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Theorem kqfvima 17421
Description: When the image set is open, the quotient map satisfies a partial converse to fnfvima 5756, which is normally only true for injective functions. (Contributed by Mario Carneiro, 25-Aug-2015.)
Hypothesis
Ref Expression
kqval.2  |-  F  =  ( x  e.  X  |->  { y  e.  J  |  x  e.  y } )
Assertion
Ref Expression
kqfvima  |-  ( ( J  e.  (TopOn `  X )  /\  U  e.  J  /\  A  e.  X )  ->  ( A  e.  U  <->  ( F `  A )  e.  ( F " U ) ) )
Distinct variable groups:    x, y, A    x, J, y    x, X, y
Allowed substitution hints:    U( x, y)    F( x, y)

Proof of Theorem kqfvima
Dummy variables  w  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 kqval.2 . . . . 5  |-  F  =  ( x  e.  X  |->  { y  e.  J  |  x  e.  y } )
21kqffn 17416 . . . 4  |-  ( J  e.  (TopOn `  X
)  ->  F  Fn  X )
323ad2ant1 976 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  U  e.  J  /\  A  e.  X )  ->  F  Fn  X )
4 toponss 16667 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  U  e.  J )  ->  U  C_  X )
543adant3 975 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  U  e.  J  /\  A  e.  X )  ->  U  C_  X )
6 fnfvima 5756 . . . 4  |-  ( ( F  Fn  X  /\  U  C_  X  /\  A  e.  U )  ->  ( F `  A )  e.  ( F " U
) )
763expia 1153 . . 3  |-  ( ( F  Fn  X  /\  U  C_  X )  -> 
( A  e.  U  ->  ( F `  A
)  e.  ( F
" U ) ) )
83, 5, 7syl2anc 642 . 2  |-  ( ( J  e.  (TopOn `  X )  /\  U  e.  J  /\  A  e.  X )  ->  ( A  e.  U  ->  ( F `  A )  e.  ( F " U ) ) )
9 fnfun 5341 . . . 4  |-  ( F  Fn  X  ->  Fun  F )
10 fvelima 5574 . . . . 5  |-  ( ( Fun  F  /\  ( F `  A )  e.  ( F " U
) )  ->  E. z  e.  U  ( F `  z )  =  ( F `  A ) )
1110ex 423 . . . 4  |-  ( Fun 
F  ->  ( ( F `  A )  e.  ( F " U
)  ->  E. z  e.  U  ( F `  z )  =  ( F `  A ) ) )
123, 9, 113syl 18 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  U  e.  J  /\  A  e.  X )  ->  (
( F `  A
)  e.  ( F
" U )  ->  E. z  e.  U  ( F `  z )  =  ( F `  A ) ) )
13 simpr 447 . . . . 5  |-  ( ( ( J  e.  (TopOn `  X )  /\  U  e.  J  /\  A  e.  X )  /\  z  e.  U )  ->  z  e.  U )
14 simpl1 958 . . . . . . . . 9  |-  ( ( ( J  e.  (TopOn `  X )  /\  U  e.  J  /\  A  e.  X )  /\  z  e.  U )  ->  J  e.  (TopOn `  X )
)
155sselda 3180 . . . . . . . . 9  |-  ( ( ( J  e.  (TopOn `  X )  /\  U  e.  J  /\  A  e.  X )  /\  z  e.  U )  ->  z  e.  X )
16 simpl3 960 . . . . . . . . 9  |-  ( ( ( J  e.  (TopOn `  X )  /\  U  e.  J  /\  A  e.  X )  /\  z  e.  U )  ->  A  e.  X )
171kqfeq 17415 . . . . . . . . 9  |-  ( ( J  e.  (TopOn `  X )  /\  z  e.  X  /\  A  e.  X )  ->  (
( F `  z
)  =  ( F `
 A )  <->  A. y  e.  J  ( z  e.  y  <->  A  e.  y
) ) )
1814, 15, 16, 17syl3anc 1182 . . . . . . . 8  |-  ( ( ( J  e.  (TopOn `  X )  /\  U  e.  J  /\  A  e.  X )  /\  z  e.  U )  ->  (
( F `  z
)  =  ( F `
 A )  <->  A. y  e.  J  ( z  e.  y  <->  A  e.  y
