Theorem afv0fv0 39878

Description: If the value of the alternative function at an argument is the empty set, the function's value at this argument is the empty set. (Contributed by Alexander van der Vekens, 25-May-2017.)

Assertion

Ref	Expression
afv0fv0	⊢ ((𝐹'''𝐴) = ∅ → (𝐹‘𝐴) = ∅)

Proof of Theorem afv0fv0

Step	Hyp	Ref	Expression
1		0ex 4718	. . 3 ⊢ ∅ ∈ V
2		eleq1a 2683	. . 3 ⊢ (∅ ∈ V → ((𝐹'''𝐴) = ∅ → (𝐹'''𝐴) ∈ V))
3	1, 2	ax-mp 5	. 2 ⊢ ((𝐹'''𝐴) = ∅ → (𝐹'''𝐴) ∈ V)
4		afvvfveq 39877	. . 3 ⊢ ((𝐹'''𝐴) ∈ V → (𝐹'''𝐴) = (𝐹‘𝐴))
5		eqeq1 2614	. . . 4 ⊢ ((𝐹'''𝐴) = (𝐹‘𝐴) → ((𝐹'''𝐴) = ∅ ↔ (𝐹‘𝐴) = ∅))
6	5	biimpd 218	. . 3 ⊢ ((𝐹'''𝐴) = (𝐹‘𝐴) → ((𝐹'''𝐴) = ∅ → (𝐹‘𝐴) = ∅))
7	4, 6	syl 17	. 2 ⊢ ((𝐹'''𝐴) ∈ V → ((𝐹'''𝐴) = ∅ → (𝐹‘𝐴) = ∅))
8	3, 7	mpcom 37	1 ⊢ ((𝐹'''𝐴) = ∅ → (𝐹‘𝐴) = ∅)

Syntax hints:   → wi 4   = wceq 1475   ∈ wcel 1977  Vcvv 3173  ∅c0 3874  ‘cfv 5804  '''cafv 39843

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-nul 3875  df-if 4037  df-fv 5812  df-afv 39846

This theorem is referenced by:  afvfv0bi  39881  aov0ov0  39922