Theorem ssrnres 4989

Description: Subset of the range of a restriction. (Contributed by NM, 16-Jan-2006.)

Assertion

Ref	Expression
ssrnres	|- ( B C_ ran ( C |` A ) <-> ran ( C i^i ( A X. B ) ) = B )

Proof of Theorem ssrnres

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

Step	Hyp	Ref	Expression
1		inss2 3302	. . . . 5 |- ( C i^i ( A X. B ) ) C_ ( A X. B )
2		rnss 4777	. . . . 5 |- ( ( C i^i ( A X. B ) ) C_ ( A X. B ) -> ran ( C i^i ( A X. B ) ) C_ ran ( A X. B ) )
3	1, 2	ax-mp 5	. . . 4 |- ran ( C i^i ( A X. B ) ) C_ ran ( A X. B )
4		rnxpss 4978	. . . 4 |- ran ( A X. B ) C_ B
5	3, 4	sstri 3111	. . 3 |- ran ( C i^i ( A X. B ) ) C_ B
6		eqss 3117	. . 3 |- ( ran ( C i^i ( A X. B ) ) = B <-> ( ran ( C i^i ( A X. B ) ) C_ B /\ B C_ ran ( C i^i ( A X. B ) ) ) )
7	5, 6	mpbiran 925	. 2 |- ( ran ( C i^i ( A X. B ) ) = B <-> B C_ ran ( C i^i ( A X. B ) ) )
8		ssid 3122	. . . . . . . 8 |- A C_ A
9		ssv 3124	. . . . . . . 8 |- B C_ _V
10		xpss12 4654	. . . . . . . 8 |- ( ( A C_ A /\ B C_ _V ) -> ( A X. B ) C_ ( A X. _V ) )
11	8, 9, 10	mp2an 423	. . . . . . 7 |- ( A X. B ) C_ ( A X. _V )
12		sslin 3307	. . . . . . 7 |- ( ( A X. B ) C_ ( A X. _V ) -> ( C i^i ( A X. B ) ) C_ ( C i^i ( A X. _V ) ) )
13	11, 12	ax-mp 5	. . . . . 6 |- ( C i^i ( A X. B ) ) C_ ( C i^i ( A X. _V ) )
14		df-res 4559	. . . . . 6 |- ( C |` A ) = ( C i^i ( A X. _V ) )
15	13, 14	sseqtrri 3137	. . . . 5 |- ( C i^i ( A X. B ) ) C_ ( C |` A )
16		rnss 4777	. . . . 5 |- ( ( C i^i ( A X. B ) ) C_ ( C |` A ) -> ran ( C i^i ( A X. B ) ) C_ ran ( C |` A ) )
17	15, 16	ax-mp 5	. . . 4 |- ran ( C i^i ( A X. B ) ) C_ ran ( C |` A )
18		sstr 3110	. . . 4 |- ( ( B C_ ran ( C i^i ( A X. B ) ) /\ ran ( C i^i ( A X. B ) ) C_ ran ( C |` A ) ) -> B C_ ran ( C |` A ) )
19	17, 18	mpan2 422	. . 3 |- ( B C_ ran ( C i^i ( A X. B ) ) -> B C_ ran ( C |` A ) )
20		ssel 3096	. . . . . . 7 |- ( B C_ ran ( C |` A ) -> ( y e. B -> y e. ran ( C |` A ) ) )
21		vex 2692	. . . . . . . 8 |- y e. _V
22	21	elrn2 4789	. . . . . . 7 |- ( y e. ran ( C |` A ) <-> E. x <. x , y >. e. ( C |` A ) )
23	20, 22	syl6ib 160	. . . . . 6 |- ( B C_ ran ( C |` A ) -> ( y e. B -> E. x <. x , y >. e. ( C |` A ) ) )
24	23	ancrd 324	. . . . 5 |- ( B C_ ran ( C |` A ) -> ( y e. B -> ( E. x <. x , y >. e. ( C |` A ) /\ y e. B ) ) )
