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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 4233 | . 2 ⊢ (𝐴 ∈ (𝐹‘𝐵) → (𝐹‘𝐵) ≠ ∅) | |
2 | fveq1 6685 | . . . 4 ⊢ (𝐹 = ∅ → (𝐹‘𝐵) = (∅‘𝐵)) | |
3 | 0fv 6725 | . . . 4 ⊢ (∅‘𝐵) = ∅ | |
4 | 2, 3 | eqtrdi 2790 | . . 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 1542 ∈ wcel 2114 ≠ wne 2935 ∅c0 4221 ‘cfv 6349 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2711 ax-sep 5177 ax-nul 5184 ax-pr 5306 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2541 df-eu 2571 df-clab 2718 df-cleq 2731 df-clel 2812 df-ne 2936 df-ral 3059 df-rex 3060 df-v 3402 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4222 df-if 4425 df-sn 4527 df-pr 4529 df-op 4533 df-uni 4807 df-br 5041 df-dm 5545 df-iota 6307 df-fv 6357 |
This theorem is referenced by: neircl 45767 |
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