Theorem brsset 31996

Description: For sets, the SSet binary relation is equivalent to the subset relationship. (Contributed by Scott Fenton, 31-Mar-2012.)

Hypothesis

Ref	Expression
brsset.1	|- B e. _V

Assertion

Ref	Expression
brsset	|- ( A SSet B <-> A C_ B )

Proof of Theorem brsset

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		relsset 31995	. . 3 |- Rel SSet
2	1	brrelexi 5158	. 2 |- ( A SSet B -> A e. _V )
3		brsset.1	. . 3 |- B e. _V
4	3	ssex 4802	. 2 |- ( A C_ B -> A e. _V )
5		breq1 4656	. . 3 |- ( x = A -> ( x SSet B <-> A SSet B ) )
6		sseq1 3626	. . 3 |- ( x = A -> ( x C_ B <-> A C_ B ) )
7		opex 4932	. . . . . . 7 |- <. x , B >. e. _V
8	7	elrn 5366	. . . . . 6 |- ( <. x , B >. e. ran ( _E (x) ( _V \ _E ) ) <-> E. y y ( _E (x) ( _V \ _E ) ) <. x , B >. )
9		vex 3203	. . . . . . . . 9 |- y e. _V
10		vex 3203	. . . . . . . . 9 |- x e. _V
11	9, 10, 3	brtxp 31987	. . . . . . . 8 |- ( y ( _E (x) ( _V \ _E ) ) <. x , B >. <-> ( y _E x /\ y ( _V \ _E ) B ) )
12		epel 5032	. . . . . . . . 9 |- ( y _E x <-> y e. x )
13		brv 4941	. . . . . . . . . . 11 |- y _V B
14		brdif 4705	. . . . . . . . . . 11 |- ( y ( _V \ _E ) B <-> ( y _V B /\ -. y _E B ) )
15	13, 14	mpbiran 953	. . . . . . . . . 10 |- ( y ( _V \ _E ) B <-> -. y _E B )
16	3	epelc 5031	. . . . . . . . . 10 |- ( y _E B <-> y e. B )
17	15, 16	xchbinx 324	. . . . . . . . 9 |- ( y ( _V \ _E ) B <-> -. y e. B )
18	12, 17	anbi12i 733	. . . . . . . 8 |- ( ( y _E x /\ y ( _V \ _E ) B ) <-> ( y e. x /\ -. y e. B ) )
19	11, 18	bitri 264	. . . . . . 7 |- ( y ( _E (x) ( _V \ _E ) ) <. x , B >. <-> ( y e. x /\ -. y e. B ) )
20	19	exbii 1774	. . . . . 6 |- ( E. y y ( _E (x) ( _V \ _E ) ) <. x , B >. <-> E. y ( y e. x /\ -. y e. B ) )
21		exanali 1786	. . . . . 6 |- ( E. y ( y e. x /\ -. y e. B ) <-> -. A. y ( y e. x -> y e. B ) )
22	8, 20, 21	3bitrri 287	. . . . 5 |- ( -. A. y ( y e. x -> y e. B ) <-> <. x , B >. e. ran ( _E (x) ( _V \ _E ) ) )
23	22	con1bii 346	. . . 4 |- ( -. <. x , B >. e. ran ( _E (x) ( _V \ _E ) ) <-> A. y ( y e. x -> y e. B ) )
24		df-br 4654	. . . . 5 |- ( x SSet B <-> <. x , B >. e. SSet )
25		df-sset 31963	. . . . . . 7 |- SSet = ( ( _V X. _V ) \ ran ( _E (x) ( _V \ _E ) ) )
26	25	eleq2i 2693	. . . . . 6 |- ( <. x , B >. e. SSet <-> <. x , B >. e. ( ( _V X. _V ) \ ran ( _E (x) ( _V \ _E ) ) ) )
27	10, 3	opelvv 5166	. . . . . . 7 |- <. x , B >. e. ( _V X. _V )
28		eldif 3584	. . . . . . 7 |- ( <. x , B >. e. ( ( _V X. _V ) \ ran ( _E (x) ( _V \ _E ) ) ) <-> ( <. x , B >. e. ( _V X. _V ) /\ -. <. x , B >. e. ran ( _E (x) ( _V \ _E ) ) ) )
29	27, 28	mpbiran 953	. . . . . 6 |- ( <. x , B >. e. ( ( _V X. _V ) \ ran ( _E (x) ( _V \ _E ) ) ) <-> -. <. x , B >. e. ran ( _E (x) ( _V \ _E ) ) )
30	26, 29	bitri 264	. . . . 5 |- ( <. x , B >. e. SSet <-> -. <. x , B >. e. ran ( _E (x) ( _V \ _E ) ) )
31	24, 30	bitri 264	. . . 4 |- ( x SSet B <-> -. <. x , B >. e. ran ( _E (x) ( _V \ _E ) ) )
32		dfss2 3591	. . . 4 |- ( x C_ B <-> A. y ( y e. x -> y e. B ) )
33	23, 31, 32	3bitr4i 292	. . 3 |- ( x SSet B <-> x C_ B )
34	5, 6, 33	vtoclbg 3267	. 2 |- ( A e. _V -> ( A SSet B <-> A C_ B ) )
35	2, 4, 34	pm5.21nii 368	1 |- ( A SSet B <-> A C_ B )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   A.wal 1481   E.wex 1704   e. wcel 1990   _Vcvv 3200   \ cdif 3571   C_ wss 3574   <.cop 4183   class class class wbr 4653   _E cep 5028   X. cxp 5112   ran crn 5115   (x) ctxp 31937   SSetcsset 31939

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-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-sset 31963

This theorem is referenced by:  idsset  31997  dfon3  31999  imagesset  32060