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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 43428 | . 2 ⊢ (𝐹''''𝐴) = if(𝐹 defAt 𝐴, (℩𝑥𝐴𝐹𝑥), 𝒫 ∪ ran 𝐹) | |
2 | iftrue 4473 | . 2 ⊢ (𝐹 defAt 𝐴 → if(𝐹 defAt 𝐴, (℩𝑥𝐴𝐹𝑥), 𝒫 ∪ ran 𝐹) = (℩𝑥𝐴𝐹𝑥)) | |
3 | 1, 2 | syl5eq 2868 | 1 ⊢ (𝐹 defAt 𝐴 → (𝐹''''𝐴) = (℩𝑥𝐴𝐹𝑥)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ifcif 4467 𝒫 cpw 4539 ∪ cuni 4838 class class class wbr 5066 ran crn 5556 ℩cio 6312 defAt wdfat 43335 ''''cafv2 43427 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-ex 1781 df-sb 2070 df-clab 2800 df-cleq 2814 df-clel 2893 df-if 4468 df-afv2 43428 |
This theorem is referenced by: dfatafv2ex 43432 funressndmafv2rn 43442 afv2eu 43457 afv2res 43458 tz6.12-afv2 43459 dfafv23 43472 rlimdmafv2 43477 |
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