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Mirrors > Home > MPE Home > Th. List > psrbaglecl | Structured version Visualization version GIF version |
Description: The set of finite bags is downward-closed. (Contributed by Mario Carneiro, 29-Dec-2014.) |
Ref | Expression |
---|---|
psrbag.d | ⊢ 𝐷 = {𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} |
Ref | Expression |
---|---|
psrbaglecl | ⊢ ((𝐼 ∈ 𝑉 ∧ (𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘𝑟 ≤ 𝐹)) → 𝐺 ∈ 𝐷) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpr2 1175 | . 2 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘𝑟 ≤ 𝐹)) → 𝐺:𝐼⟶ℕ0) | |
2 | simpr1 1174 | . . . . 5 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘𝑟 ≤ 𝐹)) → 𝐹 ∈ 𝐷) | |
3 | psrbag.d | . . . . . . 7 ⊢ 𝐷 = {𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} | |
4 | 3 | psrbag 19858 | . . . . . 6 ⊢ (𝐼 ∈ 𝑉 → (𝐹 ∈ 𝐷 ↔ (𝐹:𝐼⟶ℕ0 ∧ (◡𝐹 “ ℕ) ∈ Fin))) |
5 | 4 | adantr 473 | . . . . 5 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘𝑟 ≤ 𝐹)) → (𝐹 ∈ 𝐷 ↔ (𝐹:𝐼⟶ℕ0 ∧ (◡𝐹 “ ℕ) ∈ Fin))) |
6 | 2, 5 | mpbid 224 | . . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘𝑟 ≤ 𝐹)) → (𝐹:𝐼⟶ℕ0 ∧ (◡𝐹 “ ℕ) ∈ Fin)) |
7 | 6 | simprd 488 | . . 3 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘𝑟 ≤ 𝐹)) → (◡𝐹 “ ℕ) ∈ Fin) |
8 | 3 | psrbaglesupp 19862 | . . 3 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘𝑟 ≤ 𝐹)) → (◡𝐺 “ ℕ) ⊆ (◡𝐹 “ ℕ)) |
9 | 7, 8 | ssfid 8536 | . 2 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘𝑟 ≤ 𝐹)) → (◡𝐺 “ ℕ) ∈ Fin) |
10 | 3 | psrbag 19858 | . . 3 ⊢ (𝐼 ∈ 𝑉 → (𝐺 ∈ 𝐷 ↔ (𝐺:𝐼⟶ℕ0 ∧ (◡𝐺 “ ℕ) ∈ Fin))) |
11 | 10 | adantr 473 | . 2 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘𝑟 ≤ 𝐹)) → (𝐺 ∈ 𝐷 ↔ (𝐺:𝐼⟶ℕ0 ∧ (◡𝐺 “ ℕ) ∈ Fin))) |
12 | 1, 9, 11 | mpbir2and 700 | 1 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘𝑟 ≤ 𝐹)) → 𝐺 ∈ 𝐷) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 387 ∧ w3a 1068 = wceq 1507 ∈ wcel 2050 {crab 3093 class class class wbr 4929 ◡ccnv 5406 “ cima 5410 ⟶wf 6184 (class class class)co 6976 ∘𝑟 cofr 7226 ↑𝑚 cmap 8206 Fincfn 8306 ≤ cle 10475 ℕcn 11439 ℕ0cn0 11707 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2751 ax-rep 5049 ax-sep 5060 ax-nul 5067 ax-pow 5119 ax-pr 5186 ax-un 7279 ax-cnex 10391 ax-resscn 10392 ax-1cn 10393 ax-icn 10394 ax-addcl 10395 ax-addrcl 10396 ax-mulcl 10397 ax-mulrcl 10398 ax-mulcom 10399 ax-addass 10400 ax-mulass 10401 ax-distr 10402 ax-i2m1 10403 ax-1ne0 10404 ax-1rid 10405 ax-rnegex 10406 ax-rrecex 10407 ax-cnre 10408 ax-pre-lttri 10409 ax-pre-lttrn 10410 ax-pre-ltadd 10411 ax-pre-mulgt0 10412 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2584 df-clab 2760 df-cleq 2772 df-clel 2847 df-nfc 2919 df-ne 2969 df-nel 3075 df-ral 3094 df-rex 3095 df-reu 3096 df-rab 3098 df-v 3418 df-sbc 3683 df-csb 3788 df-dif 3833 df-un 3835 df-in 3837 df-ss 3844 df-pss 3846 df-nul 4180 df-if 4351 df-pw 4424 df-sn 4442 df-pr 4444 df-tp 4446 df-op 4448 df-uni 4713 df-iun 4794 df-br 4930 df-opab 4992 df-mpt 5009 df-tr 5031 df-id 5312 df-eprel 5317 df-po 5326 df-so 5327 df-fr 5366 df-we 5368 df-xp 5413 df-rel 5414 df-cnv 5415 df-co 5416 df-dm 5417 df-rn 5418 df-res 5419 df-ima 5420 df-pred 5986 df-ord 6032 df-on 6033 df-lim 6034 df-suc 6035 df-iota 6152 df-fun 6190 df-fn 6191 df-f 6192 df-f1 6193 df-fo 6194 df-f1o 6195 df-fv 6196 df-riota 6937 df-ov 6979 df-oprab 6980 df-mpo 6981 df-ofr 7228 df-om 7397 df-supp 7634 df-wrecs 7750 df-recs 7812 df-rdg 7850 df-er 8089 df-map 8208 df-en 8307 df-dom 8308 df-sdom 8309 df-fin 8310 df-pnf 10476 df-mnf 10477 df-xr 10478 df-ltxr 10479 df-le 10480 df-sub 10672 df-neg 10673 df-nn 11440 df-n0 11708 |
This theorem is referenced by: psrbaglefi 19866 |
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