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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 3381 | . 2 ⊢ ((ℕ0 ↑m 1o) = {𝑓 ∈ (ℕ0 ↑m 1o) ∣ (◡𝑓 “ ℕ) ∈ Fin} ↔ ∀𝑓 ∈ (ℕ0 ↑m 1o)(◡𝑓 “ ℕ) ∈ Fin) | |
2 | df1o2 8116 | . . . 4 ⊢ 1o = {∅} | |
3 | snfi 8594 | . . . 4 ⊢ {∅} ∈ Fin | |
4 | 2, 3 | eqeltri 2909 | . . 3 ⊢ 1o ∈ Fin |
5 | cnvimass 5949 | . . . 4 ⊢ (◡𝑓 “ ℕ) ⊆ dom 𝑓 | |
6 | elmapi 8428 | . . . 4 ⊢ (𝑓 ∈ (ℕ0 ↑m 1o) → 𝑓:1o⟶ℕ0) | |
7 | 5, 6 | fssdm 6530 | . . 3 ⊢ (𝑓 ∈ (ℕ0 ↑m 1o) → (◡𝑓 “ ℕ) ⊆ 1o) |
8 | ssfi 8738 | . . 3 ⊢ ((1o ∈ Fin ∧ (◡𝑓 “ ℕ) ⊆ 1o) → (◡𝑓 “ ℕ) ∈ Fin) | |
9 | 4, 7, 8 | sylancr 589 | . 2 ⊢ (𝑓 ∈ (ℕ0 ↑m 1o) → (◡𝑓 “ ℕ) ∈ Fin) |
10 | 1, 9 | mprgbir 3153 | 1 ⊢ (ℕ0 ↑m 1o) = {𝑓 ∈ (ℕ0 ↑m 1o) ∣ (◡𝑓 “ ℕ) ∈ Fin} |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ∈ wcel 2114 {crab 3142 ⊆ wss 3936 ∅c0 4291 {csn 4567 ◡ccnv 5554 “ cima 5558 (class class class)co 7156 1oc1o 8095 ↑m cmap 8406 Fincfn 8509 ℕcn 11638 ℕ0cn0 11898 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-1o 8102 df-er 8289 df-map 8408 df-en 8510 df-fin 8513 |
This theorem is referenced by: psr1bas 20359 ply1basf 20370 ply1plusgfvi 20410 coe1z 20431 coe1mul2 20437 coe1tm 20441 ply1coe 20464 deg1ldg 24686 deg1leb 24689 deg1val 24690 |
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