Mathbox for Alexander van der Vekens |
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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 43285 | . 2 ⊢ (𝐹''''𝐴) = if(𝐹 defAt 𝐴, (℩𝑥𝐴𝐹𝑥), 𝒫 ∪ ran 𝐹) | |
2 | iffalse 4472 | . 2 ⊢ (¬ 𝐹 defAt 𝐴 → if(𝐹 defAt 𝐴, (℩𝑥𝐴𝐹𝑥), 𝒫 ∪ ran 𝐹) = 𝒫 ∪ ran 𝐹) | |
3 | 1, 2 | syl5eq 2865 | 1 ⊢ (¬ 𝐹 defAt 𝐴 → (𝐹''''𝐴) = 𝒫 ∪ ran 𝐹) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1528 ifcif 4463 𝒫 cpw 4535 ∪ cuni 4830 class class class wbr 5057 ran crn 5549 ℩cio 6305 defAt wdfat 43192 ''''cafv2 43284 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-ext 2790 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-ex 1772 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-if 4464 df-afv2 43285 |
This theorem is referenced by: ndfatafv2undef 43288 ndfatafv2nrn 43297 afv2ndefb 43300 afv20defat 43308 |
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