Theorem ssrnres 5112

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 3384	. . . . 5 |- ( C i^i ( A X. B ) ) C_ ( A X. B )
2		rnss 4896	. . . . 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 5101	. . . 4 |- ran ( A X. B ) C_ B
5	3, 4	sstri 3192	. . 3 |- ran ( C i^i ( A X. B ) ) C_ B
6		eqss 3198	. . 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 942	. 2 |- ( ran ( C i^i ( A X. B ) ) = B <-> B C_ ran ( C i^i ( A X. B ) ) )
8		ssid 3203	. . . . . . . 8 |- A C_ A
9		ssv 3205	. . . . . . . 8 |- B C_ _V
10		xpss12 4770	. . . . . . . 8 |- ( ( A C_ A /\ B C_ _V ) -> ( A X. B ) C_ ( A X. _V ) )
11	8, 9, 10	mp2an 426	. . . . . . 7 |- ( A X. B ) C_ ( A X. _V )
12		sslin 3389	. . . . . . 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 4675	. . . . . 6 |- ( C |` A ) = ( C i^i ( A X. _V ) )
15	13, 14	sseqtrri 3218	. . . . 5 |- ( C i^i ( A X. B ) ) C_ ( C |` A )
16		rnss 4896	. . . . 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 3191	. . . 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 425	. . 3 |- ( B C_ ran ( C i^i ( A X. B ) ) -> B C_ ran ( C |` A ) )
20		ssel 3177	. . . . . . 7 |- ( B C_ ran ( C |` A ) -> ( y e. B -> y e. ran ( C |` A ) ) )
21		vex 2766	. . . . . . . 8 |- y e. _V
22	21	elrn2 4908	. . . . . . 7 |- ( y e. ran ( C |` A ) <-> E. x <. x , y >. e. ( C |` A ) )
23	20, 22	imbitrdi 161	. . . . . 6 |- ( B C_ ran ( C |` A ) -> ( y e. B -> E. x <. x , y >. e. ( C |` A ) ) )
24	23	ancrd 326	. . . . 5 |- ( B C_ ran ( C |` A ) -> ( y e. B -> ( E. x <. x , y >. e. ( C |` A ) /\ y e. B ) ) )
25	21	elrn2 4908	. . . . . 6 |- ( y e. ran ( C i^i ( A X. B ) ) <-> E. x <. x , y >. e. ( C i^i ( A X. B ) ) )
26		elin 3346	. . . . . . . 8 |- ( <. x , y >. e. ( C i^i ( A X. B ) ) <-> ( <. x , y >. e. C /\ <. x , y >. e. ( A X. B ) ) )
27		opelxp 4693	. . . . . . . . 9 |- ( <. x , y >. e. ( A X. B ) <-> ( x e. A /\ y e. B ) )
28	27	anbi2i 457	. . . . . . . 8 |- ( ( <. x , y >. e. C /\ <. x , y >. e. ( A X. B ) ) <-> ( <. x , y >. e. C /\ ( x e. A /\ y e. B ) ) )
29	21	opelres 4951	. . . . . . . . . 10 |- ( <. x , y >. e. ( C |` A ) <-> ( <. x , y >. e. C /\ x e. A ) )
30	29	anbi1i 458	. . . . . . . . 9 |- ( ( <. x , y >. e. ( C |` A ) /\ y e. B ) <-> ( ( <. x , y >. e. C /\ x e. A ) /\ y e. B ) )
31		anass 401	. . . . . . . . 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 185	. . . . . . . 8 |- ( ( <. x , y >. e. C /\ ( x e. A /\ y e. B ) ) <-> ( <. x , y >. e. ( C |` A ) /\ y e. B ) )
33	26, 28, 32	3bitri 206	. . . . . . 7 |- ( <. x , y >. e. ( C i^i ( A X. B ) ) <-> ( <. x , y >. e. ( C |` A ) /\ y e. B ) )
34	33	exbii 1619	. . . . . 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 1917	. . . . . 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 206	. . . . 5 |- ( y e. ran ( C i^i ( A X. B ) ) <-> ( E. x <. x , y >. e. ( C |` A ) /\ y e. B ) )
37	24, 36	imbitrrdi 162	. . . 4 |- ( B C_ ran ( C |` A ) -> ( y e. B -> y e. ran ( C i^i ( A X. B ) ) ) )
38	37	ssrdv 3189	. . 3 |- ( B C_ ran ( C |` A ) -> B C_ ran ( C i^i ( A X. B ) ) )
39	19, 38	impbii 126	. 2 |- ( B C_ ran ( C i^i ( A X. B ) ) <-> B C_ ran ( C |` A ) )
40	7, 39	bitr2i 185	1 |- ( B C_ ran ( C |` A ) <-> ran ( C i^i ( A X. B ) ) = B )

Syntax hints:   /\ wa 104   <-> wb 105   = wceq 1364   E.wex 1506   e. wcel 2167   _Vcvv 2763   i^i cin 3156   C_ wss 3157   <.cop 3625   X. cxp 4661   ran crn 4664   |` cres 4665

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-pow 4207  ax-pr 4242

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-v 2765  df-un 3161  df-in 3163  df-ss 3170  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-br 4034  df-opab 4095  df-xp 4669  df-rel 4670  df-cnv 4671  df-dm 4673  df-rn 4674  df-res 4675

This theorem is referenced by:  rninxp  5113