Theorem qsinxp 6558

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 6557	. . . . 5 |- ( ( ( R " A ) C_ A /\ x e. A ) -> [ x ] R = [ x ] ( R i^i ( A X. A ) ) )
2	1	eqeq2d 2169	. . . 4 |- ( ( ( R " A ) C_ A /\ x e. A ) -> ( y = [ x ] R <-> y = [ x ] ( R i^i ( A X. A ) ) ) )
3	2	rexbidva 2454	. . 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 2275	. 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 6488	. 2 |- ( A /. R ) = { y | E. x e. A y = [ x ] R }
6		df-qs 6488	. 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 2215	1 |- ( ( R " A ) C_ A -> ( A /. R ) = ( A /. ( R i^i ( A X. A ) ) ) )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1335   e. wcel 2128   {cab 2143   E.wrex 2436   i^i cin 3101   C_ wss 3102   X. cxp 4586   "cima 4591   [cec 6480   /.cqs 6481

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-14 2131  ax-ext 2139  ax-sep 4084  ax-pow 4137  ax-pr 4171

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-v 2714  df-un 3106  df-in 3108  df-ss 3115  df-pw 3546  df-sn 3567  df-pr 3568  df-op 3570  df-br 3968  df-opab 4028  df-xp 4594  df-rel 4595  df-cnv 4596  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-ec 6484  df-qs 6488

This theorem is referenced by: (None)