Theorem rnss 4906

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 4849	. . 3 |- ( A C_ B -> `' A C_ `' B )
2		dmss 4875	. . 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 4684	. 2 |- ran A = dom `' A
5		df-rn 4684	. 2 |- ran B = dom `' B
6	3, 4, 5	3sstr4g 3235	1 |- ( A C_ B -> ran A C_ ran B )

Syntax hints:   -> wi 4   C_ wss 3165   `'ccnv 4672   dom cdm 4673   ran crn 4674

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 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-ext 2186

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1375  df-nf 1483  df-sb 1785  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-v 2773  df-un 3169  df-in 3171  df-ss 3178  df-sn 3638  df-pr 3639  df-op 3641  df-br 4044  df-opab 4105  df-cnv 4681  df-dm 4683  df-rn 4684

This theorem is referenced by:  imass1  5054  imass2  5055  rnxpss2  5113  ssxpbm  5115  ssxp2  5117  ssrnres  5122  funssxp  5439  fssres  5445  dff2  5718  fliftf  5858  1stcof  6239  2ndcof  6240  smores  6368  tfrcllembfn  6433  caserel  7171  frecuzrdgtcl  10538  prdsvallem  13022  prdsval  13023  lmss  14636  txss12  14656  txbasval  14657