Theorem fnfvelrnd 7102

Description: A function's value belongs to its range. (Contributed by Glauco Siliprandi, 2-Jan-2022.)

Hypotheses

Ref	Expression
fnfvelrnd.1	⊢ (𝜑 → 𝐹 Fn 𝐴)
fnfvelrnd.2	⊢ (𝜑 → 𝐵 ∈ 𝐴)

Assertion

Ref	Expression
fnfvelrnd	⊢ (𝜑 → (𝐹‘𝐵) ∈ ran 𝐹)

Proof of Theorem fnfvelrnd

Step	Hyp	Ref	Expression
1		fnfvelrnd.1	. 2 ⊢ (𝜑 → 𝐹 Fn 𝐴)
2		fnfvelrnd.2	. 2 ⊢ (𝜑 → 𝐵 ∈ 𝐴)
3		fnfvelrn 7100	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → (𝐹‘𝐵) ∈ ran 𝐹)
4	1, 2, 3	syl2anc 584	1 ⊢ (𝜑 → (𝐹‘𝐵) ∈ ran 𝐹)

Syntax hints:   → wi 4   ∈ wcel 2108  ran crn 5686   Fn wfn 6556  ‘cfv 6561

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-iota 6514  df-fun 6563  df-fn 6564  df-fv 6569

This theorem is referenced by:  ghmqusnsg  19300  ghmquskerlem3  19304  ghmqusker  19305  noseqrdglem  28311  ssmapsn  45221  limsupgtlem  45792