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Theorem qtopomap 17425
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 17422 . . 3  |-  ( ( F  e.  ( J  Cn  K )  /\  K  e.  (TopOn `  Y
)  /\  ran  F  =  Y )  ->  K  C_  ( J qTop  F ) )
51, 2, 3, 4syl3anc 1182 . 2  |-  ( ph  ->  K  C_  ( J qTop  F ) )
6 cntop1 16986 . . . . . . 7  |-  ( F  e.  ( J  Cn  K )  ->  J  e.  Top )
71, 6syl 15 . . . . . 6  |-  ( ph  ->  J  e.  Top )
8 eqid 2296 . . . . . . 7  |-  U. J  =  U. J
98toptopon 16687 . . . . . 6  |-  ( J  e.  Top  <->  J  e.  (TopOn `  U. J ) )
107, 9sylib 188 . . . . 5  |-  ( ph  ->  J  e.  (TopOn `  U. J ) )
11 cnf2 16995 . . . . . . . 8  |-  ( ( J  e.  (TopOn `  U. J )  /\  K  e.  (TopOn `  Y )  /\  F  e.  ( J  Cn  K ) )  ->  F : U. J
--> Y )
1210, 2, 1, 11syl3anc 1182 . . . . . . 7  |-  ( ph  ->  F : U. J --> Y )
13 ffn 5405 . . . . . . 7  |-  ( F : U. J --> Y  ->  F  Fn  U. J )
1412, 13syl 15 . . . . . 6  |-  ( ph  ->  F  Fn  U. J
)
15 df-fo 5277 . . . . . 6  |-  ( F : U. J -onto-> Y  <->  ( F  Fn  U. J  /\  ran  F  =  Y ) )
1614, 3, 15sylanbrc 645 . . . . 5  |-  ( ph  ->  F : U. J -onto-> Y )
17 elqtop3 17410 . . . . 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 642 . . . 4  |-  ( ph  ->  ( y  e.  ( J qTop  F )  <->  ( y  C_  Y  /\  ( `' F " y )  e.  J ) ) )
19 foimacnv 5506 . . . . . . . 8  |-  ( ( F : U. J -onto-> Y  /\  y  C_  Y
)  ->  ( F " ( `' F "
y ) )  =  y )
2016, 19sylan 457 . . . . . . 7  |-  ( (
ph  /\  y  C_  Y )  ->  ( F " ( `' F " y ) )  =  y )
2120adantrr 697 . . . . . 6  |-  ( (
ph  /\  ( y  C_  Y  /\  ( `' F " y )  e.  J ) )  ->  ( F "
( `' F "
y ) )  =  y )
22 simprr 733 . . . . . . 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 2639 . . . . . . . 8  |-  ( ph  ->  A. x  e.  J  ( F " x )  e.  K )
2524adantr 451 . . . . . . 7  |-  ( (
ph  /\  ( y  C_  Y  /\  ( `' F " y )  e.  J ) )  ->  A. x  e.  J  ( F " x )  e.  K )
26 imaeq2 5024 . . . . . . . . 9  |-  ( x  =  ( `' F " y )  ->  ( F " x )  =  ( F " ( `' F " y ) ) )
2726eleq1d 2362 . . . . . . . 8  |-  ( x  =  ( `' F " y )  ->  (
( F " x
)  e.  K  <->  ( F " ( `' F "
y ) )  e.  K ) )
2827rspcv 2893 . . . . . . 7  |-  ( ( `' F " y )  e.  J  ->  ( A. x  e.  J  ( F " x )  e.  K  ->  ( F " ( `' F " y ) )  e.  K ) )
2922, 25, 28sylc 56 . . . . . 6  |-  ( (
ph  /\  ( y  C_  Y  /\  ( `' F " y )  e.  J ) )  ->  ( F "
( `' F "
y ) )  e.  K )
3021, 29eqeltrrd 2371 . . . . 5  |-  ( (
ph  /\  ( y  C_  Y  /\  ( `' F " y )  e.  J ) )  ->  y  e.  K
)
3130ex 423 . . . 4  |-  ( ph  ->  ( ( y  C_  Y  /\  ( `' F " y )  e.  J
)  ->  y  e.  K ) )
3218, 31sylbid 206 . . 3  |-  ( ph  ->  ( y  e.  ( J qTop  F )  -> 
y  e.  K ) )
3332ssrdv 3198 . 2  |-  ( ph  ->  ( J qTop  F ) 
C_  K )
345, 33eqssd 3209 1  |-  ( ph  ->  K  =  ( J qTop 
F ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696   A.wral 2556    C_ wss 3165   U.cuni 3843   `'ccnv 4704   ran crn 4706   "cima 4708    Fn wfn 5266   -->wf 5267   -onto->wfo 5269   ` cfv 5271  (class class class)co 5874   qTop cqtop 13422   Topctop 16647  TopOnctopon 16648    Cn ccn 16970
This theorem is referenced by:  hmeoqtop  17482
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-map 6790  df-qtop 13426  df-top 16652  df-topon 16655  df-cn 16973
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