![]() |
Mathbox for Zhi Wang |
< Previous
Next >
Nearby theorems |
|
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 4329 | . 2 ⊢ (𝐴 ∈ (𝐹‘𝐵) → (𝐹‘𝐵) ≠ ∅) | |
2 | fveq1 6883 | . . . 4 ⊢ (𝐹 = ∅ → (𝐹‘𝐵) = (∅‘𝐵)) | |
3 | 0fv 6928 | . . . 4 ⊢ (∅‘𝐵) = ∅ | |
4 | 2, 3 | eqtrdi 2782 | . . 3 ⊢ (𝐹 = ∅ → (𝐹‘𝐵) = ∅) |
5 | 4 | necon3i 2967 | . 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 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 |
Copyright terms: Public domain | W3C validator |