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Theorem rnxp 5108
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 4702 . . 3  |-  ran  ( A  X.  B )  =  dom  `' ( A  X.  B )
2 cnvxp 5099 . . . 4  |-  `' ( A  X.  B )  =  ( B  X.  A )
32dmeqi 4882 . . 3  |-  dom  `' ( A  X.  B
)  =  dom  ( B  X.  A )
41, 3eqtri 2305 . 2  |-  ran  ( A  X.  B )  =  dom  ( B  X.  A )
5 dmxp 4899 . 2  |-  ( A  =/=  (/)  ->  dom  ( B  X.  A )  =  B )
64, 5syl5eq 2329 1  |-  ( A  =/=  (/)  ->  ran  ( A  X.  B )  =  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1625    =/= wne 2448   (/)c0 3457    X. cxp 4689   `'ccnv 4690   dom cdm 4691   ran crn 4692
This theorem is referenced by:  rnxpid  5111  ssxpb  5112  xpexr  5116  xpexr2  5117  unixp  5207  fconst5  5733  fparlem3  6222  fparlem4  6223  frxp  6227  fodomr  7014  dfac5lem3  7754  fpwwe2lem13  8266  vdwlem8  13037  ramz  13074  gsumxp  15229  xkoccn  17315  txindislem  17329  ovolctb  18851  xpima  23204  axlowdimlem13  24584  axlowdim1  24589  prjcp2  25096  rngodmeqrn  25430
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-14 1690  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266  ax-sep 4143  ax-nul 4151  ax-pr 4216
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1531  df-nf 1534  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-ral 2550  df-rex 2551  df-rab 2554  df-v 2792  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-nul 3458  df-if 3568  df-sn 3648  df-pr 3649  df-op 3651  df-br 4026  df-opab 4080  df-xp 4697  df-rel 4698  df-cnv 4699  df-dm 4701  df-rn 4702
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