Theorem rnss 4858

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 4801	. . 3 |- ( A C_ B -> `' A C_ `' B )
2		dmss 4827	. . 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 4638	. 2 |- ran A = dom `' A
5		df-rn 4638	. 2 |- ran B = dom `' B
6	3, 4, 5	3sstr4g 3199	1 |- ( A C_ B -> ran A C_ ran B )

Syntax hints:   -> wi 4   C_ wss 3130   `'ccnv 4626   dom cdm 4627   ran crn 4628

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2740  df-un 3134  df-in 3136  df-ss 3143  df-sn 3599  df-pr 3600  df-op 3602  df-br 4005  df-opab 4066  df-cnv 4635  df-dm 4637  df-rn 4638

This theorem is referenced by:  imass1  5004  imass2  5005  rnxpss2  5063  ssxpbm  5065  ssxp2  5067  ssrnres  5072  funssxp  5386  fssres  5392  dff2  5661  fliftf  5800  1stcof  6164  2ndcof  6165  smores  6293  tfrcllembfn  6358  caserel  7086  frecuzrdgtcl  10412  lmss  13749  txss12  13769  txbasval  13770