Theorem brsset 30704

Description: For sets, the SSet binary relationship 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 30703	. . 3 |- Rel SSet
2	1	brrelexi 4893	. 2 |- ( A SSet B -> A e. _V )
3		brsset.1	. . 3 |- B e. _V
4	3	ssex 4560	. 2 |- ( A C_ B -> A e. _V )
5		breq1 4418	. . 3 |- ( x = A -> ( x SSet B <-> A SSet B ) )
6		sseq1 3464	. . 3 |- ( x = A -> ( x C_ B <-> A C_ B ) )
7		opex 4677	. . . . . . 7 |- <. x , B >. e. _V
8	7	elrn 5093	. . . . . 6 |- ( <. x , B >. e. ran ( _E (x) ( _V \ _E ) ) <-> E. y y ( _E (x) ( _V \ _E ) ) <. x , B >. )
9		vex 3059	. . . . . . . . 9 |- y e. _V
10		vex 3059	. . . . . . . . 9 |- x e. _V
11	9, 10, 3	brtxp 30695	. . . . . . . 8 |- ( y ( _E (x) ( _V \ _E ) ) <. x , B >. <-> ( y _E x /\ y ( _V \ _E ) B ) )
12		epel 4766	. . . . . . . . 9 |- ( y _E x <-> y e. x )
13		brv 30692	. . . . . . . . . . 11 |- y _V B
14		brdif 4466	. . . . . . . . . . 11 |- ( y ( _V \ _E ) B <-> ( y _V B /\ -. y _E B ) )
15	13, 14	mpbiran 934	. . . . . . . . . 10 |- ( y ( _V \ _E ) B <-> -. y _E B )
16	3	epelc 4765	. . . . . . . . . 10 |- ( y _E B <-> y e. B )
17	15, 16	xchbinx 316	. . . . . . . . 9 |- ( y ( _V \ _E ) B <-> -. y e. B )
18	12, 17	anbi12i 708	. . . . . . . 8 |- ( ( y _E x /\ y ( _V \ _E ) B ) <-> ( y e. x /\ -. y e. B ) )
19	11, 18	bitri 257	. . . . . . 7 |- ( y ( _E (x) ( _V \ _E ) ) <. x , B >. <-> ( y e. x /\ -. y e. B ) )
20	19	exbii 1728	. . . . . 6 |- ( E. y y ( _E (x) ( _V \ _E ) ) <. x , B >. <-> E. y ( y e. x /\ -. y e. B ) )
21		exanali 1731	. . . . . 6 |- ( E. y ( y e. x /\ -. y e. B ) <-> -. A. y ( y e. x -> y e. B ) )
22	8, 20, 21	3bitrri 280	. . . . 5 |- ( -. A. y ( y e. x -> y e. B ) <-> <. x , B >. e. ran ( _E (x) ( _V \ _E ) ) )
23	22	con1bii 337	. . . 4 |- ( -. <. x , B >. e. ran ( _E (x) ( _V \ _E ) ) <-> A. y ( y e. x -> y e. B ) )
24		df-br 4416	. . . . 5 |- ( x SSet B <-> <. x , B >. e. SSet )
25		df-sset 30670	. . . . . . 7 |- SSet = ( ( _V X. _V ) \ ran ( _E (x) ( _V \ _E ) ) )
26	25	eleq2i 2531	. . . . . 6 |- ( <. x , B >. e. SSet <-> <. x , B >. e. ( ( _V X. _V ) \ ran ( _E (x) ( _V \ _E ) ) ) )
27	10, 3	opelvv 4899	. . . . . . 7 |- <. x , B >. e. ( _V X. _V )
28		eldif 3425	. . . . . . 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 934	. . . . . 6 |- ( <. x , B >. e. ( ( _V X. _V ) \ ran ( _E (x) ( _V \ _E ) ) ) <-> -. <. x , B >. e. ran ( _E (x) ( _V \ _E ) ) )
30	26, 29	bitri 257	. . . . 5 |- ( <. x , B >. e. SSet <-> -. <. x , B >. e. ran ( _E (x) ( _V \ _E ) ) )
31	24, 30	bitri 257	. . . 4 |- ( x SSet B <-> -. <. x , B >. e. ran ( _E (x) ( _V \ _E ) ) )
32		dfss2 3432	. . . 4 |- ( x C_ B <-> A. y ( y e. x -> y e. B ) )
33	23, 31, 32	3bitr4i 285	. . 3 |- ( x SSet B <-> x C_ B )
34	5, 6, 33	vtoclbg 3119	. 2 |- ( A e. _V -> ( A SSet B <-> A C_ B ) )
35	2, 4, 34	pm5.21nii 359	1 |- ( A SSet B <-> A C_ B )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 189   /\ wa 375   A.wal 1452   E.wex 1673   e. wcel 1897   _Vcvv 3056   \ cdif 3412   C_ wss 3415   <.cop 3985   class class class wbr 4415   _E cep 4761   X. cxp 4850   ran crn 4853   (x) ctxp 30644   SSetcsset 30646

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1679  ax-4 1692  ax-5 1768  ax-6 1815  ax-7 1861  ax-8 1899  ax-9 1906  ax-10 1925  ax-11 1930  ax-12 1943  ax-13 2101  ax-ext 2441  ax-sep 4538  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6609

This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3an 993  df-tru 1457  df-ex 1674  df-nf 1678  df-sb 1808  df-eu 2313  df-mo 2314  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2591  df-ne 2634  df-ral 2753  df-rex 2754  df-rab 2757  df-v 3058  df-sbc 3279  df-dif 3418  df-un 3420  df-in 3422  df-ss 3429  df-nul 3743  df-if 3893  df-sn 3980  df-pr 3982  df-op 3986  df-uni 4212  df-br 4416  df-opab 4475  df-mpt 4476  df-eprel 4763  df-id 4767  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-iota 5564  df-fun 5602  df-fn 5603  df-f 5604  df-fo 5606  df-fv 5608  df-1st 6819  df-2nd 6820  df-txp 30668  df-sset 30670

This theorem is referenced by:  idsset  30705  dfon3  30707  imagesset  30768