Theorem elfvne0 46176

Description: If a function value has a member, then the function is not an empty set (An artifact of our function value definition.) (Contributed by Zhi Wang, 16-Sep-2024.)

Assertion

Ref	Expression
elfvne0	⊢ (𝐴 ∈ (𝐹‘𝐵) → 𝐹 ≠ ∅)

Proof of Theorem elfvne0

Step	Hyp	Ref	Expression
1		ne0i 4268	. 2 ⊢ (𝐴 ∈ (𝐹‘𝐵) → (𝐹‘𝐵) ≠ ∅)
2		fveq1 6773	. . . 4 ⊢ (𝐹 = ∅ → (𝐹‘𝐵) = (∅‘𝐵))
3		0fv 6813	. . . 4 ⊢ (∅‘𝐵) = ∅
4	2, 3	eqtrdi 2794	. . 3 ⊢ (𝐹 = ∅ → (𝐹‘𝐵) = ∅)
5	4	necon3i 2976	. 2 ⊢ ((𝐹‘𝐵) ≠ ∅ → 𝐹 ≠ ∅)
6	1, 5	syl 17	1 ⊢ (𝐴 ∈ (𝐹‘𝐵) → 𝐹 ≠ ∅)

Syntax hints:   → wi 4   = wceq 1539   ∈ wcel 2106   ≠ wne 2943  ∅c0 4256  ‘cfv 6433

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-nul 5230  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-ne 2944  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-dm 5599  df-iota 6391  df-fv 6441

This theorem is referenced by:  neircl  46198