Theorem rnss 4960

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 4901	. . 3 |- ( A C_ B -> `' A C_ `' B )
2		dmss 4928	. . 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 4734	. 2 |- ran A = dom `' A
5		df-rn 4734	. 2 |- ran B = dom `' B
6	3, 4, 5	3sstr4g 3268	1 |- ( A C_ B -> ran A C_ ran B )

Syntax hints:   -> wi 4   C_ wss 3198   `'ccnv 4722   dom cdm 4723   ran crn 4724

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2802  df-un 3202  df-in 3204  df-ss 3211  df-sn 3673  df-pr 3674  df-op 3676  df-br 4087  df-opab 4149  df-cnv 4731  df-dm 4733  df-rn 4734

This theorem is referenced by:  imass1  5109  imass2  5110  rnxpss2  5168  ssxpbm  5170  ssxp2  5172  ssrnres  5177  funssxp  5501  fssres  5509  dff2  5787  fliftf  5935  1stcof  6321  2ndcof  6322  smores  6453  tfrcllembfn  6518  caserel  7277  frecuzrdgtcl  10664  prdsvallem  13345  prdsval  13346  lmss  14960  txss12  14980  txbasval  14981