![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > psrbagfsupp | Structured version Visualization version GIF version |
Description: Finite bags have finite nonzero-support. (Contributed by Stefan O'Rear, 9-Mar-2015.) (Revised by AV, 18-Jul-2019.) |
Ref | Expression |
---|---|
psrbagfsupp.d | ⊢ 𝐷 = {ℎ ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} |
Ref | Expression |
---|---|
psrbagfsupp | ⊢ ((𝑋 ∈ 𝐷 ∧ 𝐼 ∈ 𝑉) → 𝑋 finSupp 0) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | psrbagfsupp.d | . . . . 5 ⊢ 𝐷 = {ℎ ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} | |
2 | 1 | psrbag 19725 | . . . 4 ⊢ (𝐼 ∈ 𝑉 → (𝑋 ∈ 𝐷 ↔ (𝑋:𝐼⟶ℕ0 ∧ (◡𝑋 “ ℕ) ∈ Fin))) |
3 | 2 | biimpac 472 | . . 3 ⊢ ((𝑋 ∈ 𝐷 ∧ 𝐼 ∈ 𝑉) → (𝑋:𝐼⟶ℕ0 ∧ (◡𝑋 “ ℕ) ∈ Fin)) |
4 | 3 | simprd 491 | . 2 ⊢ ((𝑋 ∈ 𝐷 ∧ 𝐼 ∈ 𝑉) → (◡𝑋 “ ℕ) ∈ Fin) |
5 | simpr 479 | . . 3 ⊢ ((𝑋 ∈ 𝐷 ∧ 𝐼 ∈ 𝑉) → 𝐼 ∈ 𝑉) | |
6 | 1 | psrbagf 19726 | . . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑋 ∈ 𝐷) → 𝑋:𝐼⟶ℕ0) |
7 | 6 | ancoms 452 | . . 3 ⊢ ((𝑋 ∈ 𝐷 ∧ 𝐼 ∈ 𝑉) → 𝑋:𝐼⟶ℕ0) |
8 | frnnn0fsupp 11677 | . . 3 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑋:𝐼⟶ℕ0) → (𝑋 finSupp 0 ↔ (◡𝑋 “ ℕ) ∈ Fin)) | |
9 | 5, 7, 8 | syl2anc 581 | . 2 ⊢ ((𝑋 ∈ 𝐷 ∧ 𝐼 ∈ 𝑉) → (𝑋 finSupp 0 ↔ (◡𝑋 “ ℕ) ∈ Fin)) |
10 | 4, 9 | mpbird 249 | 1 ⊢ ((𝑋 ∈ 𝐷 ∧ 𝐼 ∈ 𝑉) → 𝑋 finSupp 0) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 386 = wceq 1658 ∈ wcel 2166 {crab 3121 class class class wbr 4873 ◡ccnv 5341 “ cima 5345 ⟶wf 6119 (class class class)co 6905 ↑𝑚 cmap 8122 Fincfn 8222 finSupp cfsupp 8544 0cc0 10252 ℕcn 11350 ℕ0cn0 11618 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2803 ax-rep 4994 ax-sep 5005 ax-nul 5013 ax-pow 5065 ax-pr 5127 ax-un 7209 ax-cnex 10308 ax-resscn 10309 ax-1cn 10310 ax-icn 10311 ax-addcl 10312 ax-addrcl 10313 ax-mulcl 10314 ax-mulrcl 10315 ax-mulcom 10316 ax-addass 10317 ax-mulass 10318 ax-distr 10319 ax-i2m1 10320 ax-1ne0 10321 ax-1rid 10322 ax-rnegex 10323 ax-rrecex 10324 ax-cnre 10325 ax-pre-lttri 10326 ax-pre-lttrn 10327 ax-pre-ltadd 10328 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3or 1114 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-ral 3122 df-rex 3123 df-reu 3124 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-pss 3814 df-nul 4145 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-tp 4402 df-op 4404 df-uni 4659 df-iun 4742 df-br 4874 df-opab 4936 df-mpt 4953 df-tr 4976 df-id 5250 df-eprel 5255 df-po 5263 df-so 5264 df-fr 5301 df-we 5303 df-xp 5348 df-rel 5349 df-cnv 5350 df-co 5351 df-dm 5352 df-rn 5353 df-res 5354 df-ima 5355 df-pred 5920 df-ord 5966 df-on 5967 df-lim 5968 df-suc 5969 df-iota 6086 df-fun 6125 df-fn 6126 df-f 6127 df-f1 6128 df-fo 6129 df-f1o 6130 df-fv 6131 df-ov 6908 df-oprab 6909 df-mpt2 6910 df-om 7327 df-supp 7560 df-wrecs 7672 df-recs 7734 df-rdg 7772 df-er 8009 df-map 8124 df-en 8223 df-dom 8224 df-sdom 8225 df-fsupp 8545 df-pnf 10393 df-mnf 10394 df-xr 10395 df-ltxr 10396 df-le 10397 df-nn 11351 df-n0 11619 |
This theorem is referenced by: psrbagev1 19870 tdeglem1 24217 tdeglem3 24218 tdeglem4 24219 |
Copyright terms: Public domain | W3C validator |