| Mathbox for Alexander van der Vekens |
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| Mirrors > Home > MPE Home > Th. List > Mathboxes > dfatafv2eqfv | Structured version Visualization version GIF version | ||
| Description: If a function is defined at a class 𝐴, the alternate function value equals the function's value at 𝐴. (Contributed by AV, 3-Sep-2022.) |
| Ref | Expression |
|---|---|
| dfatafv2eqfv | ⊢ (𝐹 defAt 𝐴 → (𝐹''''𝐴) = (𝐹‘𝐴)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | dfafv22 47271 | . 2 ⊢ (𝐹''''𝐴) = if(𝐹 defAt 𝐴, (𝐹‘𝐴), 𝒫 ∪ ran 𝐹) | |
| 2 | iftrue 4531 | . 2 ⊢ (𝐹 defAt 𝐴 → if(𝐹 defAt 𝐴, (𝐹‘𝐴), 𝒫 ∪ ran 𝐹) = (𝐹‘𝐴)) | |
| 3 | 1, 2 | eqtrid 2789 | 1 ⊢ (𝐹 defAt 𝐴 → (𝐹''''𝐴) = (𝐹‘𝐴)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 = wceq 1540 ifcif 4525 𝒫 cpw 4600 ∪ cuni 4907 ran crn 5686 ‘cfv 6561 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-tru 1543 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-rab 3437 df-v 3482 df-un 3956 df-if 4526 df-fv 6569 df-afv2 47221 |
| This theorem is referenced by: afv2rnfveq 47274 afv20fv0 47275 afv2fvn0fveq 47276 |
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