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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 3118 | . 2 ⊢ ((ℕ0 ↑𝑚 1𝑜) = {𝑓 ∈ (ℕ0 ↑𝑚 1𝑜) ∣ (◡𝑓 “ ℕ) ∈ Fin} ↔ ∀𝑓 ∈ (ℕ0 ↑𝑚 1𝑜)(◡𝑓 “ ℕ) ∈ Fin) | |
2 | df1o2 7572 | . . . 4 ⊢ 1𝑜 = {∅} | |
3 | snfi 8038 | . . . 4 ⊢ {∅} ∈ Fin | |
4 | 2, 3 | eqeltri 2697 | . . 3 ⊢ 1𝑜 ∈ Fin |
5 | cnvimass 5485 | . . . 4 ⊢ (◡𝑓 “ ℕ) ⊆ dom 𝑓 | |
6 | elmapi 7879 | . . . . 5 ⊢ (𝑓 ∈ (ℕ0 ↑𝑚 1𝑜) → 𝑓:1𝑜⟶ℕ0) | |
7 | fdm 6051 | . . . . 5 ⊢ (𝑓:1𝑜⟶ℕ0 → dom 𝑓 = 1𝑜) | |
8 | 6, 7 | syl 17 | . . . 4 ⊢ (𝑓 ∈ (ℕ0 ↑𝑚 1𝑜) → dom 𝑓 = 1𝑜) |
9 | 5, 8 | syl5sseq 3653 | . . 3 ⊢ (𝑓 ∈ (ℕ0 ↑𝑚 1𝑜) → (◡𝑓 “ ℕ) ⊆ 1𝑜) |
10 | ssfi 8180 | . . 3 ⊢ ((1𝑜 ∈ Fin ∧ (◡𝑓 “ ℕ) ⊆ 1𝑜) → (◡𝑓 “ ℕ) ∈ Fin) | |
11 | 4, 9, 10 | sylancr 695 | . 2 ⊢ (𝑓 ∈ (ℕ0 ↑𝑚 1𝑜) → (◡𝑓 “ ℕ) ∈ Fin) |
12 | 1, 11 | mprgbir 2927 | 1 ⊢ (ℕ0 ↑𝑚 1𝑜) = {𝑓 ∈ (ℕ0 ↑𝑚 1𝑜) ∣ (◡𝑓 “ ℕ) ∈ Fin} |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1483 ∈ wcel 1990 {crab 2916 ⊆ wss 3574 ∅c0 3915 {csn 4177 ◡ccnv 5113 dom cdm 5114 “ cima 5117 ⟶wf 5884 (class class class)co 6650 1𝑜c1o 7553 ↑𝑚 cmap 7857 Fincfn 7955 ℕcn 11020 ℕ0cn0 11292 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-1o 7560 df-er 7742 df-map 7859 df-en 7956 df-fin 7959 |
This theorem is referenced by: psr1bas 19561 ply1basf 19572 ply1plusgfvi 19612 coe1z 19633 coe1mul2 19639 coe1tm 19643 ply1coe 19666 deg1ldg 23852 deg1leb 23855 deg1val 23856 |
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