Theorem rnss 4897

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 4840	. . 3 |- ( A C_ B -> `' A C_ `' B )
2		dmss 4866	. . 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 4675	. 2 |- ran A = dom `' A
5		df-rn 4675	. 2 |- ran B = dom `' B
6	3, 4, 5	3sstr4g 3227	1 |- ( A C_ B -> ran A C_ ran B )

Syntax hints:   -> wi 4   C_ wss 3157   `'ccnv 4663   dom cdm 4664   ran crn 4665

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-v 2765  df-un 3161  df-in 3163  df-ss 3170  df-sn 3629  df-pr 3630  df-op 3632  df-br 4035  df-opab 4096  df-cnv 4672  df-dm 4674  df-rn 4675

This theorem is referenced by:  imass1  5045  imass2  5046  rnxpss2  5104  ssxpbm  5106  ssxp2  5108  ssrnres  5113  funssxp  5430  fssres  5436  dff2  5709  fliftf  5849  1stcof  6230  2ndcof  6231  smores  6359  tfrcllembfn  6424  caserel  7162  frecuzrdgtcl  10521  prdsvallem  12974  prdsval  12975  lmss  14566  txss12  14586  txbasval  14587