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Theorem rnxpss 4977
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 4557 . 2  |-  ran  ( A  X.  B )  =  dom  `' ( A  X.  B )
2 cnvxp 4964 . . . 4  |-  `' ( A  X.  B )  =  ( B  X.  A )
32dmeqi 4747 . . 3  |-  dom  `' ( A  X.  B
)  =  dom  ( B  X.  A )
4 dmxpss 4976 . . 3  |-  dom  ( B  X.  A )  C_  B
53, 4eqsstri 3133 . 2  |-  dom  `' ( A  X.  B
)  C_  B
61, 5eqsstri 3133 1  |-  ran  ( A  X.  B )  C_  B
Colors of variables: wff set class
Syntax hints:    C_ wss 3075    X. cxp 4544   `'ccnv 4545   dom cdm 4546   ran crn 4547
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4053  ax-pow 4105  ax-pr 4138
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2691  df-un 3079  df-in 3081  df-ss 3088  df-pw 3516  df-sn 3537  df-pr 3538  df-op 3540  df-br 3937  df-opab 3997  df-xp 4552  df-rel 4553  df-cnv 4554  df-dm 4556  df-rn 4557
This theorem is referenced by:  rnxpss2  4979  rnxpid  4980  ssxpbm  4981  ssxp2  4983  ssrnres  4988  funssxp  5299  fconst  5325  dff2  5571  fliftf  5707  tfrcllembfn  6261  frecuzrdgtcl  10215  cnconst2  12439  lmss  12452  exmidsbthrlem  13390
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