Theorem rnss 4917

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 4859	. . 3 |- ( A C_ B -> `' A C_ `' B )
2		dmss 4886	. . 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 4694	. 2 |- ran A = dom `' A
5		df-rn 4694	. 2 |- ran B = dom `' B
6	3, 4, 5	3sstr4g 3240	1 |- ( A C_ B -> ran A C_ ran B )

Syntax hints:   -> wi 4   C_ wss 3170   `'ccnv 4682   dom cdm 4683   ran crn 4684

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2188

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-v 2775  df-un 3174  df-in 3176  df-ss 3183  df-sn 3644  df-pr 3645  df-op 3647  df-br 4052  df-opab 4114  df-cnv 4691  df-dm 4693  df-rn 4694

This theorem is referenced by:  imass1  5066  imass2  5067  rnxpss2  5125  ssxpbm  5127  ssxp2  5129  ssrnres  5134  funssxp  5455  fssres  5463  dff2  5737  fliftf  5881  1stcof  6262  2ndcof  6263  smores  6391  tfrcllembfn  6456  caserel  7204  frecuzrdgtcl  10579  prdsvallem  13179  prdsval  13180  lmss  14793  txss12  14813  txbasval  14814