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Theorem rnxpss 5055
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 4633 . 2  |-  ran  ( A  X.  B )  =  dom  `' ( A  X.  B )
2 cnvxp 5042 . . . 4  |-  `' ( A  X.  B )  =  ( B  X.  A )
32dmeqi 4823 . . 3  |-  dom  `' ( A  X.  B
)  =  dom  ( B  X.  A )
4 dmxpss 5054 . . 3  |-  dom  ( B  X.  A )  C_  B
53, 4eqsstri 3187 . 2  |-  dom  `' ( A  X.  B
)  C_  B
61, 5eqsstri 3187 1  |-  ran  ( A  X.  B )  C_  B
Colors of variables: wff set class
Syntax hints:    C_ wss 3129    X. cxp 4620   `'ccnv 4621   dom cdm 4622   ran crn 4623
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-pow 4171  ax-pr 4205
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-un 3133  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-br 4001  df-opab 4062  df-xp 4628  df-rel 4629  df-cnv 4630  df-dm 4632  df-rn 4633
This theorem is referenced by:  rnxpss2  5057  rnxpid  5058  ssxpbm  5059  ssxp2  5061  ssrnres  5066  funssxp  5380  fconst  5406  dff2  5655  fliftf  5793  tfrcllembfn  6351  frecuzrdgtcl  10385  cnconst2  13366  lmss  13379  exmidsbthrlem  14393
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