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Theorem qtopomap 17742
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 17739 . . 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 17296 . . . . . . 7  |-  ( F  e.  ( J  Cn  K )  ->  J  e.  Top )
71, 6syl 16 . . . . . 6  |-  ( ph  ->  J  e.  Top )
8 eqid 2435 . . . . . . 7  |-  U. J  =  U. J
98toptopon 16990 . . . . . 6  |-  ( J  e.  Top  <->  J  e.  (TopOn `  U. J ) )
107, 9sylib 189 . . . . 5  |-  ( ph  ->  J  e.  (TopOn `  U. J ) )
11 cnf2 17305 . . . . . . . 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 5583 . . . . . . 7  |-  ( F : U. J --> Y  ->  F  Fn  U. J )
1412, 13syl 16 . . . . . 6  |-  ( ph  ->  F  Fn  U. J
)
15 df-fo 5452 . . . . . 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 17727 . . . . 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 5684 . . . . . . . 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 2781 . . . . . . . 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 5191 . . . . . . . . 9  |-  ( x  =  ( `' F " y )  ->  ( F " x )  =  ( F " ( `' F " y ) ) )
2726eleq1d 2501 . . . . . . . 8  |-  ( x  =  ( `' F " y )  ->  (
( F " x
)  e.  K  <->  ( F " ( `' F "
y ) )  e.  K ) )
2827rspcv 3040 . . . . . . 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 2510 . . . . 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 3346 . 2  |-  ( ph  ->  ( J qTop  F ) 
C_  K )
345, 33eqssd 3357 1  |-  ( ph  ->  K  =  ( J qTop 
F ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725   A.wral 2697    C_ wss 3312   U.cuni 4007   `'ccnv 4869   ran crn 4871   "cima 4873    Fn wfn 5441   -->wf 5442   -onto->wfo 5444   ` cfv 5446  (class class class)co 6073   qTop cqtop 13721   Topctop 16950  TopOnctopon 16951    Cn ccn 17280
This theorem is referenced by:  hmeoqtop  17799
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-map 7012  df-qtop 13725  df-top 16955  df-topon 16958  df-cn 17283
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