![]() |
Mathbox for Alexander van der Vekens |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > dfatafv2iota | Structured version Visualization version GIF version |
Description: If a function is defined at a class 𝐴 the alternate function value at 𝐴 is the unique value assigned to 𝐴 by the function (analogously to (𝐹‘𝐴)). (Contributed by AV, 2-Sep-2022.) |
Ref | Expression |
---|---|
dfatafv2iota | ⊢ (𝐹 defAt 𝐴 → (𝐹''''𝐴) = (℩𝑥𝐴𝐹𝑥)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-afv2 47159 | . 2 ⊢ (𝐹''''𝐴) = if(𝐹 defAt 𝐴, (℩𝑥𝐴𝐹𝑥), 𝒫 ∪ ran 𝐹) | |
2 | iftrue 4537 | . 2 ⊢ (𝐹 defAt 𝐴 → if(𝐹 defAt 𝐴, (℩𝑥𝐴𝐹𝑥), 𝒫 ∪ ran 𝐹) = (℩𝑥𝐴𝐹𝑥)) | |
3 | 1, 2 | eqtrid 2787 | 1 ⊢ (𝐹 defAt 𝐴 → (𝐹''''𝐴) = (℩𝑥𝐴𝐹𝑥)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ifcif 4531 𝒫 cpw 4605 ∪ cuni 4912 class class class wbr 5148 ran crn 5690 ℩cio 6514 defAt wdfat 47066 ''''cafv2 47158 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-if 4532 df-afv2 47159 |
This theorem is referenced by: dfatafv2ex 47163 funressndmafv2rn 47173 afv2eu 47188 afv2res 47189 tz6.12-afv2 47190 dfafv23 47203 rlimdmafv2 47208 |
Copyright terms: Public domain | W3C validator |