Theorem qsel 6436

Description: If an element of a quotient set contains a given element, it is equal to the equivalence class of the element. (Contributed by Mario Carneiro, 12-Aug-2015.)

Assertion

Ref	Expression
qsel	|- ( ( R Er X /\ B e. ( A /. R ) /\ C e. B ) -> B = [ C ] R )

Proof of Theorem qsel

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2100	. . 3 |- ( A /. R ) = ( A /. R )
2		eleq2 2163	. . . 4 |- ( [ x ] R = B -> ( C e. [ x ] R <-> C e. B ) )
3		eqeq1 2106	. . . 4 |- ( [ x ] R = B -> ( [ x ] R = [ C ] R <-> B = [ C ] R ) )
4	2, 3	imbi12d 233	. . 3 |- ( [ x ] R = B -> ( ( C e. [ x ] R -> [ x ] R = [ C ] R ) <-> ( C e. B -> B = [ C ] R ) ) )
5		vex 2644	. . . . . 6 |- x e. _V
6		elecg 6397	. . . . . 6 |- ( ( C e. [ x ] R /\ x e. _V ) -> ( C e. [ x ] R <-> x R C ) )
7	5, 6	mpan2 419	. . . . 5 |- ( C e. [ x ] R -> ( C e. [ x ] R <-> x R C ) )
8	7	ibi 175	. . . 4 |- ( C e. [ x ] R -> x R C )
9		simpll 499	. . . . . 6 |- ( ( ( R Er X /\ x e. A ) /\ x R C ) -> R Er X )
10		simpr 109	. . . . . 6 |- ( ( ( R Er X /\ x e. A ) /\ x R C ) -> x R C )
11	9, 10	erthi 6405	. . . . 5 |- ( ( ( R Er X /\ x e. A ) /\ x R C ) -> [ x ] R = [ C ] R )
12	11	ex 114	. . . 4 |- ( ( R Er X /\ x e. A ) -> ( x R C -> [ x ] R = [ C ] R ) )
13	8, 12	syl5 32	. . 3 |- ( ( R Er X /\ x e. A ) -> ( C e. [ x ] R -> [ x ] R = [ C ] R ) )
14	1, 4, 13	ectocld 6425	. 2 |- ( ( R Er X /\ B e. ( A /. R ) ) -> ( C e. B -> B = [ C ] R ) )
15	14	3impia 1146	1 |- ( ( R Er X /\ B e. ( A /. R ) /\ C e. B ) -> B = [ C ] R )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   /\ w3a 930   = wceq 1299   e. wcel 1448   _Vcvv 2641   class class class wbr 3875   Er wer 6356   [cec 6357   /.cqs 6358

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-sep 3986  ax-pow 4038  ax-pr 4069

This theorem depends on definitions:  df-bi 116  df-3an 932  df-tru 1302  df-nf 1405  df-sb 1704  df-eu 1963  df-mo 1964  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ral 2380  df-rex 2381  df-v 2643  df-sbc 2863  df-un 3025  df-in 3027  df-ss 3034  df-pw 3459  df-sn 3480  df-pr 3481  df-op 3483  df-br 3876  df-opab 3930  df-xp 4483  df-rel 4484  df-cnv 4485  df-co 4486  df-dm 4487  df-rn 4488  df-res 4489  df-ima 4490  df-er 6359  df-ec 6361  df-qs 6365

This theorem is referenced by: (None)