Theorem qsel 7826

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 2622	. . 3 |- ( A /. R ) = ( A /. R )
2		eleq2 2690	. . . 4 |- ( [ x ] R = B -> ( C e. [ x ] R <-> C e. B ) )
3		eqeq1 2626	. . . 4 |- ( [ x ] R = B -> ( [ x ] R = [ C ] R <-> B = [ C ] R ) )
4	2, 3	imbi12d 334	. . 3 |- ( [ x ] R = B -> ( ( C e. [ x ] R -> [ x ] R = [ C ] R ) <-> ( C e. B -> B = [ C ] R ) ) )
5		vex 3203	. . . . . 6 |- x e. _V
6		elecg 7785	. . . . . 6 |- ( ( C e. [ x ] R /\ x e. _V ) -> ( C e. [ x ] R <-> x R C ) )
7	5, 6	mpan2 707	. . . . 5 |- ( C e. [ x ] R -> ( C e. [ x ] R <-> x R C ) )
8	7	ibi 256	. . . 4 |- ( C e. [ x ] R -> x R C )
9		simpll 790	. . . . . 6 |- ( ( ( R Er X /\ x e. A ) /\ x R C ) -> R Er X )
10		simpr 477	. . . . . 6 |- ( ( ( R Er X /\ x e. A ) /\ x R C ) -> x R C )
11	9, 10	erthi 7793	. . . . 5 |- ( ( ( R Er X /\ x e. A ) /\ x R C ) -> [ x ] R = [ C ] R )
12	11	ex 450	. . . 4 |- ( ( R Er X /\ x e. A ) -> ( x R C -> [ x ] R = [ C ] R ) )
13	8, 12	syl5 34	. . 3 |- ( ( R Er X /\ x e. A ) -> ( C e. [ x ] R -> [ x ] R = [ C ] R ) )
14	1, 4, 13	ectocld 7814	. 2 |- ( ( R Er X /\ B e. ( A /. R ) ) -> ( C e. B -> B = [ C ] R ) )
15	14	3impia 1261	1 |- ( ( R Er X /\ B e. ( A /. R ) /\ C e. B ) -> B = [ C ] R )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   _Vcvv 3200   class class class wbr 4653   Er wer 7739   [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-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  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-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-er 7742  df-ec 7744  df-qs 7748

This theorem is referenced by:  frgpnabllem2  18277  prter3  34167