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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 43765 | . 2 ⊢ (𝐹''''𝐴) = if(𝐹 defAt 𝐴, (℩𝑥𝐴𝐹𝑥), 𝒫 ∪ ran 𝐹) | |
2 | iftrue 4431 | . 2 ⊢ (𝐹 defAt 𝐴 → if(𝐹 defAt 𝐴, (℩𝑥𝐴𝐹𝑥), 𝒫 ∪ ran 𝐹) = (℩𝑥𝐴𝐹𝑥)) | |
3 | 1, 2 | syl5eq 2845 | 1 ⊢ (𝐹 defAt 𝐴 → (𝐹''''𝐴) = (℩𝑥𝐴𝐹𝑥)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1538 ifcif 4425 𝒫 cpw 4497 ∪ cuni 4800 class class class wbr 5030 ran crn 5520 ℩cio 6281 defAt wdfat 43672 ''''cafv2 43764 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-ex 1782 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-if 4426 df-afv2 43765 |
This theorem is referenced by: dfatafv2ex 43769 funressndmafv2rn 43779 afv2eu 43794 afv2res 43795 tz6.12-afv2 43796 dfafv23 43809 rlimdmafv2 43814 |
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