Theorem rnss 4582

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 4526	. . 3 |- ( A C_ B -> `' A C_ `' B )
2		dmss 4552	. . 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 4374	. 2 |- ran A = dom `' A
5		df-rn 4374	. 2 |- ran B = dom `' B
6	3, 4, 5	3sstr4g 3040	1 |- ( A C_ B -> ran A C_ ran B )

Syntax hints:   -> wi 4   C_ wss 2973   `'ccnv 4362   dom cdm 4363   ran crn 4364

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-v 2603  df-un 2977  df-in 2979  df-ss 2986  df-sn 3404  df-pr 3405  df-op 3407  df-br 3786  df-opab 3840  df-cnv 4371  df-dm 4373  df-rn 4374

This theorem is referenced by:  imass1  4720  imass2  4721  ssxpbm  4776  ssxp2  4778  ssrnres  4783  funssxp  5080  fssres  5086  dff2  5332  fliftf  5459  1stcof  5810  2ndcof  5811  smores  5930