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Mirrors > Home > MPE Home > Th. List > 0fsupp | Structured version Visualization version GIF version |
Description: The empty set is a finitely supported function. (Contributed by AV, 19-Jul-2019.) |
Ref | Expression |
---|---|
0fsupp | ⊢ (𝑍 ∈ 𝑉 → ∅ finSupp 𝑍) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | supp0 7834 | . . 3 ⊢ (𝑍 ∈ 𝑉 → (∅ supp 𝑍) = ∅) | |
2 | 0fin 8745 | . . 3 ⊢ ∅ ∈ Fin | |
3 | 1, 2 | eqeltrdi 2921 | . 2 ⊢ (𝑍 ∈ 𝑉 → (∅ supp 𝑍) ∈ Fin) |
4 | fun0 6418 | . . 3 ⊢ Fun ∅ | |
5 | 0ex 5210 | . . 3 ⊢ ∅ ∈ V | |
6 | funisfsupp 8837 | . . 3 ⊢ ((Fun ∅ ∧ ∅ ∈ V ∧ 𝑍 ∈ 𝑉) → (∅ finSupp 𝑍 ↔ (∅ supp 𝑍) ∈ Fin)) | |
7 | 4, 5, 6 | mp3an12 1447 | . 2 ⊢ (𝑍 ∈ 𝑉 → (∅ finSupp 𝑍 ↔ (∅ supp 𝑍) ∈ Fin)) |
8 | 3, 7 | mpbird 259 | 1 ⊢ (𝑍 ∈ 𝑉 → ∅ finSupp 𝑍) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∈ wcel 2110 Vcvv 3494 ∅c0 4290 class class class wbr 5065 Fun wfun 6348 (class class class)co 7155 supp csupp 7829 Fincfn 8508 finSupp cfsupp 8832 |
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 2157 ax-12 2173 ax-ext 2793 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3772 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4838 df-br 5066 df-opab 5128 df-tr 5172 df-id 5459 df-eprel 5464 df-po 5473 df-so 5474 df-fr 5513 df-we 5515 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-ord 6193 df-on 6194 df-lim 6195 df-suc 6196 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-ov 7158 df-oprab 7159 df-mpo 7160 df-om 7580 df-supp 7830 df-en 8509 df-fin 8512 df-fsupp 8833 |
This theorem is referenced by: lco0 44481 |
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