) ) )
19 eleq2 2344 . . . . . . . . . 10  |-  ( y  =  w  ->  (
z  e.  y  <->  z  e.  w ) )
20 eleq2 2344 . . . . . . . . . 10  |-  ( y  =  w  ->  ( A  e.  y  <->  A  e.  w ) )
2119, 20bibi12d 312 . . . . . . . . 9  |-  ( y  =  w  ->  (
( z  e.  y  <-> 
A  e.  y )  <-> 
( z  e.  w  <->  A  e.  w ) ) )
2221cbvralv 2764 . . . . . . . 8  |-  ( A. y  e.  J  (
z  e.  y  <->  A  e.  y )  <->  A. w  e.  J  ( z  e.  w  <->  A  e.  w
) )
2318, 22syl6bb 252 . . . . . . 7  |-  ( ( ( J  e.  (TopOn `  X )  /\  U  e.  J  /\  A  e.  X )  /\  z  e.  U )  ->  (
( F `  z
)  =  ( F `
 A )  <->  A. w  e.  J  ( z  e.  w  <->  A  e.  w
) ) )
24 simpl2 959 . . . . . . . 8  |-  ( ( ( J  e.  (TopOn `  X )  /\  U  e.  J  /\  A  e.  X )  /\  z  e.  U )  ->  U  e.  J )
25 eleq2 2344 . . . . . . . . . 10  |-  ( w  =  U  ->  (
z  e.  w  <->  z  e.  U ) )
26 eleq2 2344 . . . . . . . . . 10  |-  ( w  =  U  ->  ( A  e.  w  <->  A  e.  U ) )
2725, 26bibi12d 312 . . . . . . . . 9  |-  ( w  =  U  ->  (
( z  e.  w  <->  A  e.  w )  <->  ( z  e.  U  <->  A  e.  U
) ) )
2827rspcv 2880 . . . . . . . 8  |-  ( U  e.  J  ->  ( A. w  e.  J  ( z  e.  w  <->  A  e.  w )  -> 
( z  e.  U  <->  A  e.  U ) ) )
2924, 28syl 15 . . . . . . 7  |-  ( ( ( J  e.  (TopOn `  X )  /\  U  e.  J  /\  A  e.  X )  /\  z  e.  U )  ->  ( A. w  e.  J  ( z  e.  w  <->  A  e.  w )  -> 
( z  e.  U  <->  A  e.  U ) ) )
3023, 29sylbid 206 . . . . . 6  |-  ( ( ( J  e.  (TopOn `  X )  /\  U  e.  J  /\  A  e.  X )  /\  z  e.  U )  ->  (
( F `  z
)  =  ( F `
 A )  -> 
( z  e.  U  <->  A  e.  U ) ) )
31 bi1 178 . . . . . 6  |-  ( ( z  e.  U  <->  A  e.  U )  ->  (
z  e.  U  ->  A  e.  U )
)
3230, 31syl6 29 . . . . 5  |-  ( ( ( J  e.  (TopOn `  X )  /\  U  e.  J  /\  A  e.  X )  /\  z  e.  U )  ->  (
( F `  z
)  =  ( F `
 A )  -> 
( z  e.  U  ->  A  e.  U ) ) )
3313, 32mpid 37 . . . 4  |-  ( ( ( J  e.  (TopOn `  X )  /\  U  e.  J  /\  A  e.  X )  /\  z  e.  U )  ->  (
( F `  z
)  =  ( F `
 A )  ->  A  e.  U )
)
3433rexlimdva 2667 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  U  e.  J  /\  A  e.  X )  ->  ( E. z  e.  U  ( F `  z )  =  ( F `  A )  ->  A  e.  U ) )
3512, 34syld 40 . 2  |-  ( ( J  e.  (TopOn `  X )  /\  U  e.  J  /\  A  e.  X )  ->  (
( F `  A
)  e.  ( F
" U )  ->  A  e.  U )
)
368, 35impbid 183 1  |-  ( ( J  e.  (TopOn `  X )  /\  U  e.  J  /\  A  e.  X )  ->  ( A  e.  U  <->  ( F `  A )  e.  ( F " U ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   A.wral 2543   E.wrex 2544   {crab 2547    C_ wss 3152    e. cmpt 4077   "cima 4692   Fun wfun 5249    Fn wfn 5250   ` cfv 5255  TopOnctopon 16632
This theorem is referenced by:  kqsat  17422  kqdisj  17423  kqcldsat  17424  kqt0lem  17427  isr0  17428  regr1lem  17430  kqreglem1  17432  kqreglem2  17433
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-fv 5263  df-topon 16639
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