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Mirrors > Home > MPE Home > Th. List > psrbagfsupp | Structured version Visualization version GIF version |
Description: Finite bags have finite support. (Contributed by Stefan O'Rear, 9-Mar-2015.) (Revised by AV, 18-Jul-2019.) Remove a sethood antecedent. (Revised by SN, 7-Aug-2024.) |
Ref | Expression |
---|---|
psrbag.d | ⊢ 𝐷 = {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} |
Ref | Expression |
---|---|
psrbagfsupp | ⊢ (𝐹 ∈ 𝐷 → 𝐹 finSupp 0) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | id 22 | . . . . 5 ⊢ (𝐹 ∈ 𝐷 → 𝐹 ∈ 𝐷) | |
2 | psrbag.d | . . . . . . 7 ⊢ 𝐷 = {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} | |
3 | 2 | psrbagf 20849 | . . . . . 6 ⊢ (𝐹 ∈ 𝐷 → 𝐹:𝐼⟶ℕ0) |
4 | 3 | ffnd 6535 | . . . . 5 ⊢ (𝐹 ∈ 𝐷 → 𝐹 Fn 𝐼) |
5 | 1, 4 | fndmexd 7673 | . . . 4 ⊢ (𝐹 ∈ 𝐷 → 𝐼 ∈ V) |
6 | 2 | psrbag 20848 | . . . . 5 ⊢ (𝐼 ∈ V → (𝐹 ∈ 𝐷 ↔ (𝐹:𝐼⟶ℕ0 ∧ (◡𝐹 “ ℕ) ∈ Fin))) |
7 | 6 | biimpa 480 | . . . 4 ⊢ ((𝐼 ∈ V ∧ 𝐹 ∈ 𝐷) → (𝐹:𝐼⟶ℕ0 ∧ (◡𝐹 “ ℕ) ∈ Fin)) |
8 | 5, 7 | mpancom 688 | . . 3 ⊢ (𝐹 ∈ 𝐷 → (𝐹:𝐼⟶ℕ0 ∧ (◡𝐹 “ ℕ) ∈ Fin)) |
9 | 8 | simprd 499 | . 2 ⊢ (𝐹 ∈ 𝐷 → (◡𝐹 “ ℕ) ∈ Fin) |
10 | frnnn0fsuppg 12132 | . . 3 ⊢ ((𝐹 ∈ 𝐷 ∧ 𝐹:𝐼⟶ℕ0) → (𝐹 finSupp 0 ↔ (◡𝐹 “ ℕ) ∈ Fin)) | |
11 | 3, 10 | mpdan 687 | . 2 ⊢ (𝐹 ∈ 𝐷 → (𝐹 finSupp 0 ↔ (◡𝐹 “ ℕ) ∈ Fin)) |
12 | 9, 11 | mpbird 260 | 1 ⊢ (𝐹 ∈ 𝐷 → 𝐹 finSupp 0) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1543 ∈ wcel 2110 {crab 3058 Vcvv 3401 class class class wbr 5043 ◡ccnv 5539 “ cima 5543 ⟶wf 6365 (class class class)co 7202 ↑m cmap 8497 Fincfn 8615 finSupp cfsupp 8974 0cc0 10712 ℕcn 11813 ℕ0cn0 12073 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2706 ax-sep 5181 ax-nul 5188 ax-pow 5247 ax-pr 5311 ax-un 7512 ax-cnex 10768 ax-resscn 10769 ax-1cn 10770 ax-icn 10771 ax-addcl 10772 ax-addrcl 10773 ax-mulcl 10774 ax-mulrcl 10775 ax-mulcom 10776 ax-addass 10777 ax-mulass 10778 ax-distr 10779 ax-i2m1 10780 ax-1ne0 10781 ax-1rid 10782 ax-rnegex 10783 ax-rrecex 10784 ax-cnre 10785 ax-pre-lttri 10786 ax-pre-lttrn 10787 ax-pre-ltadd 10788 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2537 df-eu 2566 df-clab 2713 df-cleq 2726 df-clel 2812 df-nfc 2882 df-ne 2936 df-nel 3040 df-ral 3059 df-rex 3060 df-reu 3061 df-rab 3063 df-v 3403 df-sbc 3688 df-csb 3803 df-dif 3860 df-un 3862 df-in 3864 df-ss 3874 df-pss 3876 df-nul 4228 df-if 4430 df-pw 4505 df-sn 4532 df-pr 4534 df-tp 4536 df-op 4538 df-uni 4810 df-iun 4896 df-br 5044 df-opab 5106 df-mpt 5125 df-tr 5151 df-id 5444 df-eprel 5449 df-po 5457 df-so 5458 df-fr 5498 df-we 5500 df-xp 5546 df-rel 5547 df-cnv 5548 df-co 5549 df-dm 5550 df-rn 5551 df-res 5552 df-ima 5553 df-pred 6149 df-ord 6205 df-on 6206 df-lim 6207 df-suc 6208 df-iota 6327 df-fun 6371 df-fn 6372 df-f 6373 df-f1 6374 df-fo 6375 df-f1o 6376 df-fv 6377 df-ov 7205 df-oprab 7206 df-mpo 7207 df-om 7634 df-1st 7750 df-2nd 7751 df-supp 7893 df-wrecs 8036 df-recs 8097 df-rdg 8135 df-er 8380 df-map 8499 df-en 8616 df-dom 8617 df-sdom 8618 df-fsupp 8975 df-pnf 10852 df-mnf 10853 df-xr 10854 df-ltxr 10855 df-le 10856 df-nn 11814 df-n0 12074 |
This theorem is referenced by: psrbagaddcl 20859 psrbagev1 21007 mhpmulcl 21061 tdeglem1 24925 tdeglem3 24927 tdeglem4 24929 evlsbagval 39937 mhphflem 39946 mhphf 39947 |
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