Theorem afv20fv0 47292

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 47261	. 2 ⊢ ((𝐹''''𝐴) = ∅ → 𝐹 defAt 𝐴)
2		dfatafv2eqfv 47290	. . . . 5 ⊢ (𝐹 defAt 𝐴 → (𝐹''''𝐴) = (𝐹‘𝐴))
3	2	eqcomd 2741	. . . 4 ⊢ (𝐹 defAt 𝐴 → (𝐹‘𝐴) = (𝐹''''𝐴))
4	3	adantr 480	. . 3 ⊢ ((𝐹 defAt 𝐴 ∧ (𝐹''''𝐴) = ∅) → (𝐹‘𝐴) = (𝐹''''𝐴))
5		simpr 484	. . 3 ⊢ ((𝐹 defAt 𝐴 ∧ (𝐹''''𝐴) = ∅) → (𝐹''''𝐴) = ∅)
6	4, 5	eqtrd 2770	. 2 ⊢ ((𝐹 defAt 𝐴 ∧ (𝐹''''𝐴) = ∅) → (𝐹‘𝐴) = ∅)
7	1, 6	mpancom 688	1 ⊢ ((𝐹''''𝐴) = ∅ → (𝐹‘𝐴) = ∅)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540  ∅c0 4308  ‘cfv 6531   defAt wdfat 47145  ''''cafv2 47237

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2707  ax-nul 5276

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2714  df-cleq 2727  df-clel 2809  df-ne 2933  df-rab 3416  df-v 3461  df-dif 3929  df-un 3931  df-ss 3943  df-nul 4309  df-if 4501  df-pw 4577  df-fv 6539  df-afv2 47238

This theorem is referenced by:  afv2fv0b  47295