Theorem rnss 4907

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 4850	. . 3 |- ( A C_ B -> `' A C_ `' B )
2		dmss 4876	. . 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 4685	. 2 |- ran A = dom `' A
5		df-rn 4685	. 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 4673   dom cdm 4674   ran crn 4675

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 4682  df-dm 4684  df-rn 4685

This theorem is referenced by:  imass1  5056  imass2  5057  rnxpss2  5115  ssxpbm  5117  ssxp2  5119  ssrnres  5124  funssxp  5444  fssres  5450  dff2  5723  fliftf  5867  1stcof  6248  2ndcof  6249  smores  6377  tfrcllembfn  6442  caserel  7188  frecuzrdgtcl  10555  prdsvallem  13046  prdsval  13047  lmss  14660  txss12  14680  txbasval  14681