Theorem qsinxp 6613

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 6612	. . . . 5 |- ( ( ( R " A ) C_ A /\ x e. A ) -> [ x ] R = [ x ] ( R i^i ( A X. A ) ) )
2	1	eqeq2d 2189	. . . 4 |- ( ( ( R " A ) C_ A /\ x e. A ) -> ( y = [ x ] R <-> y = [ x ] ( R i^i ( A X. A ) ) ) )
3	2	rexbidva 2474	. . 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 2295	. 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 6543	. 2 |- ( A /. R ) = { y | E. x e. A y = [ x ] R }
6		df-qs 6543	. 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 2235	1 |- ( ( R " A ) C_ A -> ( A /. R ) = ( A /. ( R i^i ( A X. A ) ) ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1353   e. wcel 2148   {cab 2163   E.wrex 2456   i^i cin 3130   C_ wss 3131   X. cxp 4626   "cima 4631   [cec 6535   /.cqs 6536

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2741  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-br 4006  df-opab 4067  df-xp 4634  df-rel 4635  df-cnv 4636  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-ec 6539  df-qs 6543

This theorem is referenced by: (None)