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Theorem kqfeq 17415
Description: Two points in the Kolmogorov quotient are equal iff the original points are topologically indistinguishable. (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
kqfeq  |-  ( ( J  e.  V  /\  A  e.  X  /\  B  e.  X )  ->  ( ( F `  A )  =  ( F `  B )  <->  A. y  e.  J  ( A  e.  y  <->  B  e.  y ) ) )
Distinct variable groups:    x, y, A    x, B, y    x, J, y    x, X, y   
x, V
Allowed substitution hints:    F( x, y)    V( y)

Proof of Theorem kqfeq
StepHypRef Expression
1 kqval.2 . . . . 5  |-  F  =  ( x  e.  X  |->  { y  e.  J  |  x  e.  y } )
21kqfval 17414 . . . 4  |-  ( ( J  e.  V  /\  A  e.  X )  ->  ( F `  A
)  =  { y  e.  J  |  A  e.  y } )
323adant3 975 . . 3  |-  ( ( J  e.  V  /\  A  e.  X  /\  B  e.  X )  ->  ( F `  A
)  =  { y  e.  J  |  A  e.  y } )
41kqfval 17414 . . . 4  |-  ( ( J  e.  V  /\  B  e.  X )  ->  ( F `  B
)  =  { y  e.  J  |  B  e.  y } )
543adant2 974 . . 3  |-  ( ( J  e.  V  /\  A  e.  X  /\  B  e.  X )  ->  ( F `  B
)  =  { y  e.  J  |  B  e.  y } )
63, 5eqeq12d 2297 . 2  |-  ( ( J  e.  V  /\  A  e.  X  /\  B  e.  X )  ->  ( ( F `  A )  =  ( F `  B )  <->  { y  e.  J  |  A  e.  y }  =  { y  e.  J  |  B  e.  y } ) )
7 rabbi 2718 . 2  |-  ( A. y  e.  J  ( A  e.  y  <->  B  e.  y )  <->  { y  e.  J  |  A  e.  y }  =  {
y  e.  J  |  B  e.  y }
)
86, 7syl6bbr 254 1  |-  ( ( J  e.  V  /\  A  e.  X  /\  B  e.  X )  ->  ( ( F `  A )  =  ( F `  B )  <->  A. y  e.  J  ( A  e.  y  <->  B  e.  y ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ w3a 934    = wceq 1623    e. wcel 1684   A.wral 2543   {crab 2547    e. cmpt 4077   ` cfv 5255
This theorem is referenced by:  ist0-4  17420  kqfvima  17421  kqt0lem  17427  isr0  17428  r0cld  17429  regr1lem2  17431
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-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
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-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-iota 5219  df-fun 5257  df-fv 5263
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