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Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > afv20fv0 | Structured version Visualization version GIF version |
Description: If the alternate function value at an argument is the empty set, the function's value at this argument is the empty set. (Contributed by AV, 3-Sep-2022.) |
Ref | Expression |
---|---|
afv20fv0 | ⊢ ((𝐹''''𝐴) = ∅ → (𝐹‘𝐴) = ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | afv20defat 46845 | . 2 ⊢ ((𝐹''''𝐴) = ∅ → 𝐹 defAt 𝐴) | |
2 | dfatafv2eqfv 46874 | . . . . 5 ⊢ (𝐹 defAt 𝐴 → (𝐹''''𝐴) = (𝐹‘𝐴)) | |
3 | 2 | eqcomd 2732 | . . . 4 ⊢ (𝐹 defAt 𝐴 → (𝐹‘𝐴) = (𝐹''''𝐴)) |
4 | 3 | adantr 479 | . . 3 ⊢ ((𝐹 defAt 𝐴 ∧ (𝐹''''𝐴) = ∅) → (𝐹‘𝐴) = (𝐹''''𝐴)) |
5 | simpr 483 | . . 3 ⊢ ((𝐹 defAt 𝐴 ∧ (𝐹''''𝐴) = ∅) → (𝐹''''𝐴) = ∅) | |
6 | 4, 5 | eqtrd 2766 | . 2 ⊢ ((𝐹 defAt 𝐴 ∧ (𝐹''''𝐴) = ∅) → (𝐹‘𝐴) = ∅) |
7 | 1, 6 | mpancom 686 | 1 ⊢ ((𝐹''''𝐴) = ∅ → (𝐹‘𝐴) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1534 ∅c0 4325 ‘cfv 6554 defAt wdfat 46729 ''''cafv2 46821 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-ext 2697 ax-nul 5311 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-tru 1537 df-fal 1547 df-ex 1775 df-sb 2061 df-clab 2704 df-cleq 2718 df-clel 2803 df-ne 2931 df-rab 3420 df-v 3464 df-dif 3950 df-un 3952 df-ss 3964 df-nul 4326 df-if 4534 df-pw 4609 df-fv 6562 df-afv2 46822 |
This theorem is referenced by: afv2fv0b 46879 |
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