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Theorem qtopomap 19947
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 19944 . . 3  |-  ( ( F  e.  ( J  Cn  K )  /\  K  e.  (TopOn `  Y
)  /\  ran  F  =  Y )  ->  K  C_  ( J qTop  F ) )
51, 2, 3, 4syl3anc 1223 . 2  |-  ( ph  ->  K  C_  ( J qTop  F ) )
6 cntop1 19500 . . . . . . 7  |-  ( F  e.  ( J  Cn  K )  ->  J  e.  Top )
71, 6syl 16 . . . . . 6  |-  ( ph  ->  J  e.  Top )
8 eqid 2460 . . . . . . 7  |-  U. J  =  U. J
98toptopon 19194 . . . . . 6  |-  ( J  e.  Top  <->  J  e.  (TopOn `  U. J ) )
107, 9sylib 196 . . . . 5  |-  ( ph  ->  J  e.  (TopOn `  U. J ) )
11 cnf2 19509 . . . . . . . 8  |-  ( ( J  e.  (TopOn `  U. J )  /\  K  e.  (TopOn `  Y )  /\  F  e.  ( J  Cn  K ) )  ->  F : U. J
--> Y )
1210, 2, 1, 11syl3anc 1223 . . . . . . 7  |-  ( ph  ->  F : U. J --> Y )
13 ffn 5722 . . . . . . 7  |-  ( F : U. J --> Y  ->  F  Fn  U. J )
1412, 13syl 16 . . . . . 6  |-  ( ph  ->  F  Fn  U. J
)
15 df-fo 5585 . . . . . 6  |-  ( F : U. J -onto-> Y  <->  ( F  Fn  U. J  /\  ran  F  =  Y ) )
1614, 3, 15sylanbrc 664 . . . . 5  |-  ( ph  ->  F : U. J -onto-> Y )
17 elqtop3 19932 . . . . 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 661 . . . 4  |-  ( ph  ->  ( y  e.  ( J qTop  F )  <->  ( y  C_  Y  /\  ( `' F " y )  e.  J ) ) )
19 foimacnv 5824 . . . . . . . 8  |-  ( ( F : U. J -onto-> Y  /\  y  C_  Y
)  ->  ( F " ( `' F "
y ) )  =  y )
2016, 19sylan 471 . . . . . . 7  |-  ( (
ph  /\  y  C_  Y )  ->  ( F " ( `' F " y ) )  =  y )
2120adantrr 716 . . . . . 6  |-  ( (
ph  /\  ( y  C_  Y  /\  ( `' F " y )  e.  J ) )  ->  ( F "
( `' F "
y ) )  =  y )
22 simprr 756 . . . . . . 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 2871 . . . . . . . 8  |-  ( ph  ->  A. x  e.  J  ( F " x )  e.  K )
2524adantr 465 . . . . . . 7  |-  ( (
ph  /\  ( y  C_  Y  /\  ( `' F " y )  e.  J ) )  ->  A. x  e.  J  ( F " x )  e.  K )
26 imaeq2 5324 . . . . . . . . 9  |-  ( x  =  ( `' F " y )  ->  ( F " x )  =  ( F " ( `' F " y ) ) )
2726eleq1d 2529 . . . . . . . 8  |-  ( x  =  ( `' F " y )  ->  (
( F " x
)  e.  K  <->  ( F " ( `' F "
y ) )  e.  K ) )
2827rspcv 3203 . . . . . . 7  |-  ( ( `' F " y )  e.  J  ->  ( A. x  e.  J  ( F " x )  e.  K  ->  ( F " ( `' F " y ) )  e.  K ) )
2922, 25, 28sylc 60 . . . . . 6  |-  ( (
ph  /\  ( y  C_  Y  /\  ( `' F " y )  e.  J ) )  ->  ( F "
( `' F "
y ) )  e.  K )
3021, 29eqeltrrd 2549 . . . . 5  |-  ( (
ph  /\  ( y  C_  Y  /\  ( `' F " y )  e.  J ) )  ->  y  e.  K
)
3130ex 434 . . . 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 3503 . 2  |-  ( ph  ->  ( J qTop  F ) 
C_  K )
345, 33eqssd 3514 1  |-  ( ph  ->  K  =  ( J qTop 
F ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1374    e. wcel 1762   A.wral 2807    C_ wss 3469   U.cuni 4238   `'ccnv 4991   ran crn 4993   "cima 4995    Fn wfn 5574   -->wf 5575   -onto->wfo 5577   ` cfv 5579  (class class class)co 6275   qTop cqtop 14747   Topctop 19154  TopOnctopon 19155    Cn ccn 19484
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-id 4788  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-map 7412  df-qtop 14751  df-top 19159  df-topon 19162  df-cn 19487
This theorem is referenced by:  hmeoqtop  20004
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