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| 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 47221 | . 2 ⊢ (𝐹''''𝐴) = if(𝐹 defAt 𝐴, (℩𝑥𝐴𝐹𝑥), 𝒫 ∪ ran 𝐹) | |
| 2 | iftrue 4531 | . 2 ⊢ (𝐹 defAt 𝐴 → if(𝐹 defAt 𝐴, (℩𝑥𝐴𝐹𝑥), 𝒫 ∪ ran 𝐹) = (℩𝑥𝐴𝐹𝑥)) | |
| 3 | 1, 2 | eqtrid 2789 | 1 ⊢ (𝐹 defAt 𝐴 → (𝐹''''𝐴) = (℩𝑥𝐴𝐹𝑥)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → 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: dfatafv2ex 47225 funressndmafv2rn 47235 afv2eu 47250 afv2res 47251 tz6.12-afv2 47252 dfafv23 47265 rlimdmafv2 47270 | 
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