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Theorem xpima2m 4956
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 4522 . . . 4  |-  ( ( A  X.  B )
" C )  =  ran  ( ( A  X.  B )  |`  C )
2 df-res 4521 . . . . 5  |-  ( ( A  X.  B )  |`  C )  =  ( ( A  X.  B
)  i^i  ( C  X.  _V ) )
32rneqi 4737 . . . 4  |-  ran  (
( A  X.  B
)  |`  C )  =  ran  ( ( A  X.  B )  i^i  ( C  X.  _V ) )
4 inxp 4643 . . . . 5  |-  ( ( A  X.  B )  i^i  ( C  X.  _V ) )  =  ( ( A  i^i  C
)  X.  ( B  i^i  _V ) )
54rneqi 4737 . . . 4  |-  ran  (
( A  X.  B
)  i^i  ( C  X.  _V ) )  =  ran  ( ( A  i^i  C )  X.  ( B  i^i  _V ) )
61, 3, 53eqtri 2142 . . 3  |-  ( ( A  X.  B )
" C )  =  ran  ( ( A  i^i  C )  X.  ( B  i^i  _V ) )
7 rnxpm 4938 . . 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 2162 . 2  |-  ( E. x  x  e.  ( A  i^i  C )  ->  ( ( A  X.  B ) " C )  =  ( B  i^i  _V )
)
9 inv1 3369 . 2  |-  ( B  i^i  _V )  =  B
108, 9syl6eq 2166 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 1316   E.wex 1453    e. wcel 1465   _Vcvv 2660    i^i cin 3040    X. cxp 4507   ran crn 4510    |` cres 4511   "cima 4512
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  df-res 4521  df-ima 4522
This theorem is referenced by:  xpimasn  4957
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