Theorem elfvne0 50291

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 4290	. 2 ⊢ (𝐴 ∈ (𝐹‘𝐵) → (𝐹‘𝐵) ≠ ∅)
2		fveq1 6915	. . . 4 ⊢ (𝐹 = ∅ → (𝐹‘𝐵) = (∅‘𝐵))
3		0fv 6957	. . . 4 ⊢ (∅‘𝐵) = ∅
4	2, 3	eqtrdi 2814	. . 3 ⊢ (𝐹 = ∅ → (𝐹‘𝐵) = ∅)
5	4	necon3i 2991	. 2 ⊢ ((𝐹‘𝐵) ≠ ∅ → 𝐹 ≠ ∅)
6	1, 5	syl 18	1 ⊢ (𝐴 ∈ (𝐹‘𝐵) → 𝐹 ≠ ∅)

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2147   ≠ wne 2959  ∅c0 4281  ‘cfv 6566

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735  ax-nul 5266  ax-pr 5398

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-ne 2960  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-br 5109  df-dm 5668  df-iota 6520  df-fv 6574

This theorem is used by:  neircl  50350  sectrcl  50467  invrcl  50469  isorcl  50478  catcrcl  50861