Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > afvvfveq | Structured version Visualization version GIF version |
Description: The value of the alternative function at a set as argument equals the function's value at this argument. (Contributed by Alexander van der Vekens, 25-May-2017.) |
Ref | Expression |
---|---|
afvvfveq | ⊢ ((𝐹'''𝐴) ∈ 𝐵 → (𝐹'''𝐴) = (𝐹‘𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nvelim 39849 | . . 3 ⊢ ((𝐹'''𝐴) = V → ¬ (𝐹'''𝐴) ∈ 𝐵) | |
2 | 1 | necon2ai 2811 | . 2 ⊢ ((𝐹'''𝐴) ∈ 𝐵 → (𝐹'''𝐴) ≠ V) |
3 | afvnufveq 39876 | . 2 ⊢ ((𝐹'''𝐴) ≠ V → (𝐹'''𝐴) = (𝐹‘𝐴)) | |
4 | 2, 3 | syl 17 | 1 ⊢ ((𝐹'''𝐴) ∈ 𝐵 → (𝐹'''𝐴) = (𝐹‘𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 Vcvv 3173 ‘cfv 5804 '''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-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 |
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-ne 2782 df-rab 2905 df-v 3175 df-un 3545 df-if 4037 df-fv 5812 df-afv 39846 |
This theorem is referenced by: afv0fv0 39878 afv0nbfvbi 39880 aovvoveq 39921 |
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