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Theorem psrbagf 21938
Description: A finite bag is a function. (Contributed by Mario Carneiro, 29-Dec-2014.) Remove a sethood antecedent. (Revised by SN, 30-Jul-2024.)
Hypothesis
Ref Expression
psrbag.d 𝐷 = {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}
Assertion
Ref Expression
psrbagf (𝐹𝐷𝐹:𝐼⟶ℕ0)
Distinct variable groups:   𝑓,𝐹   𝑓,𝐼
Allowed substitution hint:   𝐷(𝑓)

Proof of Theorem psrbagf
StepHypRef Expression
1 psrbag.d . . 3 𝐷 = {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}
21eleq2i 2833 . 2 (𝐹𝐷𝐹 ∈ {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin})
3 elrabi 3687 . . 3 (𝐹 ∈ {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} → 𝐹 ∈ (ℕ0m 𝐼))
4 elmapi 8889 . . 3 (𝐹 ∈ (ℕ0m 𝐼) → 𝐹:𝐼⟶ℕ0)
53, 4syl 17 . 2 (𝐹 ∈ {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} → 𝐹:𝐼⟶ℕ0)
62, 5sylbi 217 1 (𝐹𝐷𝐹:𝐼⟶ℕ0)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1540  wcel 2108  {crab 3436  ccnv 5684  cima 5688  wf 6557  (class class class)co 7431  m cmap 8866  Fincfn 8985  cn 12266  0cn0 12526
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8014  df-2nd 8015  df-map 8868
This theorem is referenced by:  psrbagfsupp  21939  psrbaglesupp  21942  psrbaglecl  21943  psrbagaddcl  21944  psrbagcon  21945  psrbaglefi  21946  psrbagconcl  21947  psrbagleadd1  21948  psrbagconf1o  21949  gsumbagdiaglem  21950  psrass1lem  21952  rhmpsrlem2  21961  psrlidm  21982  psrridm  21983  psrass1  21984  psrcom  21988  mplsubrglem  22024  mplmonmul  22054  psrbagev1  22101  evlslem3  22104  evlslem1  22106  mhpmulcl  22153  psdcl  22165  psdmplcl  22166  psdadd  22167  psdvsca  22168  psdmul  22170  psdmvr  22173  psropprmul  22239  tdeglem1  26097  tdeglem3  26098  tdeglem4  26099  mdegmullem  26117  psrbagres  42556  evlsvvvallem  42571  evlsvvval  42573  selvvvval  42595  evlselvlem  42596  evlselv  42597  mhphflem  42606  mhphf  42607
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