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Mirrors > Home > MPE Home > Th. List > frnnn0fsupp | Structured version Visualization version GIF version |
Description: A function on ℕ0 is finitely supported iff its support is finite. (Contributed by AV, 8-Jul-2019.) |
Ref | Expression |
---|---|
frnnn0fsupp | ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐹:𝐼⟶ℕ0) → (𝐹 finSupp 0 ↔ (◡𝐹 “ ℕ) ∈ Fin)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | c0ex 10637 | . . . 4 ⊢ 0 ∈ V | |
2 | frnfsuppbi 8864 | . . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ 0 ∈ V) → (𝐹:𝐼⟶ℕ0 → (𝐹 finSupp 0 ↔ (◡𝐹 “ (ℕ0 ∖ {0})) ∈ Fin))) | |
3 | 1, 2 | mpan2 689 | . . 3 ⊢ (𝐼 ∈ 𝑉 → (𝐹:𝐼⟶ℕ0 → (𝐹 finSupp 0 ↔ (◡𝐹 “ (ℕ0 ∖ {0})) ∈ Fin))) |
4 | 3 | imp 409 | . 2 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐹:𝐼⟶ℕ0) → (𝐹 finSupp 0 ↔ (◡𝐹 “ (ℕ0 ∖ {0})) ∈ Fin)) |
5 | dfn2 11913 | . . . . . 6 ⊢ ℕ = (ℕ0 ∖ {0}) | |
6 | 5 | eqcomi 2832 | . . . . 5 ⊢ (ℕ0 ∖ {0}) = ℕ |
7 | 6 | a1i 11 | . . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐹:𝐼⟶ℕ0) → (ℕ0 ∖ {0}) = ℕ) |
8 | 7 | imaeq2d 5931 | . . 3 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐹:𝐼⟶ℕ0) → (◡𝐹 “ (ℕ0 ∖ {0})) = (◡𝐹 “ ℕ)) |
9 | 8 | eleq1d 2899 | . 2 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐹:𝐼⟶ℕ0) → ((◡𝐹 “ (ℕ0 ∖ {0})) ∈ Fin ↔ (◡𝐹 “ ℕ) ∈ Fin)) |
10 | 4, 9 | bitrd 281 | 1 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐹:𝐼⟶ℕ0) → (𝐹 finSupp 0 ↔ (◡𝐹 “ ℕ) ∈ Fin)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 Vcvv 3496 ∖ cdif 3935 {csn 4569 class class class wbr 5068 ◡ccnv 5556 “ cima 5560 ⟶wf 6353 Fincfn 8511 finSupp cfsupp 8835 0cc0 10539 ℕcn 11640 ℕ0cn0 11900 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-supp 7833 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-fsupp 8836 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-nn 11641 df-n0 11901 |
This theorem is referenced by: snifpsrbag 20148 psrbagfsupp 20291 |
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