoqtop  17729
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-rep 4262  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-ral 2655  df-rex 2656  df-reu 2657  df-rab 2659  df-v 2902  df-sbc 3106  df-csb 3196  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-op 3767  df-uni 3959  df-iun 4038  df-br 4155  df-opab 4209  df-mpt 4210  df-id 4440  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-f1 5400  df-fo 5401  df-f1o 5402  df-fv 5403  df-ov 6024  df-oprab 6025  df-mpt2 6026  df-map 6957  df-qtop 13661  df-top 16887  df-topon 16890  df-cn 17214
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