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Mirrors > Home > MPE Home > Th. List > psrbag | Structured version Visualization version GIF version |
Description: Elementhood in the set of finite bags. (Contributed by Mario Carneiro, 29-Dec-2014.) |
Ref | Expression |
---|---|
psrbag.d | ⊢ 𝐷 = {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} |
Ref | Expression |
---|---|
psrbag | ⊢ (𝐼 ∈ 𝑉 → (𝐹 ∈ 𝐷 ↔ (𝐹:𝐼⟶ℕ0 ∧ (◡𝐹 “ ℕ) ∈ Fin))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cnveq 5708 | . . . . 5 ⊢ (𝑓 = 𝐹 → ◡𝑓 = ◡𝐹) | |
2 | 1 | imaeq1d 5895 | . . . 4 ⊢ (𝑓 = 𝐹 → (◡𝑓 “ ℕ) = (◡𝐹 “ ℕ)) |
3 | 2 | eleq1d 2874 | . . 3 ⊢ (𝑓 = 𝐹 → ((◡𝑓 “ ℕ) ∈ Fin ↔ (◡𝐹 “ ℕ) ∈ Fin)) |
4 | psrbag.d | . . 3 ⊢ 𝐷 = {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} | |
5 | 3, 4 | elrab2 3631 | . 2 ⊢ (𝐹 ∈ 𝐷 ↔ (𝐹 ∈ (ℕ0 ↑m 𝐼) ∧ (◡𝐹 “ ℕ) ∈ Fin)) |
6 | nn0ex 11891 | . . . 4 ⊢ ℕ0 ∈ V | |
7 | elmapg 8402 | . . . 4 ⊢ ((ℕ0 ∈ V ∧ 𝐼 ∈ 𝑉) → (𝐹 ∈ (ℕ0 ↑m 𝐼) ↔ 𝐹:𝐼⟶ℕ0)) | |
8 | 6, 7 | mpan 689 | . . 3 ⊢ (𝐼 ∈ 𝑉 → (𝐹 ∈ (ℕ0 ↑m 𝐼) ↔ 𝐹:𝐼⟶ℕ0)) |
9 | 8 | anbi1d 632 | . 2 ⊢ (𝐼 ∈ 𝑉 → ((𝐹 ∈ (ℕ0 ↑m 𝐼) ∧ (◡𝐹 “ ℕ) ∈ Fin) ↔ (𝐹:𝐼⟶ℕ0 ∧ (◡𝐹 “ ℕ) ∈ Fin))) |
10 | 5, 9 | syl5bb 286 | 1 ⊢ (𝐼 ∈ 𝑉 → (𝐹 ∈ 𝐷 ↔ (𝐹:𝐼⟶ℕ0 ∧ (◡𝐹 “ ℕ) ∈ Fin))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1538 ∈ wcel 2111 {crab 3110 Vcvv 3441 ◡ccnv 5518 “ cima 5522 ⟶wf 6320 (class class class)co 7135 ↑m cmap 8389 Fincfn 8492 ℕcn 11625 ℕ0cn0 11885 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-1cn 10584 ax-addcl 10586 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-map 8391 df-nn 11626 df-n0 11886 |
This theorem is referenced by: psrbagf 20603 snifpsrbag 20604 psrbaglecl 20607 psrbagaddcl 20608 psrbagcon 20609 psrbaglefi 20610 mplcoe5lem 20707 mplcoe5 20708 mplbas2 20710 psrbag0 20733 psrbagsn 20734 psrbagfsupp 20748 evlslem3 20752 |
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