Theorem brtxpsd 30647

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 4418	. . 3 |- ( A ran ( ( _V (x) _E ) /_\ ( R (x) _V ) ) B <-> <. A , B >. e. ran ( ( _V (x) _E ) /_\ ( R (x) _V ) ) )
2		opex 4678	. . . . 5 |- <. A , B >. e. _V
3	2	elrn 5087	. . . 4 |- ( <. A , B >. e. ran ( ( _V (x) _E ) /_\ ( R (x) _V ) ) <-> E. x x ( ( _V (x) _E ) /_\ ( R (x) _V ) ) <. A , B >. )
4		brsymdif 4474	. . . . . 6 |- ( x ( ( _V (x) _E ) /_\ ( R (x) _V ) ) <. A , B >. <-> -. ( x ( _V (x) _E ) <. A , B >. <-> x ( R (x) _V ) <. A , B >. ) )
5		brv 30630	. . . . . . . . 9 |- x _V A
6		vex 3081	. . . . . . . . . 10 |- x e. _V
7		brtxpsd.1	. . . . . . . . . 10 |- A e. _V
8		brtxpsd.2	. . . . . . . . . 10 |- B e. _V
9	6, 7, 8	brtxp 30633	. . . . . . . . 9 |- ( x ( _V (x) _E ) <. A , B >. <-> ( x _V A /\ x _E B ) )
10	5, 9	mpbiran 926	. . . . . . . 8 |- ( x ( _V (x) _E ) <. A , B >. <-> x _E B )
11	8	epelc 4759	. . . . . . . 8 |- ( x _E B <-> x e. B )
12	10, 11	bitri 252	. . . . . . 7 |- ( x ( _V (x) _E ) <. A , B >. <-> x e. B )
13		brv 30630	. . . . . . . 8 |- x _V B
14	6, 7, 8	brtxp 30633	. . . . . . . 8 |- ( x ( R (x) _V ) <. A , B >. <-> ( x R A /\ x _V B ) )
15	13, 14	mpbiran2 927	. . . . . . 7 |- ( x ( R (x) _V ) <. A , B >. <-> x R A )
16	12, 15	bibi12i 316	. . . . . 6 |- ( ( x ( _V (x) _E ) <. A , B >. <-> x ( R (x) _V ) <. A , B >. ) <-> ( x e. B <-> x R A ) )
17	4, 16	xchbinx 311	. . . . 5 |- ( x ( ( _V (x) _E ) /_\ ( R (x) _V ) ) <. A , B >. <-> -. ( x e. B <-> x R A ) )
18	17	exbii 1712	. . . 4 |- ( E. x x ( ( _V (x) _E ) /_\ ( R (x) _V ) ) <. A , B >. <-> E. x -. ( x e. B <-> x R A ) )
19	3, 18	bitri 252	. . 3 |- ( <. A , B >. e. ran ( ( _V (x) _E ) /_\ ( R (x) _V ) ) <-> E. x -. ( x e. B <-> x R A ) )
20		exnal 1695	. . 3 |- ( E. x -. ( x e. B <-> x R A ) <-> -. A. x ( x e. B <-> x R A ) )
21	1, 19, 20	3bitrri 275	. 2 |- ( -. A. x ( x e. B <-> x R A ) <-> A ran ( ( _V (x) _E ) /_\ ( R (x) _V ) ) B )
22	21	con1bii 332	1 |- ( -. A ran ( ( _V (x) _E ) /_\ ( R (x) _V ) ) B <-> A. x ( x e. B <-> x R A ) )

Syntax hints:   -. wn 3   <-> wb 187   A.wal 1435   E.wex 1659   e. wcel 1867   _Vcvv 3078   /_\ csymdif 3689   <.cop 3999   class class class wbr 4417   _E cep 4755   ran crn 4847   (x) ctxp 30582

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-8 1869  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-sep 4540  ax-nul 4548  ax-pow 4595  ax-pr 4653  ax-un 6589

This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-ral 2778  df-rex 2779  df-rab 2782  df-v 3080  df-sbc 3297  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-symdif 3690  df-nul 3759  df-if 3907  df-sn 3994  df-pr 3996  df-op 4000  df-uni 4214  df-br 4418  df-opab 4477  df-mpt 4478  df-eprel 4757  df-id 4761  df-xp 4852  df-rel 4853  df-cnv 4854  df-co 4855  df-dm 4856  df-rn 4857  df-res 4858  df-iota 5557  df-fun 5595  df-fn 5596  df-f 5597  df-fo 5599  df-fv 5601  df-1st 6799  df-2nd 6800  df-txp 30606

This theorem is referenced by:  brtxpsd2  30648