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Theorem rnxpm 4938
Description: The range of a cross product. Part of Theorem 3.13(x) of [Monk1] p. 37, with nonempty changed to inhabited. (Contributed by Jim Kingdon, 12-Dec-2018.)
Assertion
Ref Expression
rnxpm  |-  ( E. x  x  e.  A  ->  ran  ( A  X.  B )  =  B )
Distinct variable group:    x, A
Allowed substitution hint:    B( x)

Proof of Theorem rnxpm
StepHypRef Expression
1 df-rn 4520 . . 3  |-  ran  ( A  X.  B )  =  dom  `' ( A  X.  B )
2 cnvxp 4927 . . . 4  |-  `' ( A  X.  B )  =  ( B  X.  A )
32dmeqi 4710 . . 3  |-  dom  `' ( A  X.  B
)  =  dom  ( B  X.  A )
41, 3eqtri 2138 . 2  |-  ran  ( A  X.  B )  =  dom  ( B  X.  A )
5 dmxpm 4729 . 2  |-  ( E. x  x  e.  A  ->  dom  ( B  X.  A )  =  B )
64, 5syl5eq 2162 1  |-  ( E. x  x  e.  A  ->  ran  ( A  X.  B )  =  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1316   E.wex 1453    e. wcel 1465    X. cxp 4507   `'ccnv 4508   dom cdm 4509   ran crn 4510
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-v 2662  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-br 3900  df-opab 3960  df-xp 4515  df-rel 4516  df-cnv 4517  df-dm 4519  df-rn 4520
This theorem is referenced by:  ssxpbm  4944  ssxp2  4946  xpexr2m  4950  xpima2m  4956  unixpm  5044  djuexb  6897  exmidfodomrlemim  7025
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