Theorem rnss 4953

Description: Subset theorem for range. (Contributed by NM, 22-Mar-1998.)

Assertion

Ref	Expression
rnss	|- ( A C_ B -> ran A C_ ran B )

Proof of Theorem rnss

Step	Hyp	Ref	Expression
1		cnvss 4894	. . 3 |- ( A C_ B -> `' A C_ `' B )
2		dmss 4921	. . 3 |- ( `' A C_ `' B -> dom `' A C_ dom `' B )
3	1, 2	syl 14	. 2 |- ( A C_ B -> dom `' A C_ dom `' B )
4		df-rn 4729	. 2 |- ran A = dom `' A
5		df-rn 4729	. 2 |- ran B = dom `' B
6	3, 4, 5	3sstr4g 3267	1 |- ( A C_ B -> ran A C_ ran B )

Syntax hints:   -> wi 4   C_ wss 3197   `'ccnv 4717   dom cdm 4718   ran crn 4719

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2801  df-un 3201  df-in 3203  df-ss 3210  df-sn 3672  df-pr 3673  df-op 3675  df-br 4083  df-opab 4145  df-cnv 4726  df-dm 4728  df-rn 4729

This theorem is referenced by:  imass1  5102  imass2  5103  rnxpss2  5161  ssxpbm  5163  ssxp2  5165  ssrnres  5170  funssxp  5492  fssres  5500  dff2  5778  fliftf  5922  1stcof  6307  2ndcof  6308  smores  6436  tfrcllembfn  6501  caserel  7250  frecuzrdgtcl  10629  prdsvallem  13300  prdsval  13301  lmss  14914  txss12  14934  txbasval  14935