Theorem ssrnres 5046

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 3343	. . . . 5 |- ( C i^i ( A X. B ) ) C_ ( A X. B )
2		rnss 4834	. . . . 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 5035	. . . 4 |- ran ( A X. B ) C_ B
5	3, 4	sstri 3151	. . 3 |- ran ( C i^i ( A X. B ) ) C_ B
6		eqss 3157	. . 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 930	. 2 |- ( ran ( C i^i ( A X. B ) ) = B <-> B C_ ran ( C i^i ( A X. B ) ) )
8		ssid 3162	. . . . . . . 8 |- A C_ A
9		ssv 3164	. . . . . . . 8 |- B C_ _V
10		xpss12 4711	. . . . . . . 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 3348	. . . . . . 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 4616	. . . . . 6 |- ( C |` A ) = ( C i^i ( A X. _V ) )
15	13, 14	sseqtrri 3177	. . . . 5 |- ( C i^i ( A X. B ) ) C_ ( C |` A )
16		rnss 4834	. . . . 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 3150	. . . 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 3136	. . . . . . 7 |- ( B C_ ran ( C |` A ) -> ( y e. B -> y e. ran ( C |` A ) ) )
21		vex 2729	. . . . . . . 8 |- y e. _V
22	21	elrn2 4846	. . . . . . 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 4846	. . . . . 6 |- ( y e. ran ( C i^i ( A X. B ) ) <-> E. x <. x , y >. e. ( C i^i ( A X. B ) ) )
26		elin 3305	. . . . . . . 8 |- ( <. x , y >. e. ( C i^i ( A X. B ) ) <-> ( <. x , y >. e. C /\ <. x , y >. e. ( A X. B ) ) )
27		opelxp 4634	. . . . . . . . 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 4889	. . . . . . . . . 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 1593	. . . . . 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 1890	. . . . . 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 3148	. . 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 1343   E.wex 1480   e. wcel 2136   _Vcvv 2726   i^i cin 3115   C_ wss 3116   <.cop 3579   X. cxp 4602   ran crn 4605   |` cres 4606

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-br 3983  df-opab 4044  df-xp 4610  df-rel 4611  df-cnv 4612  df-dm 4614  df-rn 4615  df-res 4616

This theorem is referenced by:  rninxp  5047