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Theorem rnxp 5302
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 4892 . . 3  |-  ran  ( A  X.  B )  =  dom  `' ( A  X.  B )
2 cnvxp 5293 . . . 4  |-  `' ( A  X.  B )  =  ( B  X.  A )
32dmeqi 5074 . . 3  |-  dom  `' ( A  X.  B
)  =  dom  ( B  X.  A )
41, 3eqtri 2458 . 2  |-  ran  ( A  X.  B )  =  dom  ( B  X.  A )
5 dmxp 5091 . 2  |-  ( A  =/=  (/)  ->  dom  ( B  X.  A )  =  B )
64, 5syl5eq 2482 1  |-  ( A  =/=  (/)  ->  ran  ( A  X.  B )  =  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1653    =/= wne 2601   (/)c0 3630    X. cxp 4879   `'ccnv 4880   dom cdm 4881   ran crn 4882
This theorem is referenced by:  rnxpid  5305  ssxpb  5306  xpexr  5310  xpexr2  5311  xpima  5316  unixp  5405  fconst5  5952  fparlem3  6451  fparlem4  6452  frxp  6459  fodomr  7261  dfac5lem3  8011  fpwwe2lem13  8522  vdwlem8  13361  ramz  13398  gsumxp  15555  xkoccn  17656  txindislem  17670  cnextf  18102  metustexhalfOLD  18598  metustexhalf  18599  ovolctb  19391  imadifxp  24043  sibf0  24654  axlowdimlem13  25898  axlowdim1  25903  ovoliunnfl  26260  voliunnfl  26262
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 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 2419  ax-sep 4333  ax-nul 4341  ax-pr 4406
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 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-br 4216  df-opab 4270  df-xp 4887  df-rel 4888  df-cnv 4889  df-dm 4891  df-rn 4892
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