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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 5202 | . . 3 ⊢ ∅ ∈ V | |
2 | eleq1a 2905 | . . 3 ⊢ (∅ ∈ V → ((𝐹'''𝐴) = ∅ → (𝐹'''𝐴) ∈ V)) | |
3 | 1, 2 | ax-mp 5 | . 2 ⊢ ((𝐹'''𝐴) = ∅ → (𝐹'''𝐴) ∈ V) |
4 | afvvfveq 43224 | . . 3 ⊢ ((𝐹'''𝐴) ∈ V → (𝐹'''𝐴) = (𝐹‘𝐴)) | |
5 | eqeq1 2822 | . . . 4 ⊢ ((𝐹'''𝐴) = (𝐹‘𝐴) → ((𝐹'''𝐴) = ∅ ↔ (𝐹‘𝐴) = ∅)) | |
6 | 5 | biimpd 230 | . . 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 1528 ∈ wcel 2105 Vcvv 3492 ∅c0 4288 ‘cfv 6348 '''cafv 43193 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-fal 1541 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-int 4868 df-br 5058 df-opab 5120 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-res 5560 df-iota 6307 df-fun 6350 df-fv 6356 df-aiota 43162 df-dfat 43195 df-afv 43196 |
This theorem is referenced by: afvfv0bi 43228 aov0ov0 43269 |
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