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Theorem psrbagfsupp 20851

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 20849	. . . . . 6 ⊢ (𝐹 ∈ 𝐷 → 𝐹:𝐼⟶ℕ₀)
4	3	ffnd 6535	. . . . 5 ⊢ (𝐹 ∈ 𝐷 → 𝐹 Fn 𝐼)
5	1, 4	fndmexd 7673	. . . 4 ⊢ (𝐹 ∈ 𝐷 → 𝐼 ∈ V)
6	2	psrbag 20848	. . . . 5 ⊢ (𝐼 ∈ V → (𝐹 ∈ 𝐷 ↔ (𝐹:𝐼⟶ℕ₀ ∧ (^◡𝐹 “ ℕ) ∈ Fin)))
7	6	biimpa 480	. . . 4 ⊢ ((𝐼 ∈ V ∧ 𝐹 ∈ 𝐷) → (𝐹:𝐼⟶ℕ₀ ∧ (^◡𝐹 “ ℕ) ∈ Fin))
8	5, 7	mpancom 688	. . 3 ⊢ (𝐹 ∈ 𝐷 → (𝐹:𝐼⟶ℕ₀ ∧ (^◡𝐹 “ ℕ) ∈ Fin))
9	8	simprd 499	. 2 ⊢ (𝐹 ∈ 𝐷 → (^◡𝐹 “ ℕ) ∈ Fin)
10		frnnn0fsuppg 12132	. . 3 ⊢ ((𝐹 ∈ 𝐷 ∧ 𝐹:𝐼⟶ℕ₀) → (𝐹 finSupp 0 ↔ (^◡𝐹 “ ℕ) ∈ Fin))
11	3, 10	mpdan 687	. 2 ⊢ (𝐹 ∈ 𝐷 → (𝐹 finSupp 0 ↔ (^◡𝐹 “ ℕ) ∈ Fin))
12	9, 11	mpbird 260	1 ⊢ (𝐹 ∈ 𝐷 → 𝐹 finSupp 0)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1543 ∈ wcel 2110 {crab 3058 Vcvv 3401 class class class wbr 5043 ^◡ccnv 5539 “ cima 5543 ⟶wf 6365 (class class class)co 7202 ↑_m cmap 8497 Fincfn 8615 finSupp cfsupp 8974 0cc0 10712 ℕcn 11813 ℕ₀cn0 12073

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2706 ax-sep 5181 ax-nul 5188 ax-pow 5247 ax-pr 5311 ax-un 7512 ax-cnex 10768 ax-resscn 10769 ax-1cn 10770 ax-icn 10771 ax-addcl 10772 ax-addrcl 10773 ax-mulcl 10774 ax-mulrcl 10775 ax-mulcom 10776 ax-addass 10777 ax-mulass 10778 ax-distr 10779 ax-i2m1 10780 ax-1ne0 10781 ax-1rid 10782 ax-rnegex 10783 ax-rrecex 10784 ax-cnre 10785 ax-pre-lttri 10786 ax-pre-lttrn 10787 ax-pre-ltadd 10788

This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2537 df-eu 2566 df-clab 2713 df-cleq 2726 df-clel 2812 df-nfc 2882 df-ne 2936 df-nel 3040 df-ral 3059 df-rex 3060 df-reu 3061 df-rab 3063 df-v 3403 df-sbc 3688 df-csb 3803 df-dif 3860 df-un 3862 df-in 3864 df-ss 3874 df-pss 3876 df-nul 4228 df-if 4430 df-pw 4505 df-sn 4532 df-pr 4534 df-tp 4536 df-op 4538 df-uni 4810 df-iun 4896 df-br 5044 df-opab 5106 df-mpt 5125 df-tr 5151 df-id 5444 df-eprel 5449 df-po 5457 df-so 5458 df-fr 5498 df-we 5500 df-xp 5546 df-rel 5547 df-cnv 5548 df-co 5549 df-dm 5550 df-rn 5551 df-res 5552 df-ima 5553 df-pred 6149 df-ord 6205 df-on 6206 df-lim 6207 df-suc 6208 df-iota 6327 df-fun 6371 df-fn 6372 df-f 6373 df-f1 6374 df-fo 6375 df-f1o 6376 df-fv 6377 df-ov 7205 df-oprab 7206 df-mpo 7207 df-om 7634 df-1st 7750 df-2nd 7751 df-supp 7893 df-wrecs 8036 df-recs 8097 df-rdg 8135 df-er 8380 df-map 8499 df-en 8616 df-dom 8617 df-sdom 8618 df-fsupp 8975 df-pnf 10852 df-mnf 10853 df-xr 10854 df-ltxr 10855 df-le 10856 df-nn 11814 df-n0 12074

This theorem is referenced by: psrbagaddcl 20859 psrbagev1 21007 mhpmulcl 21061 tdeglem1 24925 tdeglem3 24927 tdeglem4 24929 evlsbagval 39937 mhphflem 39946 mhphf 39947

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