Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > afv0fv0 | Structured version Visualization version GIF version |
Description: If the value of the alternative function at an argument is the empty set, the function's value at this argument is the empty set. (Contributed by Alexander van der Vekens, 25-May-2017.) |
Ref | Expression |
---|---|
afv0fv0 | ⊢ ((𝐹'''𝐴) = ∅ → (𝐹‘𝐴) = ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0ex 4790 | . . 3 ⊢ ∅ ∈ V | |
2 | eleq1a 2696 | . . 3 ⊢ (∅ ∈ V → ((𝐹'''𝐴) = ∅ → (𝐹'''𝐴) ∈ V)) | |
3 | 1, 2 | ax-mp 5 | . 2 ⊢ ((𝐹'''𝐴) = ∅ → (𝐹'''𝐴) ∈ V) |
4 | afvvfveq 41228 | . . 3 ⊢ ((𝐹'''𝐴) ∈ V → (𝐹'''𝐴) = (𝐹‘𝐴)) | |
5 | eqeq1 2626 | . . . 4 ⊢ ((𝐹'''𝐴) = (𝐹‘𝐴) → ((𝐹'''𝐴) = ∅ ↔ (𝐹‘𝐴) = ∅)) | |
6 | 5 | biimpd 219 | . . 3 ⊢ ((𝐹'''𝐴) = (𝐹‘𝐴) → ((𝐹'''𝐴) = ∅ → (𝐹‘𝐴) = ∅)) |
7 | 4, 6 | syl 17 | . 2 ⊢ ((𝐹'''𝐴) ∈ V → ((𝐹'''𝐴) = ∅ → (𝐹‘𝐴) = ∅)) |
8 | 3, 7 | mpcom 38 | 1 ⊢ ((𝐹'''𝐴) = ∅ → (𝐹‘𝐴) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1483 ∈ wcel 1990 Vcvv 3200 ∅c0 3915 ‘cfv 5888 '''cafv 41194 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-nul 3916 df-if 4087 df-fv 5896 df-afv 41197 |
This theorem is referenced by: afvfv0bi 41232 aov0ov0 41273 |
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