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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 48056 | . 2 ⊢ (𝐹''''𝐴) = if(𝐹 defAt 𝐴, (𝐹‘𝐴), 𝒫 ∪ ran 𝐹) | |
| 2 | iftrue 4495 | . 2 ⊢ (𝐹 defAt 𝐴 → if(𝐹 defAt 𝐴, (𝐹‘𝐴), 𝒫 ∪ ran 𝐹) = (𝐹‘𝐴)) | |
| 3 | 1, 2 | eqtrid 2812 | 1 ⊢ (𝐹 defAt 𝐴 → (𝐹''''𝐴) = (𝐹‘𝐴)) |
| Colors of variables: wff setvar class |
| This proof depends on syntax axioms: → wi 4 = wceq 1570 ifcif 4489 𝒫 cpw 4564 ∪ cuni 4874 ran crn 5664 ‘cfv 6540 defAt wdfat 47913 ''''cafv2 48005 |
| This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2148 ax-9 2156 ax-ext 2737 |
| This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-tru 1573 df-ex 1813 df-sb 2100 df-clab 2744 df-cleq 2757 df-clel 2840 df-rab 3419 df-v 3459 df-un 3911 df-if 4490 df-fv 6548 df-afv2 48006 |
| This theorem is used by: afv2rnfveq 48059 afv20fv0 48060 afv2fvn0fveq 48061 |
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