Theorem fnfvelrnd 42790

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 6955	. 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 2110  ran crn 5591   Fn wfn 6427  ‘cfv 6432

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 2015  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2711  ax-sep 5227  ax-nul 5234  ax-pr 5356

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2072  df-mo 2542  df-eu 2571  df-clab 2718  df-cleq 2732  df-clel 2818  df-nfc 2891  df-ral 3071  df-rex 3072  df-rab 3075  df-v 3433  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-nul 4263  df-if 4466  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4846  df-br 5080  df-opab 5142  df-id 5490  df-xp 5596  df-rel 5597  df-cnv 5598  df-co 5599  df-dm 5600  df-rn 5601  df-iota 6390  df-fun 6434  df-fn 6435  df-fv 6440

This theorem is referenced by:  limsupgtlem  43300