Theorem fnafv2elrn 47270

Description: An alternate function value belongs to the range of the function, analogous to fnfvelrn 7013. (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 47268	. 2 ⊢ ((Fun 𝐹 ∧ 𝐵 ∈ dom 𝐹) → (𝐹''''𝐵) ∈ ran 𝐹)
2	1	funfni 6587	1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → (𝐹''''𝐵) ∈ ran 𝐹)

Syntax hints:   → wi 4   ∧ wa 395   ∈ wcel 2111  ran crn 5617   Fn wfn 6476  ''''cafv2 47245

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 2113  ax-9 2121  ax-10 2144  ax-12 2180  ax-ext 2703  ax-sep 5234  ax-nul 5244  ax-pr 5370  ax-un 7668

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 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-ne 2929  df-nel 3033  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4476  df-pw 4552  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-br 5092  df-opab 5154  df-id 5511  df-xp 5622  df-rel 5623  df-cnv 5624  df-co 5625  df-dm 5626  df-rn 5627  df-res 5628  df-iota 6437  df-fun 6483  df-fn 6484  df-dfat 47156  df-afv2 47246

This theorem is referenced by:  fafv2elcdm  47271  fafv2elrnb  47272