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Theorem fvconst 5916
Description: The value of a constant function. (Contributed by NM, 30-May-1999.)
Assertion
Ref Expression
fvconst  |-  ( ( F : A --> { B }  /\  C  e.  A
)  ->  ( F `  C )  =  B )

Proof of Theorem fvconst
StepHypRef Expression
1 ffvelrn 5860 . 2  |-  ( ( F : A --> { B }  /\  C  e.  A
)  ->  ( F `  C )  e.  { B } )
2 elsni 3921 . 2  |-  ( ( F `  C )  e.  { B }  ->  ( F `  C
)  =  B )
31, 2syl 16 1  |-  ( ( F : A --> { B }  /\  C  e.  A
)  ->  ( F `  C )  =  B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756   {csn 3896   -->wf 5433   ` cfv 5437
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4432  ax-nul 4440  ax-pr 4550
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-ral 2739  df-rex 2740  df-rab 2743  df-v 2993  df-sbc 3206  df-dif 3350  df-un 3352  df-in 3354  df-ss 3361  df-nul 3657  df-if 3811  df-sn 3897  df-pr 3899  df-op 3903  df-uni 4111  df-br 4312  df-opab 4370  df-id 4655  df-xp 4865  df-rel 4866  df-cnv 4867  df-co 4868  df-dm 4869  df-rn 4870  df-iota 5400  df-fun 5439  df-fn 5440  df-f 5441  df-fv 5445
This theorem is referenced by:  fvconst2g  5950  fconst2g  5951  fconstfv  5959  ipasslem9  24257  resf1o  26049  ccatmulgnn0dir  26959  plymul02  26966
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