Theorem brsset 28056

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 28055	. . 3 |- Rel SSet
2	1	brrelexi 4979	. 2 |- ( A SSet B -> A e. _V )
3		brsset.1	. . 3 |- B e. _V
4	3	ssex 4536	. 2 |- ( A C_ B -> A e. _V )
5		breq1 4395	. . 3 |- ( x = A -> ( x SSet B <-> A SSet B ) )
6		sseq1 3477	. . 3 |- ( x = A -> ( x C_ B <-> A C_ B ) )
7		opex 4656	. . . . . . 7 |- <. x , B >. e. _V
8	7	elrn 5180	. . . . . 6 |- ( <. x , B >. e. ran ( _E (x) ( _V \ _E ) ) <-> E. y y ( _E (x) ( _V \ _E ) ) <. x , B >. )
9		vex 3073	. . . . . . . . 9 |- y e. _V
10		vex 3073	. . . . . . . . 9 |- x e. _V
11	9, 10, 3	brtxp 28047	. . . . . . . 8 |- ( y ( _E (x) ( _V \ _E ) ) <. x , B >. <-> ( y _E x /\ y ( _V \ _E ) B ) )
12		epel 4735	. . . . . . . . 9 |- ( y _E x <-> y e. x )
13		brv 28044	. . . . . . . . . . 11 |- y _V B
14		brdif 4442	. . . . . . . . . . 11 |- ( y ( _V \ _E ) B <-> ( y _V B /\ -. y _E B ) )
15	13, 14	mpbiran 909	. . . . . . . . . 10 |- ( y ( _V \ _E ) B <-> -. y _E B )
16	3	epelc 4734	. . . . . . . . . 10 |- ( y _E B <-> y e. B )
17	15, 16	xchbinx 310	. . . . . . . . 9 |- ( y ( _V \ _E ) B <-> -. y e. B )
18	12, 17	anbi12i 697	. . . . . . . 8 |- ( ( y _E x /\ y ( _V \ _E ) B ) <-> ( y e. x /\ -. y e. B ) )
19	11, 18	bitri 249	. . . . . . 7 |- ( y ( _E (x) ( _V \ _E ) ) <. x , B >. <-> ( y e. x /\ -. y e. B ) )
20	19	exbii 1635	. . . . . 6 |- ( E. y y ( _E (x) ( _V \ _E ) ) <. x , B >. <-> E. y ( y e. x /\ -. y e. B ) )
21		exanali 1638	. . . . . 6 |- ( E. y ( y e. x /\ -. y e. B ) <-> -. A. y ( y e. x -> y e. B ) )
22	8, 20, 21	3bitrri 272	. . . . 5 |- ( -. A. y ( y e. x -> y e. B ) <-> <. x , B >. e. ran ( _E (x) ( _V \ _E ) ) )
23	22	con1bii 331	. . . 4 |- ( -. <. x , B >. e. ran ( _E (x) ( _V \ _E ) ) <-> A. y ( y e. x -> y e. B ) )
24		df-br 4393	. . . . 5 |- ( x SSet B <-> <. x , B >. e. SSet )
25		df-sset 28022	. . . . . . 7 |- SSet = ( ( _V X. _V ) \ ran ( _E (x) ( _V \ _E ) ) )
26	25	eleq2i 2529	. . . . . 6 |- ( <. x , B >. e. SSet <-> <. x , B >. e. ( ( _V X. _V ) \ ran ( _E (x) ( _V \ _E ) ) ) )
27	10, 3	opelvv 4985	. . . . . . 7 |- <. x , B >. e. ( _V X. _V )
28		eldif 3438	. . . . . . 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 909	. . . . . 6 |- ( <. x , B >. e. ( ( _V X. _V ) \ ran ( _E (x) ( _V \ _E ) ) ) <-> -. <. x , B >. e. ran ( _E (x) ( _V \ _E ) ) )
30	26, 29	bitri 249	. . . . 5 |- ( <. x , B >. e. SSet <-> -. <. x , B >. e. ran ( _E (x) ( _V \ _E ) ) )
31	24, 30	bitri 249	. . . 4 |- ( x SSet B <-> -. <. x , B >. e. ran ( _E (x) ( _V \ _E ) ) )
32		dfss2 3445	. . . 4 |- ( x C_ B <-> A. y ( y e. x -> y e. B ) )
33	23, 31, 32	3bitr4i 277	. . 3 |- ( x SSet B <-> x C_ B )
34	5, 6, 33	vtoclbg 3129	. 2 |- ( A e. _V -> ( A SSet B <-> A C_ B ) )
35	2, 4, 34	pm5.21nii 353	1 |- ( A SSet B <-> A C_ B )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 184   /\ wa 369   A.wal 1368   E.wex 1587   e. wcel 1758   _Vcvv 3070   \ cdif 3425   C_ wss 3428   <.cop 3983   class class class wbr 4392   _E cep 4730   X. cxp 4938   ran crn 4941   (x) ctxp 27996   SSetcsset 27998

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4513  ax-nul 4521  ax-pow 4570  ax-pr 4631  ax-un 6474

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3072  df-sbc 3287  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-nul 3738  df-if 3892  df-sn 3978  df-pr 3980  df-op 3984  df-uni 4192  df-br 4393  df-opab 4451  df-mpt 4452  df-eprel 4732  df-id 4736  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-iota 5481  df-fun 5520  df-fn 5521  df-f 5522  df-fo 5524  df-fv 5526  df-1st 6679  df-2nd 6680  df-txp 28020  df-sset 28022

This theorem is referenced by:  idsset  28057  dfon3  28059  imagesset  28120