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Mirrors > Home > MPE Home > Th. List > psr1baslem | Structured version Visualization version GIF version |
Description: The set of finite bags on 1o is just the set of all functions from 1o to ℕ0. (Contributed by Mario Carneiro, 9-Feb-2015.) |
Ref | Expression |
---|---|
psr1baslem | ⊢ (ℕ0 ↑m 1o) = {𝑓 ∈ (ℕ0 ↑m 1o) ∣ (◡𝑓 “ ℕ) ∈ Fin} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rabid2 3334 | . 2 ⊢ ((ℕ0 ↑m 1o) = {𝑓 ∈ (ℕ0 ↑m 1o) ∣ (◡𝑓 “ ℕ) ∈ Fin} ↔ ∀𝑓 ∈ (ℕ0 ↑m 1o)(◡𝑓 “ ℕ) ∈ Fin) | |
2 | df1o2 8099 | . . . 4 ⊢ 1o = {∅} | |
3 | snfi 8577 | . . . 4 ⊢ {∅} ∈ Fin | |
4 | 2, 3 | eqeltri 2886 | . . 3 ⊢ 1o ∈ Fin |
5 | cnvimass 5916 | . . . 4 ⊢ (◡𝑓 “ ℕ) ⊆ dom 𝑓 | |
6 | elmapi 8411 | . . . 4 ⊢ (𝑓 ∈ (ℕ0 ↑m 1o) → 𝑓:1o⟶ℕ0) | |
7 | 5, 6 | fssdm 6504 | . . 3 ⊢ (𝑓 ∈ (ℕ0 ↑m 1o) → (◡𝑓 “ ℕ) ⊆ 1o) |
8 | ssfi 8722 | . . 3 ⊢ ((1o ∈ Fin ∧ (◡𝑓 “ ℕ) ⊆ 1o) → (◡𝑓 “ ℕ) ∈ Fin) | |
9 | 4, 7, 8 | sylancr 590 | . 2 ⊢ (𝑓 ∈ (ℕ0 ↑m 1o) → (◡𝑓 “ ℕ) ∈ Fin) |
10 | 1, 9 | mprgbir 3121 | 1 ⊢ (ℕ0 ↑m 1o) = {𝑓 ∈ (ℕ0 ↑m 1o) ∣ (◡𝑓 “ ℕ) ∈ Fin} |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1538 ∈ wcel 2111 {crab 3110 ⊆ wss 3881 ∅c0 4243 {csn 4525 ◡ccnv 5518 “ cima 5522 (class class class)co 7135 1oc1o 8078 ↑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 |
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-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-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-1st 7671 df-2nd 7672 df-1o 8085 df-er 8272 df-map 8391 df-en 8493 df-fin 8496 |
This theorem is referenced by: psr1bas 20820 ply1basf 20831 ply1plusgfvi 20871 coe1z 20892 coe1mul2 20898 coe1tm 20902 ply1coe 20925 deg1ldg 24693 deg1leb 24696 deg1val 24697 |
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