Theorem qsinxp 6697

Description: Restrict the equivalence relation in a quotient set to the base set. (Contributed by Mario Carneiro, 23-Feb-2015.)

Assertion

Ref	Expression
qsinxp	|- ( ( R " A ) C_ A -> ( A /. R ) = ( A /. ( R i^i ( A X. A ) ) ) )

Proof of Theorem qsinxp

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ecinxp 6696	. . . . 5 |- ( ( ( R " A ) C_ A /\ x e. A ) -> [ x ] R = [ x ] ( R i^i ( A X. A ) ) )
2	1	eqeq2d 2216	. . . 4 |- ( ( ( R " A ) C_ A /\ x e. A ) -> ( y = [ x ] R <-> y = [ x ] ( R i^i ( A X. A ) ) ) )
3	2	rexbidva 2502	. . 3 |- ( ( R " A ) C_ A -> ( E. x e. A y = [ x ] R <-> E. x e. A y = [ x ] ( R i^i ( A X. A ) ) ) )
4	3	abbidv 2322	. 2 |- ( ( R " A ) C_ A -> { y | E. x e. A y = [ x ] R } = { y | E. x e. A y = [ x ] ( R i^i ( A X. A ) ) } )
5		df-qs 6625	. 2 |- ( A /. R ) = { y | E. x e. A y = [ x ] R }
6		df-qs 6625	. 2 |- ( A /. ( R i^i ( A X. A ) ) ) = { y | E. x e. A y = [ x ] ( R i^i ( A X. A ) ) }
7	4, 5, 6	3eqtr4g 2262	1 |- ( ( R " A ) C_ A -> ( A /. R ) = ( A /. ( R i^i ( A X. A ) ) ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1372   e. wcel 2175   {cab 2190   E.wrex 2484   i^i cin 3164   C_ wss 3165   X. cxp 4672   "cima 4677   [cec 6617   /.cqs 6618

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-14 2178  ax-ext 2186  ax-sep 4161  ax-pow 4217  ax-pr 4252

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1375  df-nf 1483  df-sb 1785  df-eu 2056  df-mo 2057  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ral 2488  df-rex 2489  df-v 2773  df-un 3169  df-in 3171  df-ss 3178  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-br 4044  df-opab 4105  df-xp 4680  df-rel 4681  df-cnv 4682  df-dm 4684  df-rn 4685  df-res 4686  df-ima 4687  df-ec 6621  df-qs 6625

This theorem is referenced by: (None)