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Theorem kqfeq 17471
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 17470 . . . 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 17470 . . . 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 2330 . 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 2752 . 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 1633    e. wcel 1701   A.wral 2577   {crab 2581    e. cmpt 4114   ` cfv 5292
This theorem is referenced by:  ist0-4  17476  kqfvima  17477  kqt0lem  17483  isr0  17484  r0cld  17485  regr1lem2  17487
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-sep 4178  ax-nul 4186  ax-pr 4251
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-ral 2582  df-rex 2583  df-rab 2586  df-v 2824  df-sbc 3026  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-nul 3490  df-if 3600  df-sn 3680  df-pr 3681  df-op 3683  df-uni 3865  df-br 4061  df-opab 4115  df-mpt 4116  df-id 4346  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-iota 5256  df-fun 5294  df-fv 5300
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