Theorem qsel 6590

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 2170	. . 3 |- ( A /. R ) = ( A /. R )
2		eleq2 2234	. . . 4 |- ( [ x ] R = B -> ( C e. [ x ] R <-> C e. B ) )
3		eqeq1 2177	. . . 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 2733	. . . . . 6 |- x e. _V
6		elecg 6551	. . . . . 6 |- ( ( C e. [ x ] R /\ x e. _V ) -> ( C e. [ x ] R <-> x R C ) )
7	5, 6	mpan2 423	. . . . 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 524	. . . . . 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 6559	. . . . 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 6579	. 2 |- ( ( R Er X /\ B e. ( A /. R ) ) -> ( C e. B -> B = [ C ] R ) )
15	14	3impia 1195	1 |- ( ( R Er X /\ B e. ( A /. R ) /\ C e. B ) -> B = [ C ] R )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   /\ w3a 973   = wceq 1348   e. wcel 2141   _Vcvv 2730   class class class wbr 3989   Er wer 6510   [cec 6511   /.cqs 6512

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-sbc 2956  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-br 3990  df-opab 4051  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-er 6513  df-ec 6515  df-qs 6519

This theorem is referenced by: (None)