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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 45396 | . 2 ⊢ ((𝐹''''𝐴) = ∅ → 𝐹 defAt 𝐴) | |
2 | dfatafv2eqfv 45425 | . . . . 5 ⊢ (𝐹 defAt 𝐴 → (𝐹''''𝐴) = (𝐹‘𝐴)) | |
3 | 2 | eqcomd 2742 | . . . 4 ⊢ (𝐹 defAt 𝐴 → (𝐹‘𝐴) = (𝐹''''𝐴)) |
4 | 3 | adantr 481 | . . 3 ⊢ ((𝐹 defAt 𝐴 ∧ (𝐹''''𝐴) = ∅) → (𝐹‘𝐴) = (𝐹''''𝐴)) |
5 | simpr 485 | . . 3 ⊢ ((𝐹 defAt 𝐴 ∧ (𝐹''''𝐴) = ∅) → (𝐹''''𝐴) = ∅) | |
6 | 4, 5 | eqtrd 2776 | . 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 396 = wceq 1541 ∅c0 4280 ‘cfv 6493 defAt wdfat 45280 ''''cafv2 45372 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2707 ax-nul 5261 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-tru 1544 df-fal 1554 df-ex 1782 df-sb 2068 df-clab 2714 df-cleq 2728 df-clel 2814 df-ne 2942 df-rab 3406 df-v 3445 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-nul 4281 df-if 4485 df-pw 4560 df-fv 6501 df-afv2 45373 |
This theorem is referenced by: afv2fv0b 45430 |
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