Theorem fvconst 6687

Description: The value of a constant function. (Contributed by NM, 30-May-1999.)

Assertion

Ref	Expression
fvconst	⊢ ((𝐹:𝐴⟶{𝐵} ∧ 𝐶 ∈ 𝐴) → (𝐹‘𝐶) = 𝐵)

Proof of Theorem fvconst

Step	Hyp	Ref	Expression
1		ffvelrn 6611	. 2 ⊢ ((𝐹:𝐴⟶{𝐵} ∧ 𝐶 ∈ 𝐴) → (𝐹‘𝐶) ∈ {𝐵})
2		elsni 4416	. 2 ⊢ ((𝐹‘𝐶) ∈ {𝐵} → (𝐹‘𝐶) = 𝐵)
3	1, 2	syl 17	1 ⊢ ((𝐹:𝐴⟶{𝐵} ∧ 𝐶 ∈ 𝐴) → (𝐹‘𝐶) = 𝐵)

Syntax hints:   → wi 4   ∧ wa 386   = wceq 1656   ∈ wcel 2164  {csn 4399  ⟶wf 6123  ‘cfv 6127

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-sep 5007  ax-nul 5015  ax-pr 5129

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-sbc 3663  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4147  df-if 4309  df-sn 4400  df-pr 4402  df-op 4406  df-uni 4661  df-br 4876  df-opab 4938  df-id 5252  df-xp 5352  df-rel 5353  df-cnv 5354  df-co 5355  df-dm 5356  df-rn 5357  df-iota 6090  df-fun 6129  df-fn 6130  df-f 6131  df-fv 6135

This theorem is referenced by:  fvconst2g  6728  fconst2g  6729  ipasslem9  28244  resf1o  30049  ccatmulgnn0dir  31162  zrtermorngc  42862  zrtermoringc  42931