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Theorem ipfeq 18214
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.  =  ( .if `  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 6311 . . 3  |-  (  .,  Fn  ( V  X.  V
)  <->  .,  =  ( x  e.  V ,  y  e.  V  |->  ( x 
.,  y ) ) )
21biimpi 194 . 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.  =  ( .if `  W
)
63, 4, 5ipffval 18212 . 2  |-  .x.  =  ( x  e.  V ,  y  e.  V  |->  ( x  .,  y
) )
72, 6syl6reqr 2514 1  |-  (  .,  Fn  ( V  X.  V
)  ->  .x.  =  .,  )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1370    X. cxp 4949    Fn wfn 5524   ` cfv 5529  (class class class)co 6203    |-> cmpt2 6205   Basecbs 14296   .icip 14366   .ifcipf 18189
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-id 4747  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-fv 5537  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-1st 6690  df-2nd 6691  df-ipf 18191
This theorem is referenced by: (None)
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