Theorem rnss 4665

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 4609	. . 3 |- ( A C_ B -> `' A C_ `' B )
2		dmss 4635	. . 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 4449	. 2 |- ran A = dom `' A
5		df-rn 4449	. 2 |- ran B = dom `' B
6	3, 4, 5	3sstr4g 3067	1 |- ( A C_ B -> ran A C_ ran B )

Syntax hints:   -> wi 4   C_ wss 2999   `'ccnv 4437   dom cdm 4438   ran crn 4439

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070

This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621  df-un 3003  df-in 3005  df-ss 3012  df-sn 3452  df-pr 3453  df-op 3455  df-br 3846  df-opab 3900  df-cnv 4446  df-dm 4448  df-rn 4449

This theorem is referenced by:  imass1  4807  imass2  4808  rnxpss2  4864  ssxpbm  4866  ssxp2  4868  ssrnres  4873  funssxp  5180  fssres  5186  dff2  5443  fliftf  5578  1stcof  5934  2ndcof  5935  smores  6057  tfrcllembfn  6122  caserel  6776  frecuzrdgtcl  9815