Theorem fnafv2elrn 43426

Description: An alternate function value belongs to the range of the function, analogous to fnfvelrn 6842. (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 43424	. 2 ⊢ ((Fun 𝐹 ∧ 𝐵 ∈ dom 𝐹) → (𝐹''''𝐵) ∈ ran 𝐹)
2	1	funfni 6451	1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → (𝐹''''𝐵) ∈ ran 𝐹)

Syntax hints:   → wi 4   ∧ wa 398   ∈ wcel 2110  ran crn 5550   Fn wfn 6344  ''''cafv2 43401

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5195  ax-nul 5202  ax-pr 5321  ax-un 7455

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-nel 3124  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-sbc 3772  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4561  df-pr 4563  df-op 4567  df-uni 4832  df-br 5059  df-opab 5121  df-id 5454  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-rn 5560  df-res 5561  df-iota 6308  df-fun 6351  df-fn 6352  df-dfat 43312  df-afv2 43402

This theorem is referenced by:  fafv2elrn  43427  fafv2elrnb  43428