Theorem rnss 4857

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 4800	. . 3 |- ( A C_ B -> `' A C_ `' B )
2		dmss 4826	. . 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 4637	. 2 |- ran A = dom `' A
5		df-rn 4637	. 2 |- ran B = dom `' B
6	3, 4, 5	3sstr4g 3198	1 |- ( A C_ B -> ran A C_ ran B )

Syntax hints:   -> wi 4   C_ wss 3129   `'ccnv 4625   dom cdm 4626   ran crn 4627

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2739  df-un 3133  df-in 3135  df-ss 3142  df-sn 3598  df-pr 3599  df-op 3601  df-br 4004  df-opab 4065  df-cnv 4634  df-dm 4636  df-rn 4637

This theorem is referenced by:  imass1  5003  imass2  5004  rnxpss2  5062  ssxpbm  5064  ssxp2  5066  ssrnres  5071  funssxp  5385  fssres  5391  dff2  5660  fliftf  5799  1stcof  6163  2ndcof  6164  smores  6292  tfrcllembfn  6357  caserel  7085  frecuzrdgtcl  10411  lmss  13716  txss12  13736  txbasval  13737