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Theorem rnxp 5232
Description: The range of a cross product. Part of Theorem 3.13(x) of [Monk1] p. 37. (Contributed by NM, 12-Apr-2004.)
Assertion
Ref Expression
rnxp  |-  ( A  =/=  (/)  ->  ran  ( A  X.  B )  =  B )

Proof of Theorem rnxp
StepHypRef Expression
1 df-rn 4822 . . 3  |-  ran  ( A  X.  B )  =  dom  `' ( A  X.  B )
2 cnvxp 5223 . . . 4  |-  `' ( A  X.  B )  =  ( B  X.  A )
32dmeqi 5004 . . 3  |-  dom  `' ( A  X.  B
)  =  dom  ( B  X.  A )
41, 3eqtri 2400 . 2  |-  ran  ( A  X.  B )  =  dom  ( B  X.  A )
5 dmxp 5021 . 2  |-  ( A  =/=  (/)  ->  dom  ( B  X.  A )  =  B )
64, 5syl5eq 2424 1  |-  ( A  =/=  (/)  ->  ran  ( A  X.  B )  =  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1649    =/= wne 2543   (/)c0 3564    X. cxp 4809   `'ccnv 4810   dom cdm 4811   ran crn 4812
This theorem is referenced by:  rnxpid  5235  ssxpb  5236  xpexr  5240  xpexr2  5241  xpima  5246  unixp  5335  fconst5  5881  fparlem3  6380  fparlem4  6381  frxp  6385  fodomr  7187  dfac5lem3  7932  fpwwe2lem13  8443  vdwlem8  13276  ramz  13313  gsumxp  15470  xkoccn  17565  txindislem  17579  cnextf  18011  metustexhalf  18469  ovolctb  19246  imadifxp  23874  axlowdimlem13  25600  axlowdim1  25605  ovoliunnfl  25946  voliunnfl  25948
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-sep 4264  ax-nul 4272  ax-pr 4337
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-ral 2647  df-rex 2648  df-rab 2651  df-v 2894  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-nul 3565  df-if 3676  df-sn 3756  df-pr 3757  df-op 3759  df-br 4147  df-opab 4201  df-xp 4817  df-rel 4818  df-cnv 4819  df-dm 4821  df-rn 4822
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