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Theorem kqfvima 20743
Description: When the image set is open, the quotient map satisfies a partial converse to fnfvima 6158, 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 20738 . . . 4  |-  ( J  e.  (TopOn `  X
)  ->  F  Fn  X )
323ad2ant1 1026 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  U  e.  J  /\  A  e.  X )  ->  F  Fn  X )
4 toponss 19942 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  U  e.  J )  ->  U  C_  X )
543adant3 1025 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  U  e.  J  /\  A  e.  X )  ->  U  C_  X )
6 fnfvima 6158 . . . 4  |-  ( ( F  Fn  X  /\  U  C_  X  /\  A  e.  U )  ->  ( F `  A )  e.  ( F " U
) )
763expia 1207 . . 3  |-  ( ( F  Fn  X  /\  U  C_  X )  -> 
( A  e.  U  ->  ( F `  A
)  e.  ( F
" U ) ) )
83, 5, 7syl2anc 665 . 2  |-  ( ( J  e.  (TopOn `  X )  /\  U  e.  J  /\  A  e.  X )  ->  ( A  e.  U  ->  ( F `  A )  e.  ( F " U ) ) )
9 fnfun 5691 . . . 4  |-  ( F  Fn  X  ->  Fun  F )
10 fvelima 5933 . . . . 5  |-  ( ( Fun  F  /\  ( F `  A )  e.  ( F " U
) )  ->  E. z  e.  U  ( F `  z )  =  ( F `  A ) )
1110ex 435 . . . 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 simpl1 1008 . . . . . . . 8  |-  ( ( ( J  e.  (TopOn `  X )  /\  U  e.  J  /\  A  e.  X )  /\  z  e.  U )  ->  J  e.  (TopOn `  X )
)
145sselda 3464 . . . . . . . 8  |-  ( ( ( J  e.  (TopOn `  X )  /\  U  e.  J  /\  A  e.  X )  /\  z  e.  U )  ->  z  e.  X )
15 simpl3 1010 . . . . . . . 8  |-  ( ( ( J  e.  (TopOn `  X )  /\  U  e.  J  /\  A  e.  X )  /\  z  e.  U )  ->  A  e.  X )
161kqfeq 20737 . . . . . . . 8  |-  ( ( J  e.  (TopOn `  X )  /\  z  e.  X  /\  A  e.  X )  ->  (
( F `  z
)  =  ( F `
 A )  <->  A. y  e.  J  ( z  e.  y  <->  A  e.  y
) ) )
1713, 14, 15, 16syl3anc 1264 . . . . . . 7  |-  ( ( ( 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
) ) )
18 eleq2 2496 . . . . . . . . 9  |-  ( y  =  w  ->  (
z  e.  y  <->  z  e.  w ) )
19 eleq2 2496 . . . . . . . . 9  |-  ( y  =  w  ->  ( A  e.  y  <->  A  e.  w ) )
2018, 19bibi12d 322 . . . . . . . 8  |-  ( y  =  w  ->  (
( z  e.  y  <-> 
A  e.  y )  <-> 
( z  e.  w  <->  A  e.  w ) ) )
2120cbvralv 3054 . . . . . . 7  |-  ( A. y  e.  J  (
z  e.  y  <->  A  e.  y )  <->  A. w  e.  J  ( z  e.  w  <->  A  e.  w
) )
2217, 21syl6bb 264 . . . . . 6  |-  ( ( ( 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
) ) )
23 simpl2 1009 . . . . . . 7  |-  ( ( ( J  e.  (TopOn `  X )  /\  U  e.  J  /\  A  e.  X )  /\  z  e.  U )  ->  U  e.  J )
24 eleq2 2496 . . . . . . . . 9  |-  ( w  =  U  ->  (
z  e.  w  <->  z  e.  U ) )
25 eleq2 2496 . . . . . . . . 9  |-  ( w  =  U  ->  ( A  e.  w  <->  A  e.  U ) )
2624, 25bibi12d 322 . . . . . . . 8  |-  ( w  =  U  ->  (
( z  e.  w  <->  A  e.  w )  <->  ( z  e.  U  <->  A  e.  U
) ) )
2726rspcv 3178 . . . . . . 7  |-  ( U  e.  J  ->  ( A. w  e.  J  ( z  e.  w  <->  A  e.  w )  -> 
( z  e.  U  <->  A  e.  U ) ) )
2823, 27syl 17 . . . . . 6  |-  ( ( ( 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 ) ) )
2922, 28sylbid 218 . . . . 5  |-  ( ( ( J  e.  (TopOn `  X )  /\  U  e.  J  /\  A  e.  X )  /\  z  e.  U )  ->  (
( F `  z
)  =  ( F `
 A )  -> 
( z  e.  U  <->  A  e.  U ) ) )
30 simpr 462 . . . . 5  |-  ( ( ( J  e.  (TopOn `  X )  /\  U  e.  J  /\  A  e.  X )  /\  z  e.  U )  ->  z  e.  U )
31 biimp 196 . . . . 5  |-  ( ( z  e.  U  <->  A  e.  U )  ->  (
z  e.  U  ->  A  e.  U )
)
3229, 30, 31syl6ci 67 . . . 4  |-  ( ( ( J  e.  (TopOn `  X )  /\  U  e.  J  /\  A  e.  X )  /\  z  e.  U )  ->  (
( F `  z
)  =  ( F `
 A )  ->  A  e.  U )
)
3332rexlimdva 2914 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  U  e.  J  /\  A  e.  X )  ->  ( E. z  e.  U  ( F `  z )  =  ( F `  A )  ->  A  e.  U ) )
3412, 33syld 45 . 2  |-  ( ( J  e.  (TopOn `  X )  /\  U  e.  J  /\  A  e.  X )  ->  (
( F `  A
)  e.  ( F
" U )  ->  A  e.  U )
)
358, 34impbid 193 1  |-  ( ( J  e.  (TopOn `  X )  /\  U  e.  J  /\  A  e.  X )  ->  ( A  e.  U  <->  ( F `  A )  e.  ( F " U ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1872   A.wral 2771   E.wrex 2772   {crab 2775    C_ wss 3436    |-> cmpt 4482   "cima 4856   Fun wfun 5595    Fn wfn 5596   ` cfv 5601  TopOnctopon 19916
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-sep 4546  ax-nul 4555  ax-pow 4602  ax-pr 4660  ax-un 6597
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2273  df-mo 2274  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-ral 2776  df-rex 2777  df-rab 2780  df-v 3082  df-sbc 3300  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-if 3912  df-pw 3983  df-sn 3999  df-pr 4001  df-op 4005  df-uni 4220  df-br 4424  df-opab 4483  df-mpt 4484  df-id 4768  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-fv 5609  df-topon 19921
This theorem is referenced by:  kqsat  20744  kqdisj  20745  kqcldsat  20746  kqt0lem  20749  isr0  20750  regr1lem  20752  kqreglem1  20754  kqreglem2  20755
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