Theorem fnfvelrnd 7077

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 7075	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → (𝐹‘𝐵) ∈ ran 𝐹)
4	1, 2, 3	syl2anc 595	1 ⊢ (𝜑 → (𝐹‘𝐵) ∈ ran 𝐹)

Syntax hints:   → wi 4   ∈ wcel 2143  ran crn 5662   Fn wfn 6531  ‘cfv 6536

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-12 2213  ax-ext 2735  ax-sep 5257  ax-nul 5269  ax-pr 5404

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-br 5110  df-opab 5174  df-id 5556  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-iota 6492  df-fun 6538  df-fn 6539  df-fv 6544

This theorem is referenced by:  ghmqusnsg  19347  ghmquskerlem3  19351  ghmqusker  19352  noseqrdglem  28498  tgelrnpln  29058  esplyfvaln  33964  esplyind  33965  mh-inf3f1  37052  ssmapsn  45932  limsupgtlem  46491