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Theorem rnxp 5266
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 4856 . . 3  |-  ran  ( A  X.  B )  =  dom  `' ( A  X.  B )
2 cnvxp 5257 . . . 4  |-  `' ( A  X.  B )  =  ( B  X.  A )
32dmeqi 5038 . . 3  |-  dom  `' ( A  X.  B
)  =  dom  ( B  X.  A )
41, 3eqtri 2432 . 2  |-  ran  ( A  X.  B )  =  dom  ( B  X.  A )
5 dmxp 5055 . 2  |-  ( A  =/=  (/)  ->  dom  ( B  X.  A )  =  B )
64, 5syl5eq 2456 1  |-  ( A  =/=  (/)  ->  ran  ( A  X.  B )  =  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1649    =/= wne 2575   (/)c0 3596    X. cxp 4843   `'ccnv 4844   dom cdm 4845   ran crn 4846
This theorem is referenced by:  rnxpid  5269  ssxpb  5270  xpexr  5274  xpexr2  5275  xpima  5280  unixp  5369  fconst5  5916  fparlem3  6415  fparlem4  6416  frxp  6423  fodomr  7225  dfac5lem3  7970  fpwwe2lem13  8481  vdwlem8  13319  ramz  13356  gsumxp  15513  xkoccn  17612  txindislem  17626  cnextf  18058  metustexhalfOLD  18554  metustexhalf  18555  ovolctb  19347  imadifxp  23999  sibf0  24610  axlowdimlem13  25805  axlowdim1  25810  ovoliunnfl  26155  voliunnfl  26157
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 2393  ax-sep 4298  ax-nul 4306  ax-pr 4371
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 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-ral 2679  df-rex 2680  df-rab 2683  df-v 2926  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-sn 3788  df-pr 3789  df-op 3791  df-br 4181  df-opab 4235  df-xp 4851  df-rel 4852  df-cnv 4853  df-dm 4855  df-rn 4856
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