Theorem fnafv2elrn 46402

Description: An alternate function value belongs to the range of the function, analogous to fnfvelrn 7082. (Contributed by AV, 2-Sep-2022.)

Assertion

Ref	Expression
fnafv2elrn	⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → (𝐹''''𝐵) ∈ ran 𝐹)

Proof of Theorem fnafv2elrn

Step	Hyp	Ref	Expression
1		afv2elrn 46400	. 2 ⊢ ((Fun 𝐹 ∧ 𝐵 ∈ dom 𝐹) → (𝐹''''𝐵) ∈ ran 𝐹)
2	1	funfni 6655	1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → (𝐹''''𝐵) ∈ ran 𝐹)

Syntax hints:   → wi 4   ∧ wa 395   ∈ wcel 2105  ran crn 5677   Fn wfn 6538  ''''cafv2 46377

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 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-12 2170  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pr 5427  ax-un 7729

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-iota 6495  df-fun 6545  df-fn 6546  df-dfat 46288  df-afv2 46378

This theorem is referenced by:  fafv2elcdm  46403  fafv2elrnb  46404