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Theorem rnxpss 4862
Description: The range of a cross product is a subclass of the second factor. (Contributed by NM, 16-Jan-2006.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
Assertion
Ref Expression
rnxpss  |-  ran  ( A  X.  B )  C_  B

Proof of Theorem rnxpss
StepHypRef Expression
1 df-rn 4449 . 2  |-  ran  ( A  X.  B )  =  dom  `' ( A  X.  B )
2 cnvxp 4850 . . . 4  |-  `' ( A  X.  B )  =  ( B  X.  A )
32dmeqi 4637 . . 3  |-  dom  `' ( A  X.  B
)  =  dom  ( B  X.  A )
4 dmxpss 4861 . . 3  |-  dom  ( B  X.  A )  C_  B
53, 4eqsstri 3056 . 2  |-  dom  `' ( A  X.  B
)  C_  B
61, 5eqsstri 3056 1  |-  ran  ( A  X.  B )  C_  B
Colors of variables: wff set class
Syntax hints:    C_ wss 2999    X. cxp 4436   `'ccnv 4437   dom cdm 4438   ran crn 4439
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 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pow 4009  ax-pr 4036
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-un 3003  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-br 3846  df-opab 3900  df-xp 4444  df-rel 4445  df-cnv 4446  df-dm 4448  df-rn 4449
This theorem is referenced by:  rnxpss2  4864  rnxpid  4865  ssxpbm  4866  ssxp2  4868  ssrnres  4873  funssxp  5180  fconst  5206  dff2  5443  fliftf  5578  tfrcllembfn  6122  frecuzrdgtcl  9815  exmidsbthrlem  11867
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