Theorem rnss 4892

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 4835	. . 3 |- ( A C_ B -> `' A C_ `' B )
2		dmss 4861	. . 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 4670	. 2 |- ran A = dom `' A
5		df-rn 4670	. 2 |- ran B = dom `' B
6	3, 4, 5	3sstr4g 3222	1 |- ( A C_ B -> ran A C_ ran B )

Syntax hints:   -> wi 4   C_ wss 3153   `'ccnv 4658   dom cdm 4659   ran crn 4660

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-v 2762  df-un 3157  df-in 3159  df-ss 3166  df-sn 3624  df-pr 3625  df-op 3627  df-br 4030  df-opab 4091  df-cnv 4667  df-dm 4669  df-rn 4670

This theorem is referenced by:  imass1  5040  imass2  5041  rnxpss2  5099  ssxpbm  5101  ssxp2  5103  ssrnres  5108  funssxp  5423  fssres  5429  dff2  5702  fliftf  5842  1stcof  6216  2ndcof  6217  smores  6345  tfrcllembfn  6410  caserel  7146  frecuzrdgtcl  10483  lmss  14414  txss12  14434  txbasval  14435