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 43424 | . 2 ⊢ ((𝐹''''𝐴) = ∅ → 𝐹 defAt 𝐴) | |
2 | dfatafv2eqfv 43453 | . . . . 5 ⊢ (𝐹 defAt 𝐴 → (𝐹''''𝐴) = (𝐹‘𝐴)) | |
3 | 2 | eqcomd 2827 | . . . 4 ⊢ (𝐹 defAt 𝐴 → (𝐹‘𝐴) = (𝐹''''𝐴)) |
4 | 3 | adantr 483 | . . 3 ⊢ ((𝐹 defAt 𝐴 ∧ (𝐹''''𝐴) = ∅) → (𝐹‘𝐴) = (𝐹''''𝐴)) |
5 | simpr 487 | . . 3 ⊢ ((𝐹 defAt 𝐴 ∧ (𝐹''''𝐴) = ∅) → (𝐹''''𝐴) = ∅) | |
6 | 4, 5 | eqtrd 2856 | . 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 398 = wceq 1533 ∅c0 4291 ‘cfv 6350 defAt wdfat 43308 ''''cafv2 43400 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-nul 5203 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-rab 3147 df-v 3497 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-fv 6358 df-afv2 43401 |
This theorem is referenced by: afv2fv0b 43458 |
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