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Theorem kqfvima 17676
Description: When the image set is open, the quotient map satisfies a partial converse to fnfvima 5908, 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 17671 . . . 4  |-  ( J  e.  (TopOn `  X
)  ->  F  Fn  X )
323ad2ant1 978 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  U  e.  J  /\  A  e.  X )  ->  F  Fn  X )
4 toponss 16910 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  U  e.  J )  ->  U  C_  X )
543adant3 977 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  U  e.  J  /\  A  e.  X )  ->  U  C_  X )
6 fnfvima 5908 . . . 4  |-  ( ( F  Fn  X  /\  U  C_  X  /\  A  e.  U )  ->  ( F `  A )  e.  ( F " U
) )
763expia 1155 . . 3  |-  ( ( F  Fn  X  /\  U  C_  X )  -> 
( A  e.  U  ->  ( F `  A
)  e.  ( F
" U ) ) )
83, 5, 7syl2anc 643 . 2  |-  ( ( J  e.  (TopOn `  X )  /\  U  e.  J  /\  A  e.  X )  ->  ( A  e.  U  ->  ( F `  A )  e.  ( F " U ) ) )
9 fnfun 5475 . . . 4  |-  ( F  Fn  X  ->  Fun  F )
10 fvelima 5710 . . . . 5  |-  ( ( Fun  F  /\  ( F `  A )  e.  ( F " U
) )  ->  E. z  e.  U  ( F `  z )  =  ( F `  A ) )
1110ex 424 . . . 4  |-  ( Fun 
F  ->  ( ( F `  A )  e.  ( F " U
)  ->  E. z  e.  U  ( F `  z )  =  ( F `  A ) ) )
123, 9, 113syl 19 . . 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 448 . . . . 5  |-  ( ( ( J  e.  (TopOn `  X )  /\  U  e.  J  /\  A  e.  X )  /\  z  e.  U )  ->  z  e.  U )
14 simpl1 960 . . . . . . . . 9  |-  ( ( ( J  e.  (TopOn `  X )  /\  U  e.  J  /\  A  e.  X )  /\  z  e.  U )  ->  J  e.  (TopOn `  X )
)
155sselda 3284 . . . . . . . . 9  |-  ( ( ( J  e.  (TopOn `  X )  /\  U  e.  J  /\  A  e.  X )  /\  z  e.  U )  ->  z  e.  X )
16 simpl3 962 . . . . . . . . 9  |-  ( ( ( J  e.  (TopOn `  X )  /\  U  e.  J  /\  A  e.  X )  /\  z  e.  U )  ->  A  e.  X )
171kqfeq 17670 . . . . . . . . 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 1184 . . . . . . . 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 2441 . . . . . . . . . 10  |-  ( y  =  w  ->  (
z  e.  y  <->  z  e.  w ) )
20 eleq2 2441 . . . . . . . . . 10  |-  ( y  =  w  ->  ( A  e.  y  <->  A  e.  w ) )
2119, 20bibi12d 313 . . . . . . . . 9  |-  ( y  =  w  ->  (
( z  e.  y  <-> 
A  e.  y )  <-> 
( z  e.  w  <->  A  e.  w ) ) )
2221cbvralv 2868 . . . . . . . 8  |-  ( A. y  e.  J  (
z  e.  y  <->  A  e.  y )  <->  A. w  e.  J  ( z  e.  w  <->  A  e.  w
) )
2318, 22syl6bb 253 . . . . . . 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 961 . . . . . . . 8  |-  ( ( ( J  e.  (TopOn `  X )  /\  U  e.  J  /\  A  e.  X )  /\  z  e.  U )  ->  U  e.  J )
25 eleq2 2441 . . . . . . . . . 10  |-  ( w  =  U  ->  (
z  e.  w  <->  z  e.  U ) )
26 eleq2 2441 . . . . . . . . . 10  |-  ( w  =  U  ->  ( A  e.  w  <->  A  e.  U ) )
2725, 26bibi12d 313 . . . . . . . . 9  |-  ( w  =  U  ->  (
( z  e.  w  <->  A  e.  w )  <->  ( z  e.  U  <->  A  e.  U
) ) )
2827rspcv 2984 . . . . . . . 8  |-  ( U  e.  J  ->  ( A. w  e.  J  ( z  e.  w  <->  A  e.  w )  -> 
( z  e.  U  <->  A  e.  U ) ) )
2924, 28syl 16 . . . . . . 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 207 . . . . . 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 179 . . . . . 6  |-  ( ( z  e.  U  <->  A  e.  U )  ->  (
z  e.  U  ->  A  e.  U )
)
3230, 31syl6 31 . . . . 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 39 . . . 4  |-  ( ( ( J  e.  (TopOn `  X )  /\  U  e.  J  /\  A  e.  X )  /\  z  e.  U )  ->  (
( F `  z
)  =  ( F `
 A )  ->  A  e.  U )
)
3433rexlimdva 2766 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  U  e.  J  /\  A  e.  X )  ->  ( E. z  e.  U  ( F `  z )  =  ( F `  A )  ->  A  e.  U ) )
3512, 34syld 42 . 2  |-  ( ( J  e.  (TopOn `  X )  /\  U  e.  J  /\  A  e.  X )  ->  (
( F `  A
)  e.  ( F
" U )  ->  A  e.  U )
)
368, 35impbid 184 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 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717   A.wral 2642   E.wrex 2643   {crab 2646    C_ wss 3256    e. cmpt 4200   "cima 4814   Fun wfun 5381    Fn wfn 5382   ` cfv 5387  TopOnctopon 16875
This theorem is referenced by:  kqsat  17677  kqdisj  17678  kqcldsat  17679  kqt0lem  17682  isr0  17683  regr1lem  17685  kqreglem1  17687  kqreglem2  17688
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337  ax-un 4634
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-ral 2647  df-rex 2648  df-rab 2651  df-v 2894  df-sbc 3098  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-op 3759  df-uni 3951  df-br 4147  df-opab 4201  df-mpt 4202  df-id 4432  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-fv 5395  df-topon 16882
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