Theorem ssrnres 5230

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 3452	. . . . 5 |- ( C i^i ( A X. B ) ) C_ ( A X. B )
2		rnss 5012	. . . . 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 5219	. . . 4 |- ran ( A X. B ) C_ B
5	3, 4	sstri 3257	. . 3 |- ran ( C i^i ( A X. B ) ) C_ B
6		eqss 3263	. . 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 953	. 2 |- ( ran ( C i^i ( A X. B ) ) = B <-> B C_ ran ( C i^i ( A X. B ) ) )
8		ssid 3268	. . . . . . . 8 |- A C_ A
9		ssv 3270	. . . . . . . 8 |- B C_ _V
10		xpss12 4882	. . . . . . . 8 |- ( ( A C_ A /\ B C_ _V ) -> ( A X. B ) C_ ( A X. _V ) )
11	8, 9, 10	mp2an 430	. . . . . . 7 |- ( A X. B ) C_ ( A X. _V )
12		sslin 3457	. . . . . . 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 4786	. . . . . 6 |- ( C |` A ) = ( C i^i ( A X. _V ) )
15	13, 14	sseqtrri 3283	. . . . 5 |- ( C i^i ( A X. B ) ) C_ ( C |` A )
16		rnss 5012	. . . . 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 3256	. . . 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 429	. . 3 |- ( B C_ ran ( C i^i ( A X. B ) ) -> B C_ ran ( C |` A ) )
20		ssel 3242	. . . . . . 7 |- ( B C_ ran ( C |` A ) -> ( y e. B -> y e. ran ( C |` A ) ) )
21		vex 2824	. . . . . . . 8 |- y e. _V
22	21	elrn2 5024	. . . . . . 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 5024	. . . . . 6 |- ( y e. ran ( C i^i ( A X. B ) ) <-> E. x <. x , y >. e. ( C i^i ( A X. B ) ) )
26		elin 3412	. . . . . . . 8 |- ( <. x , y >. e. ( C i^i ( A X. B ) ) <-> ( <. x , y >. e. C /\ <. x , y >. e. ( A X. B ) ) )
27		opelxp 4804	. . . . . . . . 9 |- ( <. x , y >. e. ( A X. B ) <-> ( x e. A /\ y e. B ) )
28	27	anbi2i 461	. . . . . . . 8 |- ( ( <. x , y >. e. C /\ <. x , y >. e. ( A X. B ) ) <-> ( <. x , y >. e. C /\ ( x e. A /\ y e. B ) ) )
29	21	opelres 5068	. . . . . . . . . 10 |- ( <. x , y >. e. ( C |` A ) <-> ( <. x , y >. e. C /\ x e. A ) )
30	29	anbi1i 462	. . . . . . . . 9 |- ( ( <. x , y >. e. ( C |` A ) /\ y e. B ) <-> ( ( <. x , y >. e. C /\ x e. A ) /\ y e. B ) )
31		anass 405	. . . . . . . . 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 1658	. . . . . 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 1958	. . . . . 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 3254	. . 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 )

This proof depends on syntax axioms:   /\ wa 104   <-> wb 105   = wceq 1402   E.wex 1545   e. wcel 2209   _Vcvv 2821   i^i cin 3219   C_ wss 3220   <.cop 3712   X. cxp 4772   ran crn 4775   |` cres 4776

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-pow 4311  ax-pr 4346

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-br 4131  df-opab 4193  df-xp 4780  df-rel 4781  df-cnv 4782  df-dm 4784  df-rn 4785  df-res 4786

This theorem is used by:  rninxp  5231