Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > afvfundmfveq | Structured version Visualization version GIF version |
Description: If a class is a function restricted to a member of its domain, then the function value for this member is equal for both definitions. (Contributed by Alexander van der Vekens, 25-May-2017.) |
Ref | Expression |
---|---|
afvfundmfveq | ⊢ (𝐹 defAt 𝐴 → (𝐹'''𝐴) = (𝐹‘𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfafv2 39861 | . 2 ⊢ (𝐹'''𝐴) = if(𝐹 defAt 𝐴, (𝐹‘𝐴), V) | |
2 | iftrue 4042 | . 2 ⊢ (𝐹 defAt 𝐴 → if(𝐹 defAt 𝐴, (𝐹‘𝐴), V) = (𝐹‘𝐴)) | |
3 | 1, 2 | syl5eq 2656 | 1 ⊢ (𝐹 defAt 𝐴 → (𝐹'''𝐴) = (𝐹‘𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 Vcvv 3173 ifcif 4036 ‘cfv 5804 defAt wdfat 39842 '''cafv 39843 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-rab 2905 df-v 3175 df-un 3545 df-if 4037 df-fv 5812 df-afv 39846 |
This theorem is referenced by: afvnufveq 39876 afvfvn0fveq 39879 afv0nbfvbi 39880 afveu 39882 fnbrafvb 39883 afvelrn 39897 afvres 39901 tz6.12-afv 39902 dmfcoafv 39904 afvco2 39905 rlimdmafv 39906 aovfundmoveq 39910 |
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