MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  fvconst Structured version   Unicode version

Theorem fvconst 6069
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 6007 . 2  |-  ( ( F : A --> { B }  /\  C  e.  A
)  ->  ( F `  C )  e.  { B } )
2 elsni 3997 . 2  |-  ( ( F `  C )  e.  { B }  ->  ( F `  C
)  =  B )
31, 2syl 17 1  |-  ( ( F : A --> { B }  /\  C  e.  A
)  ->  ( F `  C )  =  B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1405    e. wcel 1842   {csn 3972   -->wf 5565   ` cfv 5569
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4517  ax-nul 4525  ax-pr 4630
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2759  df-rex 2760  df-rab 2763  df-v 3061  df-sbc 3278  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-nul 3739  df-if 3886  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4192  df-br 4396  df-opab 4454  df-id 4738  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-fv 5577
This theorem is referenced by:  fvconst2g  6105  fconst2g  6106  fconstfvOLD  6115  ipasslem9  26167  resf1o  28000  ccatmulgnn0dir  29002  zrtermorngc  38320  zrtermoringc  38389
  Copyright terms: Public domain W3C validator