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Theorem elfvne0 47771
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 ∈ (𝐹𝐵) → 𝐹 ≠ ∅)

Proof of Theorem elfvne0
StepHypRef Expression
1 ne0i 4329 . 2 (𝐴 ∈ (𝐹𝐵) → (𝐹𝐵) ≠ ∅)
2 fveq1 6883 . . . 4 (𝐹 = ∅ → (𝐹𝐵) = (∅‘𝐵))
3 0fv 6928 . . . 4 (∅‘𝐵) = ∅
42, 3eqtrdi 2782 . . 3 (𝐹 = ∅ → (𝐹𝐵) = ∅)
54necon3i 2967 . 2 ((𝐹𝐵) ≠ ∅ → 𝐹 ≠ ∅)
61, 5syl 17 1 (𝐴 ∈ (𝐹𝐵) → 𝐹 ≠ ∅)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1533  wcel 2098  wne 2934  c0 4317  cfv 6536
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-dm 5679  df-iota 6488  df-fv 6544
This theorem is referenced by:  neircl  47793
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