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Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > afv20defat | Structured version Visualization version GIF version |
Description: If the alternate function value at an argument is the empty set, the function is defined at this argument. (Contributed by AV, 3-Sep-2022.) |
Ref | Expression |
---|---|
afv20defat | ⊢ ((𝐹''''𝐴) = ∅ → 𝐹 defAt 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ndfatafv2 43767 | . . 3 ⊢ (¬ 𝐹 defAt 𝐴 → (𝐹''''𝐴) = 𝒫 ∪ ran 𝐹) | |
2 | pwne0 5222 | . . . . 5 ⊢ 𝒫 ∪ ran 𝐹 ≠ ∅ | |
3 | 2 | neii 2989 | . . . 4 ⊢ ¬ 𝒫 ∪ ran 𝐹 = ∅ |
4 | eqeq1 2802 | . . . 4 ⊢ ((𝐹''''𝐴) = 𝒫 ∪ ran 𝐹 → ((𝐹''''𝐴) = ∅ ↔ 𝒫 ∪ ran 𝐹 = ∅)) | |
5 | 3, 4 | mtbiri 330 | . . 3 ⊢ ((𝐹''''𝐴) = 𝒫 ∪ ran 𝐹 → ¬ (𝐹''''𝐴) = ∅) |
6 | 1, 5 | syl 17 | . 2 ⊢ (¬ 𝐹 defAt 𝐴 → ¬ (𝐹''''𝐴) = ∅) |
7 | 6 | con4i 114 | 1 ⊢ ((𝐹''''𝐴) = ∅ → 𝐹 defAt 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1538 ∅c0 4243 𝒫 cpw 4497 ∪ cuni 4800 ran crn 5520 defAt wdfat 43672 ''''cafv2 43764 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-ext 2770 ax-nul 5174 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-ex 1782 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-ne 2988 df-v 3443 df-dif 3884 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-afv2 43765 |
This theorem is referenced by: afv20fv0 43819 |
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