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Theorem psrbagfsupp 21033

Description: Finite bags have finite support. (Contributed by Stefan O'Rear, 9-Mar-2015.) (Revised by AV, 18-Jul-2019.) Remove a sethood antecedent. (Revised by SN, 7-Aug-2024.)

**Hypothesis**
Ref	Expression
psrbag.d	⊢ 𝐷 = {𝑓 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑓 “ ℕ) ∈ Fin}

**Assertion**
Ref	Expression
psrbagfsupp	⊢ (𝐹 ∈ 𝐷 → 𝐹 finSupp 0)

Distinct variable groups: 𝑓,𝐹 𝑓,𝐼 Allowed substitution hint: 𝐷(𝑓)

**Proof of Theorem psrbagfsupp**
Step	Hyp	Ref	Expression
1		id 22	. . . . 5 ⊢ (𝐹 ∈ 𝐷 → 𝐹 ∈ 𝐷)
2		psrbag.d	. . . . . . 7 ⊢ 𝐷 = {𝑓 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑓 “ ℕ) ∈ Fin}
3	2	psrbagf 21031	. . . . . 6 ⊢ (𝐹 ∈ 𝐷 → 𝐹:𝐼⟶ℕ₀)
4	3	ffnd 6585	. . . . 5 ⊢ (𝐹 ∈ 𝐷 → 𝐹 Fn 𝐼)
5	1, 4	fndmexd 7727	. . . 4 ⊢ (𝐹 ∈ 𝐷 → 𝐼 ∈ V)
6	2	psrbag 21030	. . . . 5 ⊢ (𝐼 ∈ V → (𝐹 ∈ 𝐷 ↔ (𝐹:𝐼⟶ℕ₀ ∧ (^◡𝐹 “ ℕ) ∈ Fin)))
7	6	biimpa 476	. . . 4 ⊢ ((𝐼 ∈ V ∧ 𝐹 ∈ 𝐷) → (𝐹:𝐼⟶ℕ₀ ∧ (^◡𝐹 “ ℕ) ∈ Fin))
8	5, 7	mpancom 684	. . 3 ⊢ (𝐹 ∈ 𝐷 → (𝐹:𝐼⟶ℕ₀ ∧ (^◡𝐹 “ ℕ) ∈ Fin))
9	8	simprd 495	. 2 ⊢ (𝐹 ∈ 𝐷 → (^◡𝐹 “ ℕ) ∈ Fin)
10		frnnn0fsuppg 12222	. . 3 ⊢ ((𝐹 ∈ 𝐷 ∧ 𝐹:𝐼⟶ℕ₀) → (𝐹 finSupp 0 ↔ (^◡𝐹 “ ℕ) ∈ Fin))
11	3, 10	mpdan 683	. 2 ⊢ (𝐹 ∈ 𝐷 → (𝐹 finSupp 0 ↔ (^◡𝐹 “ ℕ) ∈ Fin))
12	9, 11	mpbird 256	1 ⊢ (𝐹 ∈ 𝐷 → 𝐹 finSupp 0)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1539 ∈ wcel 2108 {crab 3067 Vcvv 3422 class class class wbr 5070 ^◡ccnv 5579 “ cima 5583 ⟶wf 6414 (class class class)co 7255 ↑_m cmap 8573 Fincfn 8691 finSupp cfsupp 9058 0cc0 10802 ℕcn 11903 ℕ₀cn0 12163

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566 ax-cnex 10858 ax-resscn 10859 ax-1cn 10860 ax-icn 10861 ax-addcl 10862 ax-addrcl 10863 ax-mulcl 10864 ax-mulrcl 10865 ax-mulcom 10866 ax-addass 10867 ax-mulass 10868 ax-distr 10869 ax-i2m1 10870 ax-1ne0 10871 ax-1rid 10872 ax-rnegex 10873 ax-rrecex 10874 ax-cnre 10875 ax-pre-lttri 10876 ax-pre-lttrn 10877 ax-pre-ltadd 10878

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3068 df-rex 3069 df-reu 3070 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3902 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-uni 4837 df-iun 4923 df-br 5071 df-opab 5133 df-mpt 5154 df-tr 5188 df-id 5480 df-eprel 5486 df-po 5494 df-so 5495 df-fr 5535 df-we 5537 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-pred 6191 df-ord 6254 df-on 6255 df-lim 6256 df-suc 6257 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-ov 7258 df-oprab 7259 df-mpo 7260 df-om 7688 df-1st 7804 df-2nd 7805 df-supp 7949 df-frecs 8068 df-wrecs 8099 df-recs 8173 df-rdg 8212 df-er 8456 df-map 8575 df-en 8692 df-dom 8693 df-sdom 8694 df-fsupp 9059 df-pnf 10942 df-mnf 10943 df-xr 10944 df-ltxr 10945 df-le 10946 df-nn 11904 df-n0 12164

This theorem is referenced by: psrbagaddcl 21041 psrbagev1 21195 mhpmulcl 21249 tdeglem1 25125 tdeglem3 25127 tdeglem4 25129 evlsbagval 40198 mhphflem 40207 mhphf 40208

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