Theorem qsinxp 6845

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 6844	. . . . 5 |- ( ( ( R " A ) C_ A /\ x e. A ) -> [ x ] R = [ x ] ( R i^i ( A X. A ) ) )
2	1	eqeq2d 2244	. . . 4 |- ( ( ( R " A ) C_ A /\ x e. A ) -> ( y = [ x ] R <-> y = [ x ] ( R i^i ( A X. A ) ) ) )
3	2	rexbidva 2539	. . 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 2352	. 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 6773	. 2 |- ( A /. R ) = { y | E. x e. A y = [ x ] R }
6		df-qs 6773	. 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 2290	1 |- ( ( R " A ) C_ A -> ( A /. R ) = ( A /. ( R i^i ( A X. A ) ) ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1398   e. wcel 2203   {cab 2218   E.wrex 2521   i^i cin 3210   C_ wss 3211   X. cxp 4747   "cima 4752   [cec 6765   /.cqs 6766

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-pow 4287  ax-pr 4322

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-v 2815  df-un 3215  df-in 3217  df-ss 3224  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-br 4110  df-opab 4172  df-xp 4755  df-rel 4756  df-cnv 4757  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-ec 6769  df-qs 6773

This theorem is referenced by: (None)