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Theorem fvconst 6936
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 ffvelrn 6859 . 2 ((𝐹:𝐴⟶{𝐵} ∧ 𝐶𝐴) → (𝐹𝐶) ∈ {𝐵})
2 elsni 4533 . 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 399   = wceq 1542  wcel 2114  {csn 4516  wf 6335  cfv 6339
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2710  ax-sep 5167  ax-nul 5174  ax-pr 5296
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ral 3058  df-rex 3059  df-v 3400  df-sbc 3681  df-dif 3846  df-un 3848  df-in 3850  df-ss 3860  df-nul 4212  df-if 4415  df-sn 4517  df-pr 4519  df-op 4523  df-uni 4797  df-br 5031  df-opab 5093  df-id 5429  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-iota 6297  df-fun 6341  df-fn 6342  df-f 6343  df-fv 6347
This theorem is referenced by:  fvconst2g  6974  fconst2g  6975  nf1const  7071  ipasslem9  28773  resf1o  30640  ccatmulgnn0dir  32091  prv1n  32964  zrtermorngc  45093  zrtermoringc  45162
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