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Theorem ipfeq 16806
Description: If the inner product operation is already a function, the functionalization of it is equal to the original operation. (Contributed by Mario Carneiro, 14-Aug-2015.)
Hypotheses
Ref Expression
ipffval.1  |-  V  =  ( Base `  W
)
ipffval.2  |-  .,  =  ( .i `  W )
ipffval.3  |-  .x.  =  ( .i f `  W
)
Assertion
Ref Expression
ipfeq  |-  (  .,  Fn  ( V  X.  V
)  ->  .x.  =  .,  )

Proof of Theorem ipfeq
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fnov 6119 . . 3  |-  (  .,  Fn  ( V  X.  V
)  <->  .,  =  ( x  e.  V ,  y  e.  V  |->  ( x 
.,  y ) ) )
21biimpi 187 . 2  |-  (  .,  Fn  ( V  X.  V
)  ->  .,  =  ( x  e.  V , 
y  e.  V  |->  ( x  .,  y ) ) )
3 ipffval.1 . . 3  |-  V  =  ( Base `  W
)
4 ipffval.2 . . 3  |-  .,  =  ( .i `  W )
5 ipffval.3 . . 3  |-  .x.  =  ( .i f `  W
)
63, 4, 5ipffval 16804 . 2  |-  .x.  =  ( x  e.  V ,  y  e.  V  |->  ( x  .,  y
) )
72, 6syl6reqr 2440 1  |-  (  .,  Fn  ( V  X.  V
)  ->  .x.  =  .,  )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1649    X. cxp 4818    Fn wfn 5391   ` cfv 5396  (class class class)co 6022    e. cmpt2 6024   Basecbs 13398   .icip 13463   .i fcipf 16781
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346  ax-un 4643
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-ral 2656  df-rex 2657  df-rab 2660  df-v 2903  df-sbc 3107  df-csb 3197  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-nul 3574  df-if 3685  df-pw 3746  df-sn 3765  df-pr 3766  df-op 3768  df-uni 3960  df-iun 4039  df-br 4156  df-opab 4210  df-mpt 4211  df-id 4441  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833  df-iota 5360  df-fun 5398  df-fn 5399  df-f 5400  df-fv 5404  df-ov 6025  df-oprab 6026  df-mpt2 6027  df-1st 6290  df-2nd 6291  df-ipf 16783
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