Mathbox for Alexander van der Vekens |
< Previous
Next >
Nearby theorems |
||
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 43417 | . . 3 ⊢ (¬ 𝐹 defAt 𝐴 → (𝐹''''𝐴) = 𝒫 ∪ ran 𝐹) | |
2 | pwne0 5259 | . . . . 5 ⊢ 𝒫 ∪ ran 𝐹 ≠ ∅ | |
3 | 2 | neii 3020 | . . . 4 ⊢ ¬ 𝒫 ∪ ran 𝐹 = ∅ |
4 | eqeq1 2827 | . . . 4 ⊢ ((𝐹''''𝐴) = 𝒫 ∪ ran 𝐹 → ((𝐹''''𝐴) = ∅ ↔ 𝒫 ∪ ran 𝐹 = ∅)) | |
5 | 3, 4 | mtbiri 329 | . . 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 1537 ∅c0 4293 𝒫 cpw 4541 ∪ cuni 4840 ran crn 5558 defAt wdfat 43322 ''''cafv2 43414 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-nul 5212 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-v 3498 df-dif 3941 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-afv2 43415 |
This theorem is referenced by: afv20fv0 43469 |
Copyright terms: Public domain | W3C validator |