Theorem brtxpsd 30654

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 4402	. . 3 |- ( A ran ( ( _V (x) _E ) /_\ ( R (x) _V ) ) B <-> <. A , B >. e. ran ( ( _V (x) _E ) /_\ ( R (x) _V ) ) )
2		opex 4663	. . . . 5 |- <. A , B >. e. _V
3	2	elrn 5074	. . . 4 |- ( <. A , B >. e. ran ( ( _V (x) _E ) /_\ ( R (x) _V ) ) <-> E. x x ( ( _V (x) _E ) /_\ ( R (x) _V ) ) <. A , B >. )
4		brsymdif 4458	. . . . . 6 |- ( x ( ( _V (x) _E ) /_\ ( R (x) _V ) ) <. A , B >. <-> -. ( x ( _V (x) _E ) <. A , B >. <-> x ( R (x) _V ) <. A , B >. ) )
5		brv 30637	. . . . . . . . 9 |- x _V A
6		vex 3047	. . . . . . . . . 10 |- x e. _V
7		brtxpsd.1	. . . . . . . . . 10 |- A e. _V
8		brtxpsd.2	. . . . . . . . . 10 |- B e. _V
9	6, 7, 8	brtxp 30640	. . . . . . . . 9 |- ( x ( _V (x) _E ) <. A , B >. <-> ( x _V A /\ x _E B ) )
10	5, 9	mpbiran 928	. . . . . . . 8 |- ( x ( _V (x) _E ) <. A , B >. <-> x _E B )
11	8	epelc 4746	. . . . . . . 8 |- ( x _E B <-> x e. B )
12	10, 11	bitri 253	. . . . . . 7 |- ( x ( _V (x) _E ) <. A , B >. <-> x e. B )
13		brv 30637	. . . . . . . 8 |- x _V B
14	6, 7, 8	brtxp 30640	. . . . . . . 8 |- ( x ( R (x) _V ) <. A , B >. <-> ( x R A /\ x _V B ) )
15	13, 14	mpbiran2 929	. . . . . . 7 |- ( x ( R (x) _V ) <. A , B >. <-> x R A )
16	12, 15	bibi12i 317	. . . . . 6 |- ( ( x ( _V (x) _E ) <. A , B >. <-> x ( R (x) _V ) <. A , B >. ) <-> ( x e. B <-> x R A ) )
17	4, 16	xchbinx 312	. . . . 5 |- ( x ( ( _V (x) _E ) /_\ ( R (x) _V ) ) <. A , B >. <-> -. ( x e. B <-> x R A ) )
18	17	exbii 1717	. . . 4 |- ( E. x x ( ( _V (x) _E ) /_\ ( R (x) _V ) ) <. A , B >. <-> E. x -. ( x e. B <-> x R A ) )
19	3, 18	bitri 253	. . 3 |- ( <. A , B >. e. ran ( ( _V (x) _E ) /_\ ( R (x) _V ) ) <-> E. x -. ( x e. B <-> x R A ) )
20		exnal 1698	. . 3 |- ( E. x -. ( x e. B <-> x R A ) <-> -. A. x ( x e. B <-> x R A ) )
21	1, 19, 20	3bitrri 276	. 2 |- ( -. A. x ( x e. B <-> x R A ) <-> A ran ( ( _V (x) _E ) /_\ ( R (x) _V ) ) B )
22	21	con1bii 333	1 |- ( -. A ran ( ( _V (x) _E ) /_\ ( R (x) _V ) ) B <-> A. x ( x e. B <-> x R A ) )

Syntax hints:   -. wn 3   <-> wb 188   A.wal 1441   E.wex 1662   e. wcel 1886   _Vcvv 3044   /_\ csymdif 3661   <.cop 3973   class class class wbr 4401   _E cep 4742   ran crn 4834   (x) ctxp 30589

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-8 1888  ax-9 1895  ax-10 1914  ax-11 1919  ax-12 1932  ax-13 2090  ax-ext 2430  ax-sep 4524  ax-nul 4533  ax-pow 4580  ax-pr 4638  ax-un 6580

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 986  df-tru 1446  df-ex 1663  df-nf 1667  df-sb 1797  df-eu 2302  df-mo 2303  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2580  df-ne 2623  df-ral 2741  df-rex 2742  df-rab 2745  df-v 3046  df-sbc 3267  df-dif 3406  df-un 3408  df-in 3410  df-ss 3417  df-symdif 3662  df-nul 3731  df-if 3881  df-sn 3968  df-pr 3970  df-op 3974  df-uni 4198  df-br 4402  df-opab 4461  df-mpt 4462  df-eprel 4744  df-id 4748  df-xp 4839  df-rel 4840  df-cnv 4841  df-co 4842  df-dm 4843  df-rn 4844  df-res 4845  df-iota 5545  df-fun 5583  df-fn 5584  df-f 5585  df-fo 5587  df-fv 5589  df-1st 6790  df-2nd 6791  df-txp 30613

This theorem is referenced by:  brtxpsd2  30655