Theorem qsinxp 7823

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 7822	. . . . 5 |- ( ( ( R " A ) C_ A /\ x e. A ) -> [ x ] R = [ x ] ( R i^i ( A X. A ) ) )
2	1	eqeq2d 2632	. . . 4 |- ( ( ( R " A ) C_ A /\ x e. A ) -> ( y = [ x ] R <-> y = [ x ] ( R i^i ( A X. A ) ) ) )
3	2	rexbidva 3049	. . 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 2741	. 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 7748	. 2 |- ( A /. R ) = { y | E. x e. A y = [ x ] R }
6		df-qs 7748	. 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 2681	1 |- ( ( R " A ) C_ A -> ( A /. R ) = ( A /. ( R i^i ( A X. A ) ) ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   {cab 2608   E.wrex 2913   i^i cin 3573   C_ wss 3574   X. cxp 5112   "cima 5117   [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-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-rel 5121  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:  pi1buni  22840  pi1bas3  22843