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Theorem rnxpss 5302
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 4890 . 2  |-  ran  ( A  X.  B )  =  dom  `' ( A  X.  B )
2 cnvxp 5291 . . . 4  |-  `' ( A  X.  B )  =  ( B  X.  A )
32dmeqi 5072 . . 3  |-  dom  `' ( A  X.  B
)  =  dom  ( B  X.  A )
4 dmxpss 5301 . . 3  |-  dom  ( B  X.  A )  C_  B
53, 4eqsstri 3379 . 2  |-  dom  `' ( A  X.  B
)  C_  B
61, 5eqsstri 3379 1  |-  ran  ( A  X.  B )  C_  B
Colors of variables: wff set class
Syntax hints:    C_ wss 3321    X. cxp 4877   `'ccnv 4878   dom cdm 4879   ran crn 4880
This theorem is referenced by:  ssxpb  5304  ssrnres  5310  funssxp  5605  fconst  5630  dff2  5882  dff3  5883  fliftf  6038  marypha1lem  7439  marypha1  7440  dfac12lem2  8025  brdom4  8409  nqerf  8808  lern  14671  cnconst2  17348  lmss  17363  tsmsxplem1  18183  causs  19252  i1f0  19580  itg10  19581  taylf  20278
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-sep 4331  ax-nul 4339  ax-pr 4404
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-ral 2711  df-rex 2712  df-rab 2715  df-v 2959  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3630  df-if 3741  df-sn 3821  df-pr 3822  df-op 3824  df-br 4214  df-opab 4268  df-xp 4885  df-rel 4886  df-cnv 4887  df-dm 4889  df-rn 4890
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