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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 42266 | . 2 ⊢ ((𝐹''''𝐴) = ∅ → 𝐹 defAt 𝐴) | |
2 | dfatafv2eqfv 42295 | . . . . 5 ⊢ (𝐹 defAt 𝐴 → (𝐹''''𝐴) = (𝐹‘𝐴)) | |
3 | 2 | eqcomd 2783 | . . . 4 ⊢ (𝐹 defAt 𝐴 → (𝐹‘𝐴) = (𝐹''''𝐴)) |
4 | 3 | adantr 474 | . . 3 ⊢ ((𝐹 defAt 𝐴 ∧ (𝐹''''𝐴) = ∅) → (𝐹‘𝐴) = (𝐹''''𝐴)) |
5 | simpr 479 | . . 3 ⊢ ((𝐹 defAt 𝐴 ∧ (𝐹''''𝐴) = ∅) → (𝐹''''𝐴) = ∅) | |
6 | 4, 5 | eqtrd 2813 | . 2 ⊢ ((𝐹 defAt 𝐴 ∧ (𝐹''''𝐴) = ∅) → (𝐹‘𝐴) = ∅) |
7 | 1, 6 | mpancom 678 | 1 ⊢ ((𝐹''''𝐴) = ∅ → (𝐹‘𝐴) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 = wceq 1601 ∅c0 4140 ‘cfv 6135 defAt wdfat 42150 ''''cafv2 42242 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2054 ax-9 2115 ax-10 2134 ax-11 2149 ax-12 2162 ax-13 2333 ax-ext 2753 ax-nul 5025 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-clab 2763 df-cleq 2769 df-clel 2773 df-nfc 2920 df-ne 2969 df-rab 3098 df-v 3399 df-dif 3794 df-un 3796 df-in 3798 df-ss 3805 df-nul 4141 df-if 4307 df-pw 4380 df-fv 6143 df-afv2 42243 |
This theorem is referenced by: afv2fv0b 42300 |
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