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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 4235 | . 2 ⊢ (𝐴 ∈ (𝐹‘𝐵) → (𝐹‘𝐵) ≠ ∅) | |
2 | fveq1 6694 | . . . 4 ⊢ (𝐹 = ∅ → (𝐹‘𝐵) = (∅‘𝐵)) | |
3 | 0fv 6734 | . . . 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 ≠ ∅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1543 ∈ wcel 2112 ≠ wne 2932 ∅c0 4223 ‘cfv 6358 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pr 5307 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-ne 2933 df-ral 3056 df-rex 3057 df-rab 3060 df-v 3400 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-sn 4528 df-pr 4530 df-op 4534 df-uni 4806 df-br 5040 df-dm 5546 df-iota 6316 df-fv 6366 |
This theorem is referenced by: neircl 45814 |
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