MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  rnxp Unicode version

Theorem rnxp 5105
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 4699 . . 3  |-  ran  (  A  X.  B )  =  dom  `' ( A  X.  B )
2 cnvxp 5096 . . . 4  |-  `' ( A  X.  B )  =  ( B  X.  A )
32dmeqi 4879 . . 3  |-  dom  `' ( A  X.  B
)  =  dom  (  B  X.  A )
41, 3eqtri 2304 . 2  |-  ran  (  A  X.  B )  =  dom  (  B  X.  A )
5 dmxp 4896 . 2  |-  ( A  =/=  (/)  ->  dom  (  B  X.  A )  =  B )
64, 5syl5eq 2328 1  |-  ( A  =/=  (/)  ->  ran  (  A  X.  B )  =  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623    =/= wne 2447   (/)c0 3456    X. cxp 4686   `'ccnv 4687    dom cdm 4688   ran crn 4689
This theorem is referenced by:  rnxpid  5108  ssxpb  5109  xpexr  5113  xpexr2  5114  unixp  5203  fconst5  5693  fparlem3  6182  fparlem4  6183  frxp  6187  fodomr  7008  dfac5lem3  7748  fpwwe2lem13  8260  vdwlem8  13031  ramz  13068  gsumxp  15223  xkoccn  17309  txindislem  17323  ovolctb  18845  axlowdimlem13  23992  axlowdim1  23997  prjcp2  24495  rngodmeqrn  24830
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1636  ax-8 1644  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2265  ax-sep 4142  ax-nul 4150  ax-pr 4213
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1631  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-ral 2549  df-rex 2550  df-rab 2553  df-v 2791  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3457  df-if 3567  df-sn 3647  df-pr 3648  df-op 3650  df-br 4025  df-opab 4079  df-xp 4694  df-rel 4695  df-cnv 4696  df-dm 4698  df-rn 4699
  Copyright terms: Public domain W3C validator