Theorem rnss 4954

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 4895	. . 3 |- ( A C_ B -> `' A C_ `' B )
2		dmss 4922	. . 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 4730	. 2 |- ran A = dom `' A
5		df-rn 4730	. 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 4718   dom cdm 4719   ran crn 4720

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 4084  df-opab 4146  df-cnv 4727  df-dm 4729  df-rn 4730

This theorem is referenced by:  imass1  5103  imass2  5104  rnxpss2  5162  ssxpbm  5164  ssxp2  5166  ssrnres  5171  funssxp  5495  fssres  5503  dff2  5781  fliftf  5929  1stcof  6315  2ndcof  6316  smores  6444  tfrcllembfn  6509  caserel  7265  frecuzrdgtcl  10646  prdsvallem  13320  prdsval  13321  lmss  14935  txss12  14955  txbasval  14956