Theorem afv20fv0 47821

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 47790	. 2 ⊢ ((𝐹''''𝐴) = ∅ → 𝐹 defAt 𝐴)
2		dfatafv2eqfv 47819	. . . . 5 ⊢ (𝐹 defAt 𝐴 → (𝐹''''𝐴) = (𝐹‘𝐴))
3	2	eqcomd 2767	. . . 4 ⊢ (𝐹 defAt 𝐴 → (𝐹‘𝐴) = (𝐹''''𝐴))
4	3	adantr 484	. . 3 ⊢ ((𝐹 defAt 𝐴 ∧ (𝐹''''𝐴) = ∅) → (𝐹‘𝐴) = (𝐹''''𝐴))
5		simpr 488	. . 3 ⊢ ((𝐹 defAt 𝐴 ∧ (𝐹''''𝐴) = ∅) → (𝐹''''𝐴) = ∅)
6	4, 5	eqtrd 2796	. 2 ⊢ ((𝐹 defAt 𝐴 ∧ (𝐹''''𝐴) = ∅) → (𝐹‘𝐴) = ∅)
7	1, 6	mpancom 698	1 ⊢ ((𝐹''''𝐴) = ∅ → (𝐹‘𝐴) = ∅)

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1559  ∅c0 4285  ‘cfv 6517   defAt wdfat 47674  ''''cafv2 47766

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-ext 2733  ax-nul 5255

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-tru 1562  df-fal 1572  df-ex 1799  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-ne 2957  df-rab 3414  df-v 3455  df-dif 3907  df-un 3909  df-ss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-fv 6525  df-afv2 47767

This theorem is referenced by:  afv2fv0b  47824