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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 5328 | . . 3 ⊢ ∅ ∈ V | |
2 | eleq1a 2833 | . . 3 ⊢ (∅ ∈ V → ((𝐹'''𝐴) = ∅ → (𝐹'''𝐴) ∈ V)) | |
3 | 1, 2 | ax-mp 5 | . 2 ⊢ ((𝐹'''𝐴) = ∅ → (𝐹'''𝐴) ∈ V) |
4 | afvvfveq 46996 | . . 3 ⊢ ((𝐹'''𝐴) ∈ V → (𝐹'''𝐴) = (𝐹‘𝐴)) | |
5 | eqeq1 2738 | . . . 4 ⊢ ((𝐹'''𝐴) = (𝐹‘𝐴) → ((𝐹'''𝐴) = ∅ ↔ (𝐹‘𝐴) = ∅)) | |
6 | 5 | biimpd 229 | . . 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 1537 ∈ wcel 2103 Vcvv 3482 ∅c0 4347 ‘cfv 6572 '''cafv 46965 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2105 ax-9 2113 ax-10 2136 ax-11 2153 ax-12 2173 ax-ext 2705 ax-sep 5320 ax-nul 5327 ax-pr 5450 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2890 df-ne 2943 df-ral 3064 df-rex 3073 df-rab 3439 df-v 3484 df-sbc 3799 df-csb 3916 df-dif 3973 df-un 3975 df-in 3977 df-ss 3987 df-nul 4348 df-if 4549 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-int 4973 df-br 5170 df-opab 5232 df-id 5597 df-xp 5705 df-rel 5706 df-cnv 5707 df-co 5708 df-dm 5709 df-res 5711 df-iota 6524 df-fun 6574 df-fv 6580 df-aiota 46933 df-dfat 46967 df-afv 46968 |
This theorem is referenced by: afvfv0bi 47000 aov0ov0 47041 |
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