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Theorem xpima2m 4986
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 4552 . . . 4  |-  ( ( A  X.  B )
" C )  =  ran  ( ( A  X.  B )  |`  C )
2 df-res 4551 . . . . 5  |-  ( ( A  X.  B )  |`  C )  =  ( ( A  X.  B
)  i^i  ( C  X.  _V ) )
32rneqi 4767 . . . 4  |-  ran  (
( A  X.  B
)  |`  C )  =  ran  ( ( A  X.  B )  i^i  ( C  X.  _V ) )
4 inxp 4673 . . . . 5  |-  ( ( A  X.  B )  i^i  ( C  X.  _V ) )  =  ( ( A  i^i  C
)  X.  ( B  i^i  _V ) )
54rneqi 4767 . . . 4  |-  ran  (
( A  X.  B
)  i^i  ( C  X.  _V ) )  =  ran  ( ( A  i^i  C )  X.  ( B  i^i  _V ) )
61, 3, 53eqtri 2164 . . 3  |-  ( ( A  X.  B )
" C )  =  ran  ( ( A  i^i  C )  X.  ( B  i^i  _V ) )
7 rnxpm 4968 . . 3  |-  ( E. x  x  e.  ( A  i^i  C )  ->  ran  ( ( A  i^i  C )  X.  ( B  i^i  _V ) )  =  ( B  i^i  _V )
)
86, 7syl5eq 2184 . 2  |-  ( E. x  x  e.  ( A  i^i  C )  ->  ( ( A  X.  B ) " C )  =  ( B  i^i  _V )
)
9 inv1 3399 . 2  |-  ( B  i^i  _V )  =  B
108, 9syl6eq 2188 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 1331   E.wex 1468    e. wcel 1480   _Vcvv 2686    i^i cin 3070    X. cxp 4537   ran crn 4540    |` cres 4541   "cima 4542
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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-br 3930  df-opab 3990  df-xp 4545  df-rel 4546  df-cnv 4547  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552
This theorem is referenced by:  xpimasn  4987
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