Theorem elfvne0 49094

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 4293	. 2 ⊢ (𝐴 ∈ (𝐹‘𝐵) → (𝐹‘𝐵) ≠ ∅)
2		fveq1 6833	. . . 4 ⊢ (𝐹 = ∅ → (𝐹‘𝐵) = (∅‘𝐵))
3		0fv 6875	. . . 4 ⊢ (∅‘𝐵) = ∅
4	2, 3	eqtrdi 2787	. . 3 ⊢ (𝐹 = ∅ → (𝐹‘𝐵) = ∅)
5	4	necon3i 2964	. 2 ⊢ ((𝐹‘𝐵) ≠ ∅ → 𝐹 ≠ ∅)
6	1, 5	syl 17	1 ⊢ (𝐴 ∈ (𝐹‘𝐵) → 𝐹 ≠ ∅)

Syntax hints:   → wi 4   = wceq 1541   ∈ wcel 2113   ≠ wne 2932  ∅c0 4285  ‘cfv 6492

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 2115  ax-9 2123  ax-ext 2708  ax-nul 5251  ax-pr 5377

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 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-ne 2933  df-rab 3400  df-v 3442  df-dif 3904  df-un 3906  df-ss 3918  df-nul 4286  df-if 4480  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-br 5099  df-dm 5634  df-iota 6448  df-fv 6500

This theorem is referenced by:  neircl  49150  sectrcl  49267  invrcl  49269  isorcl  49278  catcrcl  49640