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Mathbox for Alexander van der Vekens |
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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 46215 | . 2 ⊢ (𝐹''''𝐴) = if(𝐹 defAt 𝐴, (℩𝑥𝐴𝐹𝑥), 𝒫 ∪ ran 𝐹) | |
2 | iftrue 4533 | . 2 ⊢ (𝐹 defAt 𝐴 → if(𝐹 defAt 𝐴, (℩𝑥𝐴𝐹𝑥), 𝒫 ∪ ran 𝐹) = (℩𝑥𝐴𝐹𝑥)) | |
3 | 1, 2 | eqtrid 2782 | 1 ⊢ (𝐹 defAt 𝐴 → (𝐹''''𝐴) = (℩𝑥𝐴𝐹𝑥)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1539 ifcif 4527 𝒫 cpw 4601 ∪ cuni 4907 class class class wbr 5147 ran crn 5676 ℩cio 6492 defAt wdfat 46122 ''''cafv2 46214 |
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 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-ext 2701 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-ex 1780 df-sb 2066 df-clab 2708 df-cleq 2722 df-clel 2808 df-if 4528 df-afv2 46215 |
This theorem is referenced by: dfatafv2ex 46219 funressndmafv2rn 46229 afv2eu 46244 afv2res 46245 tz6.12-afv2 46246 dfafv23 46259 rlimdmafv2 46264 |
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