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Theorem psr1baslem 20270
Description: The set of finite bags on 1o is just the set of all functions from 1o to 0. (Contributed by Mario Carneiro, 9-Feb-2015.)
Assertion
Ref Expression
psr1baslem (ℕ0m 1o) = {𝑓 ∈ (ℕ0m 1o) ∣ (𝑓 “ ℕ) ∈ Fin}

Proof of Theorem psr1baslem
StepHypRef Expression
1 rabid2 3386 . 2 ((ℕ0m 1o) = {𝑓 ∈ (ℕ0m 1o) ∣ (𝑓 “ ℕ) ∈ Fin} ↔ ∀𝑓 ∈ (ℕ0m 1o)(𝑓 “ ℕ) ∈ Fin)
2 df1o2 8110 . . . 4 1o = {∅}
3 snfi 8586 . . . 4 {∅} ∈ Fin
42, 3eqeltri 2913 . . 3 1o ∈ Fin
5 cnvimass 5946 . . . 4 (𝑓 “ ℕ) ⊆ dom 𝑓
6 elmapi 8421 . . . 4 (𝑓 ∈ (ℕ0m 1o) → 𝑓:1o⟶ℕ0)
75, 6fssdm 6526 . . 3 (𝑓 ∈ (ℕ0m 1o) → (𝑓 “ ℕ) ⊆ 1o)
8 ssfi 8730 . . 3 ((1o ∈ Fin ∧ (𝑓 “ ℕ) ⊆ 1o) → (𝑓 “ ℕ) ∈ Fin)
94, 7, 8sylancr 587 . 2 (𝑓 ∈ (ℕ0m 1o) → (𝑓 “ ℕ) ∈ Fin)
101, 9mprgbir 3157 1 (ℕ0m 1o) = {𝑓 ∈ (ℕ0m 1o) ∣ (𝑓 “ ℕ) ∈ Fin}
Colors of variables: wff setvar class
Syntax hints:   = wceq 1530  wcel 2106  {crab 3146  wss 3939  c0 4294  {csn 4563  ccnv 5552  cima 5556  (class class class)co 7151  1oc1o 8089  m cmap 8399  Fincfn 8501  cn 11630  0cn0 11889
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2152  ax-12 2167  ax-ext 2796  ax-sep 5199  ax-nul 5206  ax-pow 5262  ax-pr 5325  ax-un 7454
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2615  df-eu 2649  df-clab 2803  df-cleq 2817  df-clel 2897  df-nfc 2967  df-ne 3021  df-ral 3147  df-rex 3148  df-rab 3151  df-v 3501  df-sbc 3776  df-csb 3887  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-pss 3957  df-nul 4295  df-if 4470  df-pw 4543  df-sn 4564  df-pr 4566  df-tp 4568  df-op 4570  df-uni 4837  df-iun 4918  df-br 5063  df-opab 5125  df-mpt 5143  df-tr 5169  df-id 5458  df-eprel 5463  df-po 5472  df-so 5473  df-fr 5512  df-we 5514  df-xp 5559  df-rel 5560  df-cnv 5561  df-co 5562  df-dm 5563  df-rn 5564  df-res 5565  df-ima 5566  df-ord 6191  df-on 6192  df-lim 6193  df-suc 6194  df-iota 6311  df-fun 6353  df-fn 6354  df-f 6355  df-f1 6356  df-fo 6357  df-f1o 6358  df-fv 6359  df-ov 7154  df-oprab 7155  df-mpo 7156  df-om 7572  df-1st 7683  df-2nd 7684  df-1o 8096  df-er 8282  df-map 8401  df-en 8502  df-fin 8505
This theorem is referenced by:  psr1bas  20276  ply1basf  20287  ply1plusgfvi  20327  coe1z  20348  coe1mul2  20354  coe1tm  20358  ply1coe  20381  deg1ldg  24601  deg1leb  24604  deg1val  24605
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