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Theorem rnxp 5059
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 4645 . . 3  |-  ran  (  A  X.  B )  =  dom  `' ( A  X.  B )
2 cnvxp 5050 . . . 4  |-  `' ( A  X.  B )  =  ( B  X.  A )
32dmeqi 4833 . . 3  |-  dom  `' ( A  X.  B
)  =  dom  (  B  X.  A )
41, 3eqtri 2276 . 2  |-  ran  (  A  X.  B )  =  dom  (  B  X.  A )
5 dmxp 4850 . 2  |-  ( A  =/=  (/)  ->  dom  (  B  X.  A )  =  B )
64, 5syl5eq 2300 1  |-  ( A  =/=  (/)  ->  ran  (  A  X.  B )  =  B )
Colors of variables: wff set class
Syntax hints:    -> wi 6    = wceq 1619    =/= wne 2419   (/)c0 3397    X. cxp 4624   `'ccnv 4625   dom cdm 4626   ran crn 4627
This theorem is referenced by:  rnxpid  5062  ssxpb  5063  xpexr  5067  xpexr2  5068  unixp  5157  fconst5  5630  fparlem3  6119  fparlem4  6120  frxp  6124  fodomr  6945  dfac5lem3  7685  fpwwe2lem13  8197  vdwlem8  12962  ramz  12999  gsumxp  15154  xkoccn  17240  txindislem  17254  ovolctb  18776  axlowdimlem13  23922  axlowdim1  23927  prjcp2  24416  rngodmeqrn  24751
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237  ax-sep 4081  ax-nul 4089  ax-pr 4152
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-ral 2520  df-rex 2521  df-rab 2523  df-v 2742  df-dif 3097  df-un 3099  df-in 3101  df-ss 3108  df-nul 3398  df-if 3507  df-sn 3587  df-pr 3588  df-op 3590  df-br 3964  df-opab 4018  df-xp 4640  df-rel 4641  df-cnv 4642  df-dm 4644  df-rn 4645
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