Theorem elqsecl 7801

Description: Membership in a quotient set by an equivalence class according to .~. (Contributed by Alexander van der Vekens, 12-Apr-2018.) (Revised by AV, 30-Apr-2021.)

Assertion

Ref	Expression
elqsecl	|- ( B e. X -> ( B e. ( W /. .~ ) <-> E. x e. W B = { y | x .~ y } ) )

Distinct variable groups:   x, .~ , y   x, B   x, W   x, X

Allowed substitution hints:   B( y)   W( y)   X( y)

Proof of Theorem elqsecl

Step	Hyp	Ref	Expression
1		elqsg 7798	. 2 |- ( B e. X -> ( B e. ( W /. .~ ) <-> E. x e. W B = [ x ] .~ ) )
2		vex 3203	. . . . 5 |- x e. _V
3		dfec2 7745	. . . . 5 |- ( x e. _V -> [ x ] .~ = { y | x .~ y } )
4	2, 3	mp1i 13	. . . 4 |- ( B e. X -> [ x ] .~ = { y | x .~ y } )
5	4	eqeq2d 2632	. . 3 |- ( B e. X -> ( B = [ x ] .~ <-> B = { y | x .~ y } ) )
6	5	rexbidv 3052	. 2 |- ( B e. X -> ( E. x e. W B = [ x ] .~ <-> E. x e. W B = { y | x .~ y } ) )
7	1, 6	bitrd 268	1 |- ( B e. X -> ( B e. ( W /. .~ ) <-> E. x e. W B = { y | x .~ y } ) )

Syntax hints:   -> wi 4   <-> wb 196   = wceq 1483   e. wcel 1990   {cab 2608   E.wrex 2913   _Vcvv 3200   class class class wbr 4653   [cec 7740   /.cqs 7741

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-br 4654  df-opab 4713  df-xp 5120  df-cnv 5122  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-ec 7744  df-qs 7748

This theorem is referenced by:  eclclwwlksn1  26952