Mathbox for Alexander van der Vekens |
< Previous
Next >
Nearby theorems |
||
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 43482 | . 2 ⊢ (𝐹''''𝐴) = if(𝐹 defAt 𝐴, (℩𝑥𝐴𝐹𝑥), 𝒫 ∪ ran 𝐹) | |
2 | iffalse 4469 | . 2 ⊢ (¬ 𝐹 defAt 𝐴 → if(𝐹 defAt 𝐴, (℩𝑥𝐴𝐹𝑥), 𝒫 ∪ ran 𝐹) = 𝒫 ∪ ran 𝐹) | |
3 | 1, 2 | syl5eq 2867 | 1 ⊢ (¬ 𝐹 defAt 𝐴 → (𝐹''''𝐴) = 𝒫 ∪ ran 𝐹) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1536 ifcif 4460 𝒫 cpw 4532 ∪ cuni 4831 class class class wbr 5059 ran crn 5549 ℩cio 6305 defAt wdfat 43389 ''''cafv2 43481 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-ext 2792 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-ex 1780 df-sb 2069 df-clab 2799 df-cleq 2813 df-clel 2892 df-if 4461 df-afv2 43482 |
This theorem is referenced by: ndfatafv2undef 43485 ndfatafv2nrn 43494 afv2ndefb 43497 afv20defat 43505 |
Copyright terms: Public domain | W3C validator |