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Theorem fvconst 7111
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 7033 . 2 ((𝐹:𝐴⟶{𝐵} ∧ 𝐶𝐴) → (𝐹𝐶) ∈ {𝐵})
2 elsni 4604 . 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 397   = wceq 1542  wcel 2107  {csn 4587  wf 6493  cfv 6497
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-12 2172  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pr 5385
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-fv 6505
This theorem is referenced by:  fvconst2g  7152  fconst2g  7153  f1cdmsn  7229  nf1const  7251  ipasslem9  29822  resf1o  31694  ccatmulgnn0dir  33211  prv1n  34082  sticksstones11  40610  zrtermorngc  46385  zrtermoringc  46454
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