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Theorem fvconst 7179
Description: The value of a constant function. (Contributed by NM, 30-May-1999.)
Assertion
Ref Expression
fvconst ((𝐹:𝐴⟶{𝐵} ∧ 𝐶𝐴) → (𝐹𝐶) = 𝐵)

Proof of Theorem fvconst
StepHypRef Expression
1 ffvelcdm 7096 . 2 ((𝐹:𝐴⟶{𝐵} ∧ 𝐶𝐴) → (𝐹𝐶) ∈ {𝐵})
2 elsni 4649 . 2 ((𝐹𝐶) ∈ {𝐵} → (𝐹𝐶) = 𝐵)
31, 2syl 17 1 ((𝐹:𝐴⟶{𝐵} ∧ 𝐶𝐴) → (𝐹𝐶) = 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394   = wceq 1533  wcel 2098  {csn 4632  wf 6549  cfv 6553
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-12 2166  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pr 5433
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-br 5153  df-opab 5215  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-fv 6561
This theorem is referenced by:  fvconst2g  7220  fconst2g  7221  f1cdmsn  7297  nf1const  7319  zrtermorngc  20583  zrtermoringc  20615  ipasslem9  30668  resf1o  32533  ccatmulgnn0dir  34207  prv1n  35074  sticksstones11  41660
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