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Theorem psrbagf 21956
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 2831 . 2 (𝐹𝐷𝐹 ∈ {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin})
3 elrabi 3690 . . 3 (𝐹 ∈ {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} → 𝐹 ∈ (ℕ0m 𝐼))
4 elmapi 8888 . . 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 1537  wcel 2106  {crab 3433  ccnv 5688  cima 5692  wf 6559  (class class class)co 7431  m cmap 8865  Fincfn 8984  cn 12264  0cn0 12524
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8013  df-2nd 8014  df-map 8867
This theorem is referenced by:  psrbagfsupp  21957  psrbaglesupp  21960  psrbaglecl  21961  psrbagaddcl  21962  psrbagcon  21963  psrbaglefi  21964  psrbagconcl  21965  psrbagleadd1  21966  psrbagconf1o  21967  gsumbagdiaglem  21968  psrass1lem  21970  rhmpsrlem2  21979  psrlidm  22000  psrridm  22001  psrass1  22002  psrcom  22006  mplsubrglem  22042  mplmonmul  22072  psrbagev1  22119  evlslem3  22122  evlslem1  22124  mhpmulcl  22171  psdcl  22183  psdmplcl  22184  psdadd  22185  psdvsca  22186  psdmul  22188  psropprmul  22255  tdeglem1  26112  tdeglem3  26113  tdeglem4  26114  mdegmullem  26132  psrbagres  42533  evlsvvvallem  42548  evlsvvval  42550  selvvvval  42572  evlselvlem  42573  evlselv  42574  mhphflem  42583  mhphf  42584
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