Theorem brsingle 32024

Description: The binary relation form of the singleton function. (Contributed by Scott Fenton, 4-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

Hypotheses

Ref	Expression
brsingle.1	|- A e. _V
brsingle.2	|- B e. _V

Assertion

Ref	Expression
brsingle	|- ( ASingleton B <-> B = { A } )

Proof of Theorem brsingle

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		brsingle.1	. 2 |- A e. _V
2		brsingle.2	. 2 |- B e. _V
3		df-singleton 31969	. 2 |- Singleton = ( ( _V X. _V ) \ ran ( ( _V (x) _E ) /_\ ( _I (x) _V ) ) )
4		brxp 5147	. . 3 |- ( A ( _V X. _V ) B <-> ( A e. _V /\ B e. _V ) )
5	1, 2, 4	mpbir2an 955	. 2 |- A ( _V X. _V ) B
6		velsn 4193	. . 3 |- ( x e. { A } <-> x = A )
7	1	ideq 5274	. . 3 |- ( x _I A <-> x = A )
8	6, 7	bitr4i 267	. 2 |- ( x e. { A } <-> x _I A )
9	1, 2, 3, 5, 8	brtxpsd3 32003	1 |- ( ASingleton B <-> B = { A } )

Syntax hints:   <-> wb 196   = wceq 1483   e. wcel 1990   _Vcvv 3200   {csn 4177   class class class wbr 4653   _I cid 5023   X. cxp 5112  Singletoncsingle 31945

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-8 1992  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-pow 4843  ax-pr 4906  ax-un 6949

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-symdif 3844  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-eprel 5029  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fo 5894  df-fv 5896  df-1st 7168  df-2nd 7169  df-txp 31961  df-singleton 31969

This theorem is referenced by:  elsingles  32025  fnsingle  32026  fvsingle  32027  brapply  32045  brsuccf  32048  funpartlem  32049