Theorem fnfvelrnd 7023

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 7021	. 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 2113  ran crn 5622   Fn wfn 6483  ‘cfv 6488

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-12 2182  ax-ext 2705  ax-sep 5238  ax-nul 5248  ax-pr 5374

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-ne 2930  df-ral 3049  df-rex 3058  df-rab 3397  df-v 3439  df-dif 3901  df-un 3903  df-ss 3915  df-nul 4283  df-if 4477  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4861  df-br 5096  df-opab 5158  df-id 5516  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-iota 6444  df-fun 6490  df-fn 6491  df-fv 6496

This theorem is referenced by:  ghmqusnsg  19198  ghmquskerlem3  19202  ghmqusker  19203  noseqrdglem  28238  esplyind  33615  ssmapsn  45340  limsupgtlem  45902