Theorem brsset 29514

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 29513	. . 3 |- Rel SSet
2	1	brrelexi 5030	. 2 |- ( A SSet B -> A e. _V )
3		brsset.1	. . 3 |- B e. _V
4	3	ssex 4581	. 2 |- ( A C_ B -> A e. _V )
5		breq1 4440	. . 3 |- ( x = A -> ( x SSet B <-> A SSet B ) )
6		sseq1 3510	. . 3 |- ( x = A -> ( x C_ B <-> A C_ B ) )
7		opex 4701	. . . . . . 7 |- <. x , B >. e. _V
8	7	elrn 5233	. . . . . 6 |- ( <. x , B >. e. ran ( _E (x) ( _V \ _E ) ) <-> E. y y ( _E (x) ( _V \ _E ) ) <. x , B >. )
9		vex 3098	. . . . . . . . 9 |- y e. _V
10		vex 3098	. . . . . . . . 9 |- x e. _V
11	9, 10, 3	brtxp 29505	. . . . . . . 8 |- ( y ( _E (x) ( _V \ _E ) ) <. x , B >. <-> ( y _E x /\ y ( _V \ _E ) B ) )
12		epel 4784	. . . . . . . . 9 |- ( y _E x <-> y e. x )
13		brv 29502	. . . . . . . . . . 11 |- y _V B
14		brdif 4487	. . . . . . . . . . 11 |- ( y ( _V \ _E ) B <-> ( y _V B /\ -. y _E B ) )
15	13, 14	mpbiran 918	. . . . . . . . . 10 |- ( y ( _V \ _E ) B <-> -. y _E B )
16	3	epelc 4783	. . . . . . . . . 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 1654	. . . . . 6 |- ( E. y y ( _E (x) ( _V \ _E ) ) <. x , B >. <-> E. y ( y e. x /\ -. y e. B ) )
21		exanali 1657	. . . . . 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 4438	. . . . 5 |- ( x SSet B <-> <. x , B >. e. SSet )
25		df-sset 29480	. . . . . . 7 |- SSet = ( ( _V X. _V ) \ ran ( _E (x) ( _V \ _E ) ) )
26	25	eleq2i 2521	. . . . . 6 |- ( <. x , B >. e. SSet <-> <. x , B >. e. ( ( _V X. _V ) \ ran ( _E (x) ( _V \ _E ) ) ) )
27	10, 3	opelvv 5036	. . . . . . 7 |- <. x , B >. e. ( _V X. _V )
28		eldif 3471	. . . . . . 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 918	. . . . . 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 3478	. . . 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 3154	. 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 1381   E.wex 1599   e. wcel 1804   _Vcvv 3095   \ cdif 3458   C_ wss 3461   <.cop 4020   class class class wbr 4437   _E cep 4779   X. cxp 4987   ran crn 4990   (x) ctxp 29454   SSetcsset 29456

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-rab 2802  df-v 3097  df-sbc 3314  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-sn 4015  df-pr 4017  df-op 4021  df-uni 4235  df-br 4438  df-opab 4496  df-mpt 4497  df-eprel 4781  df-id 4785  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-fo 5584  df-fv 5586  df-1st 6785  df-2nd 6786  df-txp 29478  df-sset 29480

This theorem is referenced by:  idsset  29515  dfon3  29517  imagesset  29578