Theorem brsset 30649

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 30648	. . 3 |- Rel SSet
2	1	brrelexi 4891	. 2 |- ( A SSet B -> A e. _V )
3		brsset.1	. . 3 |- B e. _V
4	3	ssex 4565	. 2 |- ( A C_ B -> A e. _V )
5		breq1 4423	. . 3 |- ( x = A -> ( x SSet B <-> A SSet B ) )
6		sseq1 3485	. . 3 |- ( x = A -> ( x C_ B <-> A C_ B ) )
7		opex 4682	. . . . . . 7 |- <. x , B >. e. _V
8	7	elrn 5091	. . . . . 6 |- ( <. x , B >. e. ran ( _E (x) ( _V \ _E ) ) <-> E. y y ( _E (x) ( _V \ _E ) ) <. x , B >. )
9		vex 3084	. . . . . . . . 9 |- y e. _V
10		vex 3084	. . . . . . . . 9 |- x e. _V
11	9, 10, 3	brtxp 30640	. . . . . . . 8 |- ( y ( _E (x) ( _V \ _E ) ) <. x , B >. <-> ( y _E x /\ y ( _V \ _E ) B ) )
12		epel 4764	. . . . . . . . 9 |- ( y _E x <-> y e. x )
13		brv 30637	. . . . . . . . . . 11 |- y _V B
14		brdif 4471	. . . . . . . . . . 11 |- ( y ( _V \ _E ) B <-> ( y _V B /\ -. y _E B ) )
15	13, 14	mpbiran 926	. . . . . . . . . 10 |- ( y ( _V \ _E ) B <-> -. y _E B )
16	3	epelc 4763	. . . . . . . . . 10 |- ( y _E B <-> y e. B )
17	15, 16	xchbinx 311	. . . . . . . . 9 |- ( y ( _V \ _E ) B <-> -. y e. B )
18	12, 17	anbi12i 701	. . . . . . . 8 |- ( ( y _E x /\ y ( _V \ _E ) B ) <-> ( y e. x /\ -. y e. B ) )
19	11, 18	bitri 252	. . . . . . 7 |- ( y ( _E (x) ( _V \ _E ) ) <. x , B >. <-> ( y e. x /\ -. y e. B ) )
20	19	exbii 1712	. . . . . 6 |- ( E. y y ( _E (x) ( _V \ _E ) ) <. x , B >. <-> E. y ( y e. x /\ -. y e. B ) )
21		exanali 1715	. . . . . 6 |- ( E. y ( y e. x /\ -. y e. B ) <-> -. A. y ( y e. x -> y e. B ) )
22	8, 20, 21	3bitrri 275	. . . . 5 |- ( -. A. y ( y e. x -> y e. B ) <-> <. x , B >. e. ran ( _E (x) ( _V \ _E ) ) )
23	22	con1bii 332	. . . 4 |- ( -. <. x , B >. e. ran ( _E (x) ( _V \ _E ) ) <-> A. y ( y e. x -> y e. B ) )
24		df-br 4421	. . . . 5 |- ( x SSet B <-> <. x , B >. e. SSet )
25		df-sset 30615	. . . . . . 7 |- SSet = ( ( _V X. _V ) \ ran ( _E (x) ( _V \ _E ) ) )
26	25	eleq2i 2500	. . . . . 6 |- ( <. x , B >. e. SSet <-> <. x , B >. e. ( ( _V X. _V ) \ ran ( _E (x) ( _V \ _E ) ) ) )
27	10, 3	opelvv 4897	. . . . . . 7 |- <. x , B >. e. ( _V X. _V )
28		eldif 3446	. . . . . . 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 926	. . . . . 6 |- ( <. x , B >. e. ( ( _V X. _V ) \ ran ( _E (x) ( _V \ _E ) ) ) <-> -. <. x , B >. e. ran ( _E (x) ( _V \ _E ) ) )
30	26, 29	bitri 252	. . . . 5 |- ( <. x , B >. e. SSet <-> -. <. x , B >. e. ran ( _E (x) ( _V \ _E ) ) )
31	24, 30	bitri 252	. . . 4 |- ( x SSet B <-> -. <. x , B >. e. ran ( _E (x) ( _V \ _E ) ) )
32		dfss2 3453	. . . 4 |- ( x C_ B <-> A. y ( y e. x -> y e. B ) )
33	23, 31, 32	3bitr4i 280	. . 3 |- ( x SSet B <-> x C_ B )
34	5, 6, 33	vtoclbg 3140	. 2 |- ( A e. _V -> ( A SSet B <-> A C_ B ) )
35	2, 4, 34	pm5.21nii 354	1 |- ( A SSet B <-> A C_ B )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 187   /\ wa 370   A.wal 1435   E.wex 1659   e. wcel 1868   _Vcvv 3081   \ cdif 3433   C_ wss 3436   <.cop 4002   class class class wbr 4420   _E cep 4759   X. cxp 4848   ran crn 4851   (x) ctxp 30589   SSetcsset 30591

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 1839  ax-8 1870  ax-9 1872  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400  ax-sep 4543  ax-nul 4552  ax-pow 4599  ax-pr 4657  ax-un 6594

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 2269  df-mo 2270  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-ne 2620  df-ral 2780  df-rex 2781  df-rab 2784  df-v 3083  df-sbc 3300  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-if 3910  df-sn 3997  df-pr 3999  df-op 4003  df-uni 4217  df-br 4421  df-opab 4480  df-mpt 4481  df-eprel 4761  df-id 4765  df-xp 4856  df-rel 4857  df-cnv 4858  df-co 4859  df-dm 4860  df-rn 4861  df-res 4862  df-iota 5562  df-fun 5600  df-fn 5601  df-f 5602  df-fo 5604  df-fv 5606  df-1st 6804  df-2nd 6805  df-txp 30613  df-sset 30615

This theorem is referenced by:  idsset  30650  dfon3  30652  imagesset  30713