25	21	elrn2 4789	. . . . . 6 |- ( y e. ran ( C i^i ( A X. B ) ) <-> E. x <. x , y >. e. ( C i^i ( A X. B ) ) )
26		elin 3264	. . . . . . . 8 |- ( <. x , y >. e. ( C i^i ( A X. B ) ) <-> ( <. x , y >. e. C /\ <. x , y >. e. ( A X. B ) ) )
27		opelxp 4577	. . . . . . . . 9 |- ( <. x , y >. e. ( A X. B ) <-> ( x e. A /\ y e. B ) )
28	27	anbi2i 453	. . . . . . . 8 |- ( ( <. x , y >. e. C /\ <. x , y >. e. ( A X. B ) ) <-> ( <. x , y >. e. C /\ ( x e. A /\ y e. B ) ) )
29	21	opelres 4832	. . . . . . . . . 10 |- ( <. x , y >. e. ( C |` A ) <-> ( <. x , y >. e. C /\ x e. A ) )
30	29	anbi1i 454	. . . . . . . . 9 |- ( ( <. x , y >. e. ( C |` A ) /\ y e. B ) <-> ( ( <. x , y >. e. C /\ x e. A ) /\ y e. B ) )
31		anass 399	. . . . . . . . 9 |- ( ( ( <. x , y >. e. C /\ x e. A ) /\ y e. B ) <-> ( <. x , y >. e. C /\ ( x e. A /\ y e. B ) ) )
32	30, 31	bitr2i 184	. . . . . . . 8 |- ( ( <. x , y >. e. C /\ ( x e. A /\ y e. B ) ) <-> ( <. x , y >. e. ( C |` A ) /\ y e. B ) )
33	26, 28, 32	3bitri 205	. . . . . . 7 |- ( <. x , y >. e. ( C i^i ( A X. B ) ) <-> ( <. x , y >. e. ( C |` A ) /\ y e. B ) )
34	33	exbii 1585	. . . . . 6 |- ( E. x <. x , y >. e. ( C i^i ( A X. B ) ) <-> E. x ( <. x , y >. e. ( C |` A ) /\ y e. B ) )
35		19.41v 1875	. . . . . 6 |- ( E. x ( <. x , y >. e. ( C |` A ) /\ y e. B ) <-> ( E. x <. x , y >. e. ( C |` A ) /\ y e. B ) )
36	25, 34, 35	3bitri 205	. . . . 5 |- ( y e. ran ( C i^i ( A X. B ) ) <-> ( E. x <. x , y >. e. ( C |` A ) /\ y e. B ) )
37	24, 36	syl6ibr 161	. . . 4 |- ( B C_ ran ( C |` A ) -> ( y e. B -> y e. ran ( C i^i ( A X. B ) ) ) )
38	37	ssrdv 3108	. . 3 |- ( B C_ ran ( C |` A ) -> B C_ ran ( C i^i ( A X. B ) ) )
39	19, 38	impbii 125	. 2 |- ( B C_ ran ( C i^i ( A X. B ) ) <-> B C_ ran ( C |` A ) )
40	7, 39	bitr2i 184	1 |- ( B C_ ran ( C |` A ) <-> ran ( C i^i ( A X. B ) ) = B )

Syntax hints:   /\ wa 103   <-> wb 104   = wceq 1332   E.wex 1469   e. wcel 1481   _Vcvv 2689   i^i cin 3075   C_ wss 3076   <.cop 3535   X. cxp 4545   ran crn 4548   |` cres 4549

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4054  ax-pow 4106  ax-pr 4139

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2691  df-un 3080  df-in 3082  df-ss 3089  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-br 3938  df-opab 3998  df-xp 4553  df-rel 4554  df-cnv 4555  df-dm 4557  df-rn 4558  df-res 4559

This theorem is referenced by:  rninxp  4990