Theorem rnss 5012

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 4953	. . 3 |- ( A C_ B -> `' A C_ `' B )
2		dmss 4980	. . 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 4785	. 2 |- ran A = dom `' A
5		df-rn 4785	. 2 |- ran B = dom `' B
6	3, 4, 5	3sstr4g 3291	1 |- ( A C_ B -> ran A C_ ran B )

This proof depends on syntax axioms:   -> wi 4   C_ wss 3220   `'ccnv 4773   dom cdm 4774   ran crn 4775

This proof depends on 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 proof 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 3715  df-pr 3716  df-op 3718  df-br 4131  df-opab 4193  df-cnv 4782  df-dm 4784  df-rn 4785

This theorem is used by:  imass1  5162  imass2  5163  rnxpss2  5221  ssxpbm  5223  ssxp2  5225  ssrnres  5230  funssxp  5557  fssres  5565  dff2  5852  fliftf  6005  1stcof  6397  2ndcof  6398  smores  6563  tfrcllembfn  6628  caserel  7427  frecuzrdgtcl  10849  prdsvallem  13621  prdsval  14173  lmss  15347  txss12  15367  txbasval  15368  subgrprop3  16503