Theorem fexafv2ex 42123

Description: The alternate function value is always a set if the function (resp. the domain of the function) is a set. (Contributed by AV, 3-Sep-2022.)

Assertion

Ref	Expression
fexafv2ex	⊢ (𝐹 ∈ 𝑉 → (𝐹''''𝐴) ∈ V)

Proof of Theorem fexafv2ex

Step	Hyp	Ref	Expression
1		rnexg 7360	. 2 ⊢ (𝐹 ∈ 𝑉 → ran 𝐹 ∈ V)
2		afv2ex 42117	. 2 ⊢ (ran 𝐹 ∈ V → (𝐹''''𝐴) ∈ V)
3	1, 2	syl 17	1 ⊢ (𝐹 ∈ 𝑉 → (𝐹''''𝐴) ∈ V)

Syntax hints:   → wi 4   ∈ wcel 2166  Vcvv 3415  ran crn 5344  ''''cafv2 42111

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-8 2168  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2804  ax-sep 5006  ax-nul 5014  ax-pow 5066  ax-pr 5128  ax-un 7210

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2606  df-eu 2641  df-clab 2813  df-cleq 2819  df-clel 2822  df-nfc 2959  df-ral 3123  df-rex 3124  df-rab 3127  df-v 3417  df-sbc 3664  df-dif 3802  df-un 3804  df-in 3806  df-ss 3813  df-nul 4146  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4660  df-br 4875  df-opab 4937  df-cnv 5351  df-dm 5353  df-rn 5354  df-iota 6087  df-afv2 42112

This theorem is referenced by: (None)