Theorem afv20fv0 48426

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 48395	. 2 ⊢ ((𝐹''''𝐴) = ∅ → 𝐹 defAt 𝐴)
2		dfatafv2eqfv 48424	. . . . 5 ⊢ (𝐹 defAt 𝐴 → (𝐹''''𝐴) = (𝐹‘𝐴))
3	2	eqcomd 2766	. . . 4 ⊢ (𝐹 defAt 𝐴 → (𝐹‘𝐴) = (𝐹''''𝐴))
4	3	adantr 486	. . 3 ⊢ ((𝐹 defAt 𝐴 ∧ (𝐹''''𝐴) = ∅) → (𝐹‘𝐴) = (𝐹''''𝐴))
5		simpr 490	. . 3 ⊢ ((𝐹 defAt 𝐴 ∧ (𝐹''''𝐴) = ∅) → (𝐹''''𝐴) = ∅)
6	4, 5	eqtrd 2795	. 2 ⊢ ((𝐹 defAt 𝐴 ∧ (𝐹''''𝐴) = ∅) → (𝐹‘𝐴) = ∅)
7	1, 6	mpancom 701	1 ⊢ ((𝐹''''𝐴) = ∅ → (𝐹‘𝐴) = ∅)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1569  ∅c0 4278  ‘cfv 6538   defAt wdfat 48279  ''''cafv2 48371

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732  ax-nul 5259

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-ne 2956  df-rab 3413  df-v 3452  df-ss 3896  df-dif 4047  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-fv 6546  df-afv2 48372

This theorem is used by:  afv2fv0b  48429