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Theorem xpima2m 5088
Description: The image by a cross product. (Contributed by Thierry Arnoux, 16-Dec-2017.)
Assertion
Ref Expression
xpima2m  |-  ( E. x  x  e.  ( A  i^i  C )  ->  ( ( A  X.  B ) " C )  =  B )
Distinct variable groups:    x, A    x, C
Allowed substitution hint:    B( x)

Proof of Theorem xpima2m
StepHypRef Expression
1 df-ima 4651 . . . 4  |-  ( ( A  X.  B )
" C )  =  ran  ( ( A  X.  B )  |`  C )
2 df-res 4650 . . . . 5  |-  ( ( A  X.  B )  |`  C )  =  ( ( A  X.  B
)  i^i  ( C  X.  _V ) )
32rneqi 4867 . . . 4  |-  ran  (
( A  X.  B
)  |`  C )  =  ran  ( ( A  X.  B )  i^i  ( C  X.  _V ) )
4 inxp 4773 . . . . 5  |-  ( ( A  X.  B )  i^i  ( C  X.  _V ) )  =  ( ( A  i^i  C
)  X.  ( B  i^i  _V ) )
54rneqi 4867 . . . 4  |-  ran  (
( A  X.  B
)  i^i  ( C  X.  _V ) )  =  ran  ( ( A  i^i  C )  X.  ( B  i^i  _V ) )
61, 3, 53eqtri 2212 . . 3  |-  ( ( A  X.  B )
" C )  =  ran  ( ( A  i^i  C )  X.  ( B  i^i  _V ) )
7 rnxpm 5070 . . 3  |-  ( E. x  x  e.  ( A  i^i  C )  ->  ran  ( ( A  i^i  C )  X.  ( B  i^i  _V ) )  =  ( B  i^i  _V )
)
86, 7eqtrid 2232 . 2  |-  ( E. x  x  e.  ( A  i^i  C )  ->  ( ( A  X.  B ) " C )  =  ( B  i^i  _V )
)
9 inv1 3471 . 2  |-  ( B  i^i  _V )  =  B
108, 9eqtrdi 2236 1  |-  ( E. x  x  e.  ( A  i^i  C )  ->  ( ( A  X.  B ) " C )  =  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1363   E.wex 1502    e. wcel 2158   _Vcvv 2749    i^i cin 3140    X. cxp 4636   ran crn 4639    |` cres 4640   "cima 4641
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-14 2161  ax-ext 2169  ax-sep 4133  ax-pow 4186  ax-pr 4221
This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ral 2470  df-rex 2471  df-v 2751  df-un 3145  df-in 3147  df-ss 3154  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-br 4016  df-opab 4077  df-xp 4644  df-rel 4645  df-cnv 4646  df-dm 4648  df-rn 4649  df-res 4650  df-ima 4651
This theorem is referenced by:  xpimasn  5089
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