Theorem fnafvelrn 47146

Description: A function's value belongs to its range, analogous to fnfvelrn 7069. (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 47145	. 2 ⊢ ((Fun 𝐹 ∧ 𝐵 ∈ dom 𝐹) → (𝐹'''𝐵) ∈ ran 𝐹)
2	1	funfni 6643	1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → (𝐹'''𝐵) ∈ ran 𝐹)

Syntax hints:   → wi 4   ∧ wa 395   ∈ wcel 2108  ran crn 5655   Fn wfn 6525  '''cafv 47094

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-sep 5266  ax-nul 5276  ax-pr 5402

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-nul 4309  df-if 4501  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-int 4923  df-br 5120  df-opab 5182  df-id 5548  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-iota 6483  df-fun 6532  df-fn 6533  df-fv 6538  df-aiota 47062  df-dfat 47096  df-afv 47097

This theorem is referenced by:  fafvelcdm  47147