Theorem rnss 4962

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 4903	. . 3 |- ( A C_ B -> `' A C_ `' B )
2		dmss 4930	. . 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 4736	. 2 |- ran A = dom `' A
5		df-rn 4736	. 2 |- ran B = dom `' B
6	3, 4, 5	3sstr4g 3270	1 |- ( A C_ B -> ran A C_ ran B )

Syntax hints:   -> wi 4   C_ wss 3200   `'ccnv 4724   dom cdm 4725   ran crn 4726

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-v 2804  df-un 3204  df-in 3206  df-ss 3213  df-sn 3675  df-pr 3676  df-op 3678  df-br 4089  df-opab 4151  df-cnv 4733  df-dm 4735  df-rn 4736

This theorem is referenced by:  imass1  5111  imass2  5112  rnxpss2  5170  ssxpbm  5172  ssxp2  5174  ssrnres  5179  funssxp  5504  fssres  5512  dff2  5791  fliftf  5939  1stcof  6325  2ndcof  6326  smores  6457  tfrcllembfn  6522  caserel  7285  frecuzrdgtcl  10673  prdsvallem  13354  prdsval  13355  lmss  14969  txss12  14989  txbasval  14990  subgrprop3  16112