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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 4347 | . 2 ⊢ (𝐴 ∈ (𝐹‘𝐵) → (𝐹‘𝐵) ≠ ∅) | |
2 | fveq1 6906 | . . . 4 ⊢ (𝐹 = ∅ → (𝐹‘𝐵) = (∅‘𝐵)) | |
3 | 0fv 6951 | . . . 4 ⊢ (∅‘𝐵) = ∅ | |
4 | 2, 3 | eqtrdi 2791 | . . 3 ⊢ (𝐹 = ∅ → (𝐹‘𝐵) = ∅) |
5 | 4 | necon3i 2971 | . 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 1537 ∈ wcel 2106 ≠ wne 2938 ∅c0 4339 ‘cfv 6563 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-dm 5699 df-iota 6516 df-fv 6571 |
This theorem is referenced by: neircl 48701 |
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