Theorem brtxpsd 30732

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 4396	. . 3 |- ( A ran ( ( _V (x) _E ) /_\ ( R (x) _V ) ) B <-> <. A , B >. e. ran ( ( _V (x) _E ) /_\ ( R (x) _V ) ) )
2		opex 4664	. . . . 5 |- <. A , B >. e. _V
3	2	elrn 5081	. . . 4 |- ( <. A , B >. e. ran ( ( _V (x) _E ) /_\ ( R (x) _V ) ) <-> E. x x ( ( _V (x) _E ) /_\ ( R (x) _V ) ) <. A , B >. )
4		brsymdif 4452	. . . . . 6 |- ( x ( ( _V (x) _E ) /_\ ( R (x) _V ) ) <. A , B >. <-> -. ( x ( _V (x) _E ) <. A , B >. <-> x ( R (x) _V ) <. A , B >. ) )
5		brv 30715	. . . . . . . . 9 |- x _V A
6		vex 3034	. . . . . . . . . 10 |- x e. _V
7		brtxpsd.1	. . . . . . . . . 10 |- A e. _V
8		brtxpsd.2	. . . . . . . . . 10 |- B e. _V
9	6, 7, 8	brtxp 30718	. . . . . . . . 9 |- ( x ( _V (x) _E ) <. A , B >. <-> ( x _V A /\ x _E B ) )
10	5, 9	mpbiran 932	. . . . . . . 8 |- ( x ( _V (x) _E ) <. A , B >. <-> x _E B )
11	8	epelc 4752	. . . . . . . 8 |- ( x _E B <-> x e. B )
12	10, 11	bitri 257	. . . . . . 7 |- ( x ( _V (x) _E ) <. A , B >. <-> x e. B )
13		brv 30715	. . . . . . . 8 |- x _V B
14	6, 7, 8	brtxp 30718	. . . . . . . 8 |- ( x ( R (x) _V ) <. A , B >. <-> ( x R A /\ x _V B ) )
15	13, 14	mpbiran2 933	. . . . . . 7 |- ( x ( R (x) _V ) <. A , B >. <-> x R A )
16	12, 15	bibi12i 322	. . . . . 6 |- ( ( x ( _V (x) _E ) <. A , B >. <-> x ( R (x) _V ) <. A , B >. ) <-> ( x e. B <-> x R A ) )
17	4, 16	xchbinx 317	. . . . 5 |- ( x ( ( _V (x) _E ) /_\ ( R (x) _V ) ) <. A , B >. <-> -. ( x e. B <-> x R A ) )
18	17	exbii 1726	. . . 4 |- ( E. x x ( ( _V (x) _E ) /_\ ( R (x) _V ) ) <. A , B >. <-> E. x -. ( x e. B <-> x R A ) )
19	3, 18	bitri 257	. . 3 |- ( <. A , B >. e. ran ( ( _V (x) _E ) /_\ ( R (x) _V ) ) <-> E. x -. ( x e. B <-> x R A ) )
20		exnal 1707	. . 3 |- ( E. x -. ( x e. B <-> x R A ) <-> -. A. x ( x e. B <-> x R A ) )
21	1, 19, 20	3bitrri 280	. 2 |- ( -. A. x ( x e. B <-> x R A ) <-> A ran ( ( _V (x) _E ) /_\ ( R (x) _V ) ) B )
22	21	con1bii 338	1 |- ( -. A ran ( ( _V (x) _E ) /_\ ( R (x) _V ) ) B <-> A. x ( x e. B <-> x R A ) )

Syntax hints:   -. wn 3   <-> wb 189   A.wal 1450   E.wex 1671   e. wcel 1904   _Vcvv 3031   /_\ csymdif 3653   <.cop 3965   class class class wbr 4395   _E cep 4748   ran crn 4840   (x) ctxp 30667

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602

This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-sbc 3256  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-symdif 3654  df-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-br 4396  df-opab 4455  df-mpt 4456  df-eprel 4750  df-id 4754  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-fo 5595  df-fv 5597  df-1st 6812  df-2nd 6813  df-txp 30691

This theorem is referenced by:  brtxpsd2  30733