Theorem kqfeq 21527

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

Step	Hyp	Ref	Expression
1		kqval.2	. . . . 5 |- F = ( x e. X |-> { y e. J | x e. y } )
2	1	kqfval 21526	. . . 4 |- ( ( J e. V /\ A e. X ) -> ( F ` A ) = { y e. J | A e. y } )
3	2	3adant3 1081	. . 3 |- ( ( J e. V /\ A e. X /\ B e. X ) -> ( F ` A ) = { y e. J | A e. y } )
4	1	kqfval 21526	. . . 4 |- ( ( J e. V /\ B e. X ) -> ( F ` B ) = { y e. J | B e. y } )
5	4	3adant2 1080	. . 3 |- ( ( J e. V /\ A e. X /\ B e. X ) -> ( F ` B ) = { y e. J | B e. y } )
6	3, 5	eqeq12d 2637	. 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 3120	. 2 |- ( A. y e. J ( A e. y <-> B e. y ) <-> { y e. J | A e. y } = { y e. J | B e. y } )
8	6, 7	syl6bbr 278	1 |- ( ( J e. V /\ A e. X /\ B e. X ) -> ( ( F ` A ) = ( F ` B ) <-> A. y e. J ( A e. y <-> B e. y ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   {crab 2916   |-> cmpt 4729   ` cfv 5888

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fv 5896

This theorem is referenced by:  ist0-4  21532  kqfvima  21533  kqt0lem  21539  isr0  21540  r0cld  21541  regr1lem2  21543