Theorem afv20fv0 47855

Description: If the alternate function value at an argument is the empty set, the function's value at this argument is the empty set. (Contributed by AV, 3-Sep-2022.)

Assertion

Ref	Expression
afv20fv0	⊢ ((𝐹''''𝐴) = ∅ → (𝐹‘𝐴) = ∅)

Proof of Theorem afv20fv0

Step	Hyp	Ref	Expression
1		afv20defat 47824	. 2 ⊢ ((𝐹''''𝐴) = ∅ → 𝐹 defAt 𝐴)
2		dfatafv2eqfv 47853	. . . . 5 ⊢ (𝐹 defAt 𝐴 → (𝐹''''𝐴) = (𝐹‘𝐴))
3	2	eqcomd 2771	. . . 4 ⊢ (𝐹 defAt 𝐴 → (𝐹‘𝐴) = (𝐹''''𝐴))
4	3	adantr 485	. . 3 ⊢ ((𝐹 defAt 𝐴 ∧ (𝐹''''𝐴) = ∅) → (𝐹‘𝐴) = (𝐹''''𝐴))
5		simpr 489	. . 3 ⊢ ((𝐹 defAt 𝐴 ∧ (𝐹''''𝐴) = ∅) → (𝐹''''𝐴) = ∅)
6	4, 5	eqtrd 2800	. 2 ⊢ ((𝐹 defAt 𝐴 ∧ (𝐹''''𝐴) = ∅) → (𝐹‘𝐴) = ∅)
7	1, 6	mpancom 700	1 ⊢ ((𝐹''''𝐴) = ∅ → (𝐹‘𝐴) = ∅)

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1563  ∅c0 4288  ‘cfv 6525   defAt wdfat 47708  ''''cafv2 47800

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-ext 2737  ax-nul 5261

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1566  df-fal 1576  df-ex 1803  df-sb 2094  df-clab 2744  df-cleq 2757  df-clel 2840  df-ne 2961  df-rab 3418  df-v 3459  df-dif 3910  df-un 3912  df-ss 3924  df-nul 4289  df-if 4484  df-pw 4560  df-fv 6533  df-afv2 47801

This theorem is referenced by:  afv2fv0b  47858