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Theorem psrbagf 20680
 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 2843 . 2 (𝐹𝐷𝐹 ∈ {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin})
3 elrabi 3596 . . 3 (𝐹 ∈ {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} → 𝐹 ∈ (ℕ0m 𝐼))
4 elmapi 8438 . . 3 (𝐹 ∈ (ℕ0m 𝐼) → 𝐹:𝐼⟶ℕ0)
53, 4syl 17 . 2 (𝐹 ∈ {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} → 𝐹:𝐼⟶ℕ0)
62, 5sylbi 220 1 (𝐹𝐷𝐹:𝐼⟶ℕ0)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1538   ∈ wcel 2111  {crab 3074  ◡ccnv 5523   “ cima 5527  ⟶wf 6331  (class class class)co 7150   ↑m cmap 8416  Fincfn 8527  ℕcn 11674  ℕ0cn0 11934 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5169  ax-nul 5176  ax-pow 5234  ax-pr 5298  ax-un 7459 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-rab 3079  df-v 3411  df-sbc 3697  df-csb 3806  df-dif 3861  df-un 3863  df-in 3865  df-ss 3875  df-nul 4226  df-if 4421  df-pw 4496  df-sn 4523  df-pr 4525  df-op 4529  df-uni 4799  df-iun 4885  df-br 5033  df-opab 5095  df-mpt 5113  df-id 5430  df-xp 5530  df-rel 5531  df-cnv 5532  df-co 5533  df-dm 5534  df-rn 5535  df-res 5536  df-ima 5537  df-iota 6294  df-fun 6337  df-fn 6338  df-f 6339  df-fv 6343  df-ov 7153  df-oprab 7154  df-mpo 7155  df-1st 7693  df-2nd 7694  df-map 8418 This theorem is referenced by:  psrbagfsupp  20682  psrbaglesupp  20686  psrbaglecl  20688  psrbagaddcl  20690  psrbagcon  20692  psrbaglefi  20694  psrbagconcl  20696  psrbagconf1o  20698  gsumbagdiaglem  20703  psrass1lem  20705  psrmulcllem  20715  psrlidm  20731  psrridm  20732  psrass1  20733  psrcom  20737  mplsubrglem  20769  mplmonmul  20796  psrbagev1  20838  evlslem3  20843  evlslem1  20845  mhpmulcl  20892  psropprmul  20962  tdeglem1  24755  tdeglem3  24757  tdeglem4  24759  mdegmullem  24778  evlsbagval  39802  mhphflem  39811
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