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Theorem rnxp 5262
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 4852 . . 3  |-  ran  ( A  X.  B )  =  dom  `' ( A  X.  B )
2 cnvxp 5253 . . . 4  |-  `' ( A  X.  B )  =  ( B  X.  A )
32dmeqi 5034 . . 3  |-  dom  `' ( A  X.  B
)  =  dom  ( B  X.  A )
41, 3eqtri 2428 . 2  |-  ran  ( A  X.  B )  =  dom  ( B  X.  A )
5 dmxp 5051 . 2  |-  ( A  =/=  (/)  ->  dom  ( B  X.  A )  =  B )
64, 5syl5eq 2452 1  |-  ( A  =/=  (/)  ->  ran  ( A  X.  B )  =  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1649    =/= wne 2571   (/)c0 3592    X. cxp 4839   `'ccnv 4840   dom cdm 4841   ran crn 4842
This theorem is referenced by:  rnxpid  5265  ssxpb  5266  xpexr  5270  xpexr2  5271  xpima  5276  unixp  5365  fconst5  5912  fparlem3  6411  fparlem4  6412  frxp  6419  fodomr  7221  dfac5lem3  7966  fpwwe2lem13  8477  vdwlem8  13315  ramz  13352  gsumxp  15509  xkoccn  17608  txindislem  17622  cnextf  18054  metustexhalfOLD  18550  metustexhalf  18551  ovolctb  19343  imadifxp  23995  sibf0  24606  axlowdimlem13  25801  axlowdim1  25806  ovoliunnfl  26151  voliunnfl  26153
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 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-sep 4294  ax-nul 4302  ax-pr 4367
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 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-ral 2675  df-rex 2676  df-rab 2679  df-v 2922  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-nul 3593  df-if 3704  df-sn 3784  df-pr 3785  df-op 3787  df-br 4177  df-opab 4231  df-xp 4847  df-rel 4848  df-cnv 4849  df-dm 4851  df-rn 4852
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