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Mirrors > Home > MPE Home > Th. List > Mathboxes > elfvne0 | Structured version Visualization version GIF version |
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.) |
Ref | Expression |
---|---|
elfvne0 | ⊢ (𝐴 ∈ (𝐹‘𝐵) → 𝐹 ≠ ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ne0i 4268 | . 2 ⊢ (𝐴 ∈ (𝐹‘𝐵) → (𝐹‘𝐵) ≠ ∅) | |
2 | fveq1 6773 | . . . 4 ⊢ (𝐹 = ∅ → (𝐹‘𝐵) = (∅‘𝐵)) | |
3 | 0fv 6813 | . . . 4 ⊢ (∅‘𝐵) = ∅ | |
4 | 2, 3 | eqtrdi 2794 | . . 3 ⊢ (𝐹 = ∅ → (𝐹‘𝐵) = ∅) |
5 | 4 | necon3i 2976 | . 2 ⊢ ((𝐹‘𝐵) ≠ ∅ → 𝐹 ≠ ∅) |
6 | 1, 5 | syl 17 | 1 ⊢ (𝐴 ∈ (𝐹‘𝐵) → 𝐹 ≠ ∅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1539 ∈ wcel 2106 ≠ wne 2943 ∅c0 4256 ‘cfv 6433 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-nul 5230 ax-pr 5352 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-ne 2944 df-ral 3069 df-rex 3070 df-rab 3073 df-v 3434 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-nul 4257 df-if 4460 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-br 5075 df-dm 5599 df-iota 6391 df-fv 6441 |
This theorem is referenced by: neircl 46198 |
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