Theorem brtxpsd 32001

Description: Expansion of a common form used in quantifier-free definitions. (Contributed by Scott Fenton, 17-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

Hypotheses

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

Assertion

Ref	Expression
brtxpsd	|- ( -. A ran ( ( _V (x) _E ) /_\ ( R (x) _V ) ) B <-> A. x ( x e. B <-> x R A ) )

Distinct variable groups:   x, A   x, B   x, R

Proof of Theorem brtxpsd

Step	Hyp	Ref	Expression
1		df-br 4654	. . 3 |- ( A ran ( ( _V (x) _E ) /_\ ( R (x) _V ) ) B <-> <. A , B >. e. ran ( ( _V (x) _E ) /_\ ( R (x) _V ) ) )
2		opex 4932	. . . . 5 |- <. A , B >. e. _V
3	2	elrn 5366	. . . 4 |- ( <. A , B >. e. ran ( ( _V (x) _E ) /_\ ( R (x) _V ) ) <-> E. x x ( ( _V (x) _E ) /_\ ( R (x) _V ) ) <. A , B >. )
4		brsymdif 4711	. . . . . 6 |- ( x ( ( _V (x) _E ) /_\ ( R (x) _V ) ) <. A , B >. <-> -. ( x ( _V (x) _E ) <. A , B >. <-> x ( R (x) _V ) <. A , B >. ) )
5		brv 4941	. . . . . . . . 9 |- x _V A
6		vex 3203	. . . . . . . . . 10 |- x e. _V
7		brtxpsd.1	. . . . . . . . . 10 |- A e. _V
8		brtxpsd.2	. . . . . . . . . 10 |- B e. _V
9	6, 7, 8	brtxp 31987	. . . . . . . . 9 |- ( x ( _V (x) _E ) <. A , B >. <-> ( x _V A /\ x _E B ) )
10	5, 9	mpbiran 953	. . . . . . . 8 |- ( x ( _V (x) _E ) <. A , B >. <-> x _E B )
11	8	epelc 5031	. . . . . . . 8 |- ( x _E B <-> x e. B )
12	10, 11	bitri 264	. . . . . . 7 |- ( x ( _V (x) _E ) <. A , B >. <-> x e. B )
13		brv 4941	. . . . . . . 8 |- x _V B
14	6, 7, 8	brtxp 31987	. . . . . . . 8 |- ( x ( R (x) _V ) <. A , B >. <-> ( x R A /\ x _V B ) )
15	13, 14	mpbiran2 954	. . . . . . 7 |- ( x ( R (x) _V ) <. A , B >. <-> x R A )
16	12, 15	bibi12i 329	. . . . . 6 |- ( ( x ( _V (x) _E ) <. A , B >. <-> x ( R (x) _V ) <. A , B >. ) <-> ( x e. B <-> x R A ) )
17	4, 16	xchbinx 324	. . . . 5 |- ( x ( ( _V (x) _E ) /_\ ( R (x) _V ) ) <. A , B >. <-> -. ( x e. B <-> x R A ) )
18	17	exbii 1774	. . . 4 |- ( E. x x ( ( _V (x) _E ) /_\ ( R (x) _V ) ) <. A , B >. <-> E. x -. ( x e. B <-> x R A ) )
19	3, 18	bitri 264	. . 3 |- ( <. A , B >. e. ran ( ( _V (x) _E ) /_\ ( R (x) _V ) ) <-> E. x -. ( x e. B <-> x R A ) )
20		exnal 1754	. . 3 |- ( E. x -. ( x e. B <-> x R A ) <-> -. A. x ( x e. B <-> x R A ) )
21	1, 19, 20	3bitrri 287	. 2 |- ( -. A. x ( x e. B <-> x R A ) <-> A ran ( ( _V (x) _E ) /_\ ( R (x) _V ) ) B )
22	21	con1bii 346	1 |- ( -. A ran ( ( _V (x) _E ) /_\ ( R (x) _V ) ) B <-> A. x ( x e. B <-> x R A ) )

Syntax hints:   -. wn 3   <-> wb 196   A.wal 1481   E.wex 1704   e. wcel 1990   _Vcvv 3200   /_\ csymdif 3843   <.cop 4183   class class class wbr 4653   _E cep 5028   ran crn 5115   (x) ctxp 31937

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

This theorem is referenced by:  brtxpsd2  32002