Theorem elfvne0 48880

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 4286	. 2 ⊢ (𝐴 ∈ (𝐹‘𝐵) → (𝐹‘𝐵) ≠ ∅)
2		fveq1 6816	. . . 4 ⊢ (𝐹 = ∅ → (𝐹‘𝐵) = (∅‘𝐵))
3		0fv 6858	. . . 4 ⊢ (∅‘𝐵) = ∅
4	2, 3	eqtrdi 2782	. . 3 ⊢ (𝐹 = ∅ → (𝐹‘𝐵) = ∅)
5	4	necon3i 2960	. 2 ⊢ ((𝐹‘𝐵) ≠ ∅ → 𝐹 ≠ ∅)
6	1, 5	syl 17	1 ⊢ (𝐴 ∈ (𝐹‘𝐵) → 𝐹 ≠ ∅)

Syntax hints:   → wi 4   = wceq 1541   ∈ wcel 2111   ≠ wne 2928  ∅c0 4278  ‘cfv 6476

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703  ax-nul 5239  ax-pr 5365

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-ne 2929  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-ss 3914  df-nul 4279  df-if 4471  df-sn 4572  df-pr 4574  df-op 4578  df-uni 4855  df-br 5087  df-dm 5621  df-iota 6432  df-fv 6484

This theorem is referenced by:  neircl  48936  sectrcl  49054  invrcl  49056  isorcl  49065  catcrcl  49427