Theorem psrbasfsupp 33672

Description: Rewrite a finite support for nonnegative integers : For functions mapping a set 𝐼 to the nonnegative integers, having finite support can also be written as having a finite preimage of the positive integers. The latter expression is used for example in psrbas 21913, but with the former expression, theorems about finite support can be used more directly. (Contributed by Thierry Arnoux, 10-Jan-2026.)

Hypothesis

Ref	Expression
psrbasfsupp.d	⊢ 𝐷 = {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ 𝑓 finSupp 0}

Assertion

Ref	Expression
psrbasfsupp	⊢ 𝐷 = {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}

Proof of Theorem psrbasfsupp

Step	Hyp	Ref	Expression
1		psrbasfsupp.d	. 2 ⊢ 𝐷 = {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ 𝑓 finSupp 0}
2		0nn0 12452	. . . . 5 ⊢ 0 ∈ ℕ0
3		isfsupp 9278	. . . . 5 ⊢ ((𝑓 ∈ (ℕ0 ↑m 𝐼) ∧ 0 ∈ ℕ0) → (𝑓 finSupp 0 ↔ (Fun 𝑓 ∧ (𝑓 supp 0) ∈ Fin)))
4	2, 3	mpan2 692	. . . 4 ⊢ (𝑓 ∈ (ℕ0 ↑m 𝐼) → (𝑓 finSupp 0 ↔ (Fun 𝑓 ∧ (𝑓 supp 0) ∈ Fin)))
5		elmapfun 8813	. . . . 5 ⊢ (𝑓 ∈ (ℕ0 ↑m 𝐼) → Fun 𝑓)
6	5	biantrurd 532	. . . 4 ⊢ (𝑓 ∈ (ℕ0 ↑m 𝐼) → ((𝑓 supp 0) ∈ Fin ↔ (Fun 𝑓 ∧ (𝑓 supp 0) ∈ Fin)))
7		dfn2 12450	. . . . . . . . . 10 ⊢ ℕ = (ℕ0 ∖ {0})
8	7	ineq2i 4157	. . . . . . . . 9 ⊢ (ran 𝑓 ∩ ℕ) = (ran 𝑓 ∩ (ℕ0 ∖ {0}))
9		incom 4149	. . . . . . . . 9 ⊢ (ran 𝑓 ∩ ℕ) = (ℕ ∩ ran 𝑓)
10		indif2 4221	. . . . . . . . 9 ⊢ (ran 𝑓 ∩ (ℕ0 ∖ {0})) = ((ran 𝑓 ∩ ℕ0) ∖ {0})
11	8, 9, 10	3eqtr3i 2767	. . . . . . . 8 ⊢ (ℕ ∩ ran 𝑓) = ((ran 𝑓 ∩ ℕ0) ∖ {0})
12		elmapi 8796	. . . . . . . . . . 11 ⊢ (𝑓 ∈ (ℕ0 ↑m 𝐼) → 𝑓:𝐼⟶ℕ0)
13	12	frnd 6676	. . . . . . . . . 10 ⊢ (𝑓 ∈ (ℕ0 ↑m 𝐼) → ran 𝑓 ⊆ ℕ0)
14		dfss2 3907	. . . . . . . . . 10 ⊢ (ran 𝑓 ⊆ ℕ0 ↔ (ran 𝑓 ∩ ℕ0) = ran 𝑓)
15	13, 14	sylib 218	. . . . . . . . 9 ⊢ (𝑓 ∈ (ℕ0 ↑m 𝐼) → (ran 𝑓 ∩ ℕ0) = ran 𝑓)
16	15	difeq1d 4065	. . . . . . . 8 ⊢ (𝑓 ∈ (ℕ0 ↑m 𝐼) → ((ran 𝑓 ∩ ℕ0) ∖ {0}) = (ran 𝑓 ∖ {0}))
17	11, 16	eqtrid 2783	. . . . . . 7 ⊢ (𝑓 ∈ (ℕ0 ↑m 𝐼) → (ℕ ∩ ran 𝑓) = (ran 𝑓 ∖ {0}))
18	17	imaeq2d 6025	. . . . . 6 ⊢ (𝑓 ∈