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Theorem rnxp 5094
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 4680 . . 3  |-  ran  (  A  X.  B )  =  dom  `' ( A  X.  B )
2 cnvxp 5085 . . . 4  |-  `' ( A  X.  B )  =  ( B  X.  A )
32dmeqi 4868 . . 3  |-  dom  `' ( A  X.  B
)  =  dom  (  B  X.  A )
41, 3eqtri 2278 . 2  |-  ran  (  A  X.  B )  =  dom  (  B  X.  A )
5 dmxp 4885 . 2  |-  ( A  =/=  (/)  ->  dom  (  B  X.  A )  =  B )
64, 5syl5eq 2302 1  |-  ( A  =/=  (/)  ->  ran  (  A  X.  B )  =  B )
Colors of variables: wff set class
Syntax hints:    -> wi 6    = wceq 1619    =/= wne 2421   (/)c0 3430    X. cxp 4659   `'ccnv 4660   dom cdm 4661   ran crn 4662
This theorem is referenced by:  rnxpid  5097  ssxpb  5098  xpexr  5102  xpexr2  5103  unixp  5192  fconst5  5665  fparlem3  6154  fparlem4  6155  frxp  6159  fodomr  6980  dfac5lem3  7720  fpwwe2lem13  8232  vdwlem8  12998  ramz  13035  gsumxp  15190  xkoccn  17276  txindislem  17290  ovolctb  18812  axlowdimlem13  23958  axlowdim1  23963  prjcp2  24452  rngodmeqrn  24787
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 2239  ax-sep 4115  ax-nul 4123  ax-pr 4186
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 2122  df-mo 2123  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-ral 2523  df-rex 2524  df-rab 2527  df-v 2765  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-nul 3431  df-if 3540  df-sn 3620  df-pr 3621  df-op 3623  df-br 3998  df-opab 4052  df-xp 4675  df-rel 4676  df-cnv 4677  df-dm 4679  df-rn 4680
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