Theorem rnss 5007

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 4948	. . 3 |- ( A C_ B -> `' A C_ `' B )
2		dmss 4975	. . 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 4780	. 2 |- ran A = dom `' A
5		df-rn 4780	. 2 |- ran B = dom `' B
6	3, 4, 5	3sstr4g 3291	1 |- ( A C_ B -> ran A C_ ran B )

Syntax hints:   -> wi 4   C_ wss 3220   `'ccnv 4768   dom cdm 4769   ran crn 4770

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-sn 3711  df-pr 3712  df-op 3714  df-br 4126  df-opab 4188  df-cnv 4777  df-dm 4779  df-rn 4780

This theorem is referenced by:  imass1  5157  imass2  5158  rnxpss2  5216  ssxpbm  5218  ssxp2  5220  ssrnres  5225  funssxp  5552  fssres  5560  dff2  5843  fliftf  5995  1stcof  6387  2ndcof  6388  smores  6553  tfrcllembfn  6618  caserel  7417  frecuzrdgtcl  10827  prdsvallem  13598  prdsval  14150  lmss  15270  txss12  15290  txbasval  15291  subgrprop3  16417