Theorem elfvne0 49339

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 4282	. 2 ⊢ (𝐴 ∈ (𝐹‘𝐵) → (𝐹‘𝐵) ≠ ∅)
2		fveq1 6834	. . . 4 ⊢ (𝐹 = ∅ → (𝐹‘𝐵) = (∅‘𝐵))
3		0fv 6876	. . . 4 ⊢ (∅‘𝐵) = ∅
4	2, 3	eqtrdi 2788	. . 3 ⊢ (𝐹 = ∅ → (𝐹‘𝐵) = ∅)
5	4	necon3i 2965	. 2 ⊢ ((𝐹‘𝐵) ≠ ∅ → 𝐹 ≠ ∅)
6	1, 5	syl 17	1 ⊢ (𝐴 ∈ (𝐹‘𝐵) → 𝐹 ≠ ∅)

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2114   ≠ wne 2933  ∅c0 4274  ‘cfv 6493

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-nul 5242  ax-pr 5371

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-ne 2934  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-br 5087  df-dm 5635  df-iota 6449  df-fv 6501

This theorem is referenced by:  neircl  49395  sectrcl  49512  invrcl  49514  isorcl  49523  catcrcl  49885