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Theorem psrbagfsupp 21866

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 21865	. . . . . 6 ⊢ (𝐹 ∈ 𝐷 → 𝐹:𝐼⟶ℕ₀)
4	3	ffnd 6660	. . . . 5 ⊢ (𝐹 ∈ 𝐷 → 𝐹 Fn 𝐼)
5	1, 4	fndmexd 7843	. . . 4 ⊢ (𝐹 ∈ 𝐷 → 𝐼 ∈ V)
6	2	psrbag 21864	. . . . 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 688	. . 3 ⊢ (𝐹 ∈ 𝐷 → (𝐹:𝐼⟶ℕ₀ ∧ (^◡𝐹 “ ℕ) ∈ Fin))
9	8	simprd 495	. 2 ⊢ (𝐹 ∈ 𝐷 → (^◡𝐹 “ ℕ) ∈ Fin)
10		fcdmnn0fsuppg 12452	. . 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 257	1 ⊢ (𝐹 ∈ 𝐷 → 𝐹 finSupp 0)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1541 ∈ wcel 2113 {crab 3396 Vcvv 3437 class class class wbr 5095 ^◡ccnv 5620 “ cima 5624 ⟶wf 6485 (class class class)co 7355 ↑_m cmap 8759 Fincfn 8879 finSupp cfsupp 9256 0cc0 11017 ℕcn 12136 ℕ₀cn0 12392

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2182 ax-ext 2705 ax-sep 5238 ax-nul 5248 ax-pow 5307 ax-pr 5374 ax-un 7677 ax-cnex 11073 ax-resscn 11074 ax-1cn 11075 ax-icn 11076 ax-addcl 11077 ax-addrcl 11078 ax-mulcl 11079 ax-mulrcl 11080 ax-mulcom 11081 ax-addass 11082 ax-mulass 11083 ax-distr 11084 ax-i2m1 11085 ax-1ne0 11086 ax-1rid 11087 ax-rnegex 11088 ax-rrecex 11089 ax-cnre 11090 ax-pre-lttri 11091 ax-pre-lttrn 11092 ax-pre-ltadd 11093

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2725 df-clel 2808 df-nfc 2882 df-ne 2930 df-nel 3034 df-ral 3049 df-rex 3058 df-reu 3348 df-rab 3397 df-v 3439 df-sbc 3738 df-csb 3847 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-pss 3918 df-nul 4283 df-if 4477 df-pw 4553 df-sn 4578 df-pr 4580 df-op 4584 df-uni 4861 df-iun 4945 df-br 5096 df-opab 5158 df-mpt 5177 df-tr 5203 df-id 5516 df-eprel 5521 df-po 5529 df-so 5530 df-fr 5574 df-we 5576 df-xp 5627 df-rel 5628 df-cnv 5629 df-co 5630 df-dm 5631 df-rn 5632 df-res 5633 df-ima 5634 df-pred 6256 df-ord 6317 df-on 6318 df-lim 6319 df-suc 6320 df-iota 6445 df-fun 6491 df-fn 6492 df-f 6493 df-f1 6494 df-fo 6495 df-f1o 6496 df-fv 6497 df-ov 7358 df-oprab 7359 df-mpo 7360 df-om 7806 df-1st 7930 df-2nd 7931 df-supp 8100 df-frecs 8220 df-wrecs 8251 df-recs 8300 df-rdg 8338 df-er 8631 df-map 8761 df-en 8880 df-dom 8881 df-sdom 8882 df-fsupp 9257 df-pnf 11159 df-mnf 11160 df-xr 11161 df-ltxr 11162 df-le 11163 df-nn 12137 df-n0 12393

This theorem is referenced by: psrbagaddcl 21871 psrbagev1 22023 evlsvvvallem 22037 evlsvvval 22039 mhpmulcl 22083 tdeglem1 26010 tdeglem3 26011 tdeglem4 26012 extvfvcl 33629 mplvrpmfgalem 33637 mplvrpmrhm 33640 psrbagres 42714 selvvvval 42743 evlselvlem 42744 evlselv 42745 mhphflem 42754 mhphf 42755

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