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Theorem rnxpss 5108
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 4700 . 2  |-  ran  ( A  X.  B )  =  dom  `' ( A  X.  B )
2 cnvxp 5097 . . . 4  |-  `' ( A  X.  B )  =  ( B  X.  A )
32dmeqi 4880 . . 3  |-  dom  `' ( A  X.  B
)  =  dom  ( B  X.  A )
4 dmxpss 5107 . . 3  |-  dom  ( B  X.  A )  C_  B
53, 4eqsstri 3208 . 2  |-  dom  `' ( A  X.  B
)  C_  B
61, 5eqsstri 3208 1  |-  ran  ( A  X.  B )  C_  B
Colors of variables: wff set class
Syntax hints:    C_ wss 3152    X. cxp 4687   `'ccnv 4688   dom cdm 4689   ran crn 4690
This theorem is referenced by:  ssxpb  5110  ssrnres  5116  funssxp  5402  fconst  5427  dff2  5672  dff3  5673  fliftf  5814  marypha1lem  7186  marypha1  7187  dfac12lem2  7770  brdom4  8155  nqerf  8554  lern  14347  cnconst2  17011  lmss  17026  tsmsxplem1  17835  causs  18724  i1f0  19042  itg10  19043  taylf  19740  dmrngcmp  25751
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-opab 4078  df-xp 4695  df-rel 4696  df-cnv 4697  df-dm 4699  df-rn 4700
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