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Theorem fvconst 7157
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 7076 . 2 ((𝐹:𝐴⟶{𝐵} ∧ 𝐶𝐴) → (𝐹𝐶) ∈ {𝐵})
2 elsni 4640 . 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 395   = wceq 1533  wcel 2098  {csn 4623  wf 6532  cfv 6536
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 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-iota 6488  df-fun 6538  df-fn 6539  df-f 6540  df-fv 6544
This theorem is referenced by:  fvconst2g  7198  fconst2g  7199  f1cdmsn  7275  nf1const  7297  zrtermorngc  20536  zrtermoringc  20568  ipasslem9  30595  resf1o  32459  ccatmulgnn0dir  34082  prv1n  34949  sticksstones11  41515
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