Theorem afv20fv0 44642

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 44611	. 2 ⊢ ((𝐹''''𝐴) = ∅ → 𝐹 defAt 𝐴)
2		dfatafv2eqfv 44640	. . . . 5 ⊢ (𝐹 defAt 𝐴 → (𝐹''''𝐴) = (𝐹‘𝐴))
3	2	eqcomd 2744	. . . 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 2778	. 2 ⊢ ((𝐹 defAt 𝐴 ∧ (𝐹''''𝐴) = ∅) → (𝐹‘𝐴) = ∅)
7	1, 6	mpancom 684	1 ⊢ ((𝐹''''𝐴) = ∅ → (𝐹‘𝐴) = ∅)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1539  ∅c0 4253  ‘cfv 6418   defAt wdfat 44495  ''''cafv2 44587

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709  ax-nul 5225

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-tru 1542  df-fal 1552  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-ne 2943  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-pw 4532  df-fv 6426  df-afv2 44588

This theorem is referenced by:  afv2fv0b  44645