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Theorem fvconst 6070
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 6010 . 2  |-  ( ( F : A --> { B }  /\  C  e.  A
)  ->  ( F `  C )  e.  { B } )
2 elsni 4045 . 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 1374    e. wcel 1762   {csn 4020   -->wf 5575   ` cfv 5579
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-sep 4561  ax-nul 4569  ax-pr 4679
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3108  df-sbc 3325  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-br 4441  df-opab 4499  df-id 4788  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-fv 5587
This theorem is referenced by:  fvconst2g  6105  fconst2g  6106  fconstfv  6114  ipasslem9  25415  resf1o  27211  ccatmulgnn0dir  28122  plymul02  28129
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