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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 5214 | . . 3 ⊢ ∅ ∈ V | |
2 | eleq1a 2911 | . . 3 ⊢ (∅ ∈ V → ((𝐹'''𝐴) = ∅ → (𝐹'''𝐴) ∈ V)) | |
3 | 1, 2 | ax-mp 5 | . 2 ⊢ ((𝐹'''𝐴) = ∅ → (𝐹'''𝐴) ∈ V) |
4 | afvvfveq 43354 | . . 3 ⊢ ((𝐹'''𝐴) ∈ V → (𝐹'''𝐴) = (𝐹‘𝐴)) | |
5 | eqeq1 2828 | . . . 4 ⊢ ((𝐹'''𝐴) = (𝐹‘𝐴) → ((𝐹'''𝐴) = ∅ ↔ (𝐹‘𝐴) = ∅)) | |
6 | 5 | biimpd 231 | . . 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 1536 ∈ wcel 2113 Vcvv 3497 ∅c0 4294 ‘cfv 6358 '''cafv 43323 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1539 df-fal 1549 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-ral 3146 df-rex 3147 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-if 4471 df-sn 4571 df-pr 4573 df-op 4577 df-uni 4842 df-int 4880 df-br 5070 df-opab 5132 df-id 5463 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-res 5570 df-iota 6317 df-fun 6360 df-fv 6366 df-aiota 43292 df-dfat 43325 df-afv 43326 |
This theorem is referenced by: afvfv0bi 43358 aov0ov0 43399 |
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