Theorem qsel 6206

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 2081	. . 3 |- ( A /. R ) = ( A /. R )
2		eleq2 2142	. . . 4 |- ( [ x ] R = B -> ( C e. [ x ] R <-> C e. B ) )
3		eqeq1 2087	. . . 4 |- ( [ x ] R = B -> ( [ x ] R = [ C ] R <-> B = [ C ] R ) )
4	2, 3	imbi12d 232	. . 3 |- ( [ x ] R = B -> ( ( C e. [ x ] R -> [ x ] R = [ C ] R ) <-> ( C e. B -> B = [ C ] R ) ) )
5		vex 2604	. . . . . 6 |- x e. _V
6		elecg 6167	. . . . . 6 |- ( ( C e. [ x ] R /\ x e. _V ) -> ( C e. [ x ] R <-> x R C ) )
7	5, 6	mpan2 415	. . . . 5 |- ( C e. [ x ] R -> ( C e. [ x ] R <-> x R C ) )
8	7	ibi 174	. . . 4 |- ( C e. [ x ] R -> x R C )
9		simpll 495	. . . . . 6 |- ( ( ( R Er X /\ x e. A ) /\ x R C ) -> R Er X )
10		simpr 108	. . . . . 6 |- ( ( ( R Er X /\ x e. A ) /\ x R C ) -> x R C )
11	9, 10	erthi 6175	. . . . 5 |- ( ( ( R Er X /\ x e. A ) /\ x R C ) -> [ x ] R = [ C ] R )
12	11	ex 113	. . . 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 6195	. 2 |- ( ( R Er X /\ B e. ( A /. R ) ) -> ( C e. B -> B = [ C ] R ) )
15	14	3impia 1135	1 |- ( ( R Er X /\ B e. ( A /. R ) /\ C e. B ) -> B = [ C ] R )

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   /\ w3a 919   = wceq 1284   e. wcel 1433   _Vcvv 2601   class class class wbr 3785   Er wer 6126   [cec 6127   /.cqs 6128

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-v 2603  df-sbc 2816  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-br 3786  df-opab 3840  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-rn 4374  df-res 4375  df-ima 4376  df-er 6129  df-ec 6131  df-qs 6135

This theorem is referenced by: (None)