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Theorem qtopomap 20511
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 20508 . . 3  |-  ( ( F  e.  ( J  Cn  K )  /\  K  e.  (TopOn `  Y
)  /\  ran  F  =  Y )  ->  K  C_  ( J qTop  F ) )
51, 2, 3, 4syl3anc 1230 . 2  |-  ( ph  ->  K  C_  ( J qTop  F ) )
6 cntop1 20034 . . . . . . 7  |-  ( F  e.  ( J  Cn  K )  ->  J  e.  Top )
71, 6syl 17 . . . . . 6  |-  ( ph  ->  J  e.  Top )
8 eqid 2402 . . . . . . 7  |-  U. J  =  U. J
98toptopon 19726 . . . . . 6  |-  ( J  e.  Top  <->  J  e.  (TopOn `  U. J ) )
107, 9sylib 196 . . . . 5  |-  ( ph  ->  J  e.  (TopOn `  U. J ) )
11 cnf2 20043 . . . . . . . 8  |-  ( ( J  e.  (TopOn `  U. J )  /\  K  e.  (TopOn `  Y )  /\  F  e.  ( J  Cn  K ) )  ->  F : U. J
--> Y )
1210, 2, 1, 11syl3anc 1230 . . . . . . 7  |-  ( ph  ->  F : U. J --> Y )
13 ffn 5714 . . . . . . 7  |-  ( F : U. J --> Y  ->  F  Fn  U. J )
1412, 13syl 17 . . . . . 6  |-  ( ph  ->  F  Fn  U. J
)
15 df-fo 5575 . . . . . 6  |-  ( F : U. J -onto-> Y  <->  ( F  Fn  U. J  /\  ran  F  =  Y ) )
1614, 3, 15sylanbrc 662 . . . . 5  |-  ( ph  ->  F : U. J -onto-> Y )
17 elqtop3 20496 . . . . 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 659 . . . 4  |-  ( ph  ->  ( y  e.  ( J qTop  F )  <->  ( y  C_  Y  /\  ( `' F " y )  e.  J ) ) )
19 foimacnv 5816 . . . . . . . 8  |-  ( ( F : U. J -onto-> Y  /\  y  C_  Y
)  ->  ( F " ( `' F "
y ) )  =  y )
2016, 19sylan 469 . . . . . . 7  |-  ( (
ph  /\  y  C_  Y )  ->  ( F " ( `' F " y ) )  =  y )
2120adantrr 715 . . . . . 6  |-  ( (
ph  /\  ( y  C_  Y  /\  ( `' F " y )  e.  J ) )  ->  ( F "
( `' F "
y ) )  =  y )
22 simprr 758 . . . . . . 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 2818 . . . . . . . 8  |-  ( ph  ->  A. x  e.  J  ( F " x )  e.  K )
2524adantr 463 . . . . . . 7  |-  ( (
ph  /\  ( y  C_  Y  /\  ( `' F " y )  e.  J ) )  ->  A. x  e.  J  ( F " x )  e.  K )
26 imaeq2 5153 . . . . . . . . 9  |-  ( x  =  ( `' F " y )  ->  ( F " x )  =  ( F " ( `' F " y ) ) )
2726eleq1d 2471 . . . . . . . 8  |-  ( x  =  ( `' F " y )  ->  (
( F " x
)  e.  K  <->  ( F " ( `' F "
y ) )  e.  K ) )
2827rspcv 3156 . . . . . . 7  |-  ( ( `' F " y )  e.  J  ->  ( A. x  e.  J  ( F " x )  e.  K  ->  ( F " ( `' F " y ) )  e.  K ) )
2922, 25, 28sylc 59 . . . . . 6  |-  ( (
ph  /\  ( y  C_  Y  /\  ( `' F " y )  e.  J ) )  ->  ( F "
( `' F "
y ) )  e.  K )
3021, 29eqeltrrd 2491 . . . . 5  |-  ( (
ph  /\  ( y  C_  Y  /\  ( `' F " y )  e.  J ) )  ->  y  e.  K
)
3130ex 432 . . . 4  |-  ( ph  ->  ( ( y  C_  Y  /\  ( `' F " y )  e.  J
)  ->  y  e.  K ) )
3218, 31sylbid 215 . . 3  |-  ( ph  ->  ( y  e.  ( J qTop  F )  -> 
y  e.  K ) )
3332ssrdv 3448 . 2  |-  ( ph  ->  ( J qTop  F ) 
C_  K )
345, 33eqssd 3459 1  |-  ( ph  ->  K  =  ( J qTop 
F ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    = wceq 1405    e. wcel 1842   A.wral 2754    C_ wss 3414   U.cuni 4191   `'ccnv 4822   ran crn 4824   "cima 4826    Fn wfn 5564   -->wf 5565   -onto->wfo 5567   ` cfv 5569  (class class class)co 6278   qTop cqtop 15117   Topctop 19686  TopOnctopon 19687    Cn ccn 20018
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4507  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2759  df-rex 2760  df-reu 2761  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4192  df-iun 4273  df-br 4396  df-opab 4454  df-mpt 4455  df-id 4738  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-map 7459  df-qtop 15121  df-top 19691  df-topon 19694  df-cn 20021
This theorem is referenced by:  hmeoqtop  20568
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