Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 3283 | . 2 ⊢ ((ℕ0 ↑m 1o) = {𝑓 ∈ (ℕ0 ↑m 1o) ∣ (◡𝑓 “ ℕ) ∈ Fin} ↔ ∀𝑓 ∈ (ℕ0 ↑m 1o)(◡𝑓 “ ℕ) ∈ Fin) | |
2 | df1o2 8136 | . . . 4 ⊢ 1o = {∅} | |
3 | snfi 8635 | . . . 4 ⊢ {∅} ∈ Fin | |
4 | 2, 3 | eqeltri 2829 | . . 3 ⊢ 1o ∈ Fin |
5 | cnvimass 5917 | . . . 4 ⊢ (◡𝑓 “ ℕ) ⊆ dom 𝑓 | |
6 | elmapi 8452 | . . . 4 ⊢ (𝑓 ∈ (ℕ0 ↑m 1o) → 𝑓:1o⟶ℕ0) | |
7 | 5, 6 | fssdm 6518 | . . 3 ⊢ (𝑓 ∈ (ℕ0 ↑m 1o) → (◡𝑓 “ ℕ) ⊆ 1o) |
8 | ssfi 8765 | . . 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 3068 | 1 ⊢ (ℕ0 ↑m 1o) = {𝑓 ∈ (ℕ0 ↑m 1o) ∣ (◡𝑓 “ ℕ) ∈ Fin} |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1542 ∈ wcel 2113 {crab 3057 ⊆ wss 3841 ∅c0 4209 {csn 4513 ◡ccnv 5518 “ cima 5522 (class class class)co 7164 1oc1o 8117 ↑m cmap 8430 Fincfn 8548 ℕcn 11709 ℕ0cn0 11969 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1916 ax-6 1974 ax-7 2019 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2161 ax-12 2178 ax-ext 2710 ax-sep 5164 ax-nul 5171 ax-pow 5229 ax-pr 5293 ax-un 7473 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-ral 3058 df-rex 3059 df-reu 3060 df-rab 3062 df-v 3399 df-sbc 3680 df-csb 3789 df-dif 3844 df-un 3846 df-in 3848 df-ss 3858 df-pss 3860 df-nul 4210 df-if 4412 df-pw 4487 df-sn 4514 df-pr 4516 df-tp 4518 df-op 4520 df-uni 4794 df-iun 4880 df-br 5028 df-opab 5090 df-mpt 5108 df-tr 5134 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 6169 df-on 6170 df-lim 6171 df-suc 6172 df-iota 6291 df-fun 6335 df-fn 6336 df-f 6337 df-f1 6338 df-fo 6339 df-f1o 6340 df-fv 6341 df-ov 7167 df-oprab 7168 df-mpo 7169 df-om 7594 df-1st 7707 df-2nd 7708 df-1o 8124 df-map 8432 df-en 8549 df-fin 8552 |
This theorem is referenced by: psr1bas 20959 ply1basf 20970 ply1plusgfvi 21010 coe1z 21031 coe1mul2 21037 coe1tm 21041 ply1coe 21064 deg1ldg 24837 deg1leb 24840 deg1val 24841 |
Copyright terms: Public domain | W3C validator |