Theorem rnss 4992

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 4933	. . 3 |- ( A C_ B -> `' A C_ `' B )
2		dmss 4960	. . 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 4765	. 2 |- ran A = dom `' A
5		df-rn 4765	. 2 |- ran B = dom `' B
6	3, 4, 5	3sstr4g 3285	1 |- ( A C_ B -> ran A C_ ran B )

Syntax hints:   -> wi 4   C_ wss 3214   `'ccnv 4753   dom cdm 4754   ran crn 4755

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2216

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-v 2817  df-un 3218  df-in 3220  df-ss 3227  df-sn 3700  df-pr 3701  df-op 3703  df-br 4115  df-opab 4177  df-cnv 4762  df-dm 4764  df-rn 4765

This theorem is referenced by:  imass1  5142  imass2  5143  rnxpss2  5201  ssxpbm  5203  ssxp2  5205  ssrnres  5210  funssxp  5537  fssres  5545  dff2  5826  fliftf  5978  1stcof  6370  2ndcof  6371  smores  6536  tfrcllembfn  6601  caserel  7391  frecuzrdgtcl  10798  prdsvallem  13564  prdsval  14115  lmss  15237  txss12  15257  txbasval  15258  subgrprop3  16383