Theorem qsel 6668

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 2193	. . 3 |- ( A /. R ) = ( A /. R )
2		eleq2 2257	. . . 4 |- ( [ x ] R = B -> ( C e. [ x ] R <-> C e. B ) )
3		eqeq1 2200	. . . 4 |- ( [ x ] R = B -> ( [ x ] R = [ C ] R <-> B = [ C ] R ) )
4	2, 3	imbi12d 234	. . 3 |- ( [ x ] R = B -> ( ( C e. [ x ] R -> [ x ] R = [ C ] R ) <-> ( C e. B -> B = [ C ] R ) ) )
5		vex 2763	. . . . . 6 |- x e. _V
6		elecg 6629	. . . . . 6 |- ( ( C e. [ x ] R /\ x e. _V ) -> ( C e. [ x ] R <-> x R C ) )
7	5, 6	mpan2 425	. . . . 5 |- ( C e. [ x ] R -> ( C e. [ x ] R <-> x R C ) )
8	7	ibi 176	. . . 4 |- ( C e. [ x ] R -> x R C )
9		simpll 527	. . . . . 6 |- ( ( ( R Er X /\ x e. A ) /\ x R C ) -> R Er X )
10		simpr 110	. . . . . 6 |- ( ( ( R Er X /\ x e. A ) /\ x R C ) -> x R C )
11	9, 10	erthi 6637	. . . . 5 |- ( ( ( R Er X /\ x e. A ) /\ x R C ) -> [ x ] R = [ C ] R )
12	11	ex 115	. . . 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 6657	. 2 |- ( ( R Er X /\ B e. ( A /. R ) ) -> ( C e. B -> B = [ C ] R ) )
15	14	3impia 1202	1 |- ( ( R Er X /\ B e. ( A /. R ) /\ C e. B ) -> B = [ C ] R )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 980   = wceq 1364   e. wcel 2164   _Vcvv 2760   class class class wbr 4030   Er wer 6586   [cec 6587   /.cqs 6588

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4148  ax-pow 4204  ax-pr 4239

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-sbc 2987  df-un 3158  df-in 3160  df-ss 3167  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-br 4031  df-opab 4092  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-res 4672  df-ima 4673  df-er 6589  df-ec 6591  df-qs 6595

This theorem is referenced by: (None)