Theorem fnafvelrn 43388

Description: A function's value belongs to its range, analogous to fnfvelrn 6848. (Contributed by Alexander van der Vekens, 25-May-2017.)

Assertion

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

Proof of Theorem fnafvelrn

Step	Hyp	Ref	Expression
1		afvelrn 43387	. 2 ⊢ ((Fun 𝐹 ∧ 𝐵 ∈ dom 𝐹) → (𝐹'''𝐵) ∈ ran 𝐹)
2	1	funfni 6457	1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → (𝐹'''𝐵) ∈ ran 𝐹)

Syntax hints:   → wi 4   ∧ wa 398   ∈ wcel 2114  ran crn 5556   Fn wfn 6350  '''cafv 43336

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 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-sep 5203  ax-nul 5210  ax-pow 5266  ax-pr 5330

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-fal 1550  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4839  df-int 4877  df-br 5067  df-opab 5129  df-id 5460  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-rn 5566  df-res 5567  df-iota 6314  df-fun 6357  df-fn 6358  df-fv 6363  df-aiota 43305  df-dfat 43338  df-afv 43339

This theorem is referenced by:  fafvelrn  43389