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| Mirrors > Home > MPE Home > Th. List > Mathboxes > ndfatafv2 | Structured version Visualization version GIF version | ||
| Description: The alternate function value at a class 𝐴 if the function is not defined at this set 𝐴. (Contributed by AV, 2-Sep-2022.) |
| Ref | Expression |
|---|---|
| ndfatafv2 | ⊢ (¬ 𝐹 defAt 𝐴 → (𝐹''''𝐴) = 𝒫 ∪ ran 𝐹) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | df-afv2 47221 | . 2 ⊢ (𝐹''''𝐴) = if(𝐹 defAt 𝐴, (℩𝑥𝐴𝐹𝑥), 𝒫 ∪ ran 𝐹) | |
| 2 | iffalse 4534 | . 2 ⊢ (¬ 𝐹 defAt 𝐴 → if(𝐹 defAt 𝐴, (℩𝑥𝐴𝐹𝑥), 𝒫 ∪ ran 𝐹) = 𝒫 ∪ ran 𝐹) | |
| 3 | 1, 2 | eqtrid 2789 | 1 ⊢ (¬ 𝐹 defAt 𝐴 → (𝐹''''𝐴) = 𝒫 ∪ ran 𝐹) |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 → wi 4 = wceq 1540 ifcif 4525 𝒫 cpw 4600 ∪ cuni 4907 class class class wbr 5143 ran crn 5686 ℩cio 6512 defAt wdfat 47128 ''''cafv2 47220 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-if 4526 df-afv2 47221 |
| This theorem is referenced by: ndfatafv2undef 47224 ndfatafv2nrn 47233 afv2ndefb 47236 afv20defat 47244 |
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