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Theorem kqfeq 17757
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 17756 . . . 4  |-  ( ( J  e.  V  /\  A  e.  X )  ->  ( F `  A
)  =  { y  e.  J  |  A  e.  y } )
323adant3 978 . . 3  |-  ( ( J  e.  V  /\  A  e.  X  /\  B  e.  X )  ->  ( F `  A
)  =  { y  e.  J  |  A  e.  y } )
41kqfval 17756 . . . 4  |-  ( ( J  e.  V  /\  B  e.  X )  ->  ( F `  B
)  =  { y  e.  J  |  B  e.  y } )
543adant2 977 . . 3  |-  ( ( J  e.  V  /\  A  e.  X  /\  B  e.  X )  ->  ( F `  B
)  =  { y  e.  J  |  B  e.  y } )
63, 5eqeq12d 2451 . 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 2887 . 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 256 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 178    /\ w3a 937    = wceq 1653    e. wcel 1726   A.wral 2706   {crab 2710    e. cmpt 4267   ` cfv 5455
This theorem is referenced by:  ist0-4  17762  kqfvima  17763  kqt0lem  17769  isr0  17770  r0cld  17771  regr1lem2  17773
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-sep 4331  ax-nul 4339  ax-pr 4404
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-ral 2711  df-rex 2712  df-rab 2715  df-v 2959  df-sbc 3163  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3630  df-if 3741  df-sn 3821  df-pr 3822  df-op 3824  df-uni 4017  df-br 4214  df-opab 4268  df-mpt 4269  df-id 4499  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-iota 5419  df-fun 5457  df-fv 5463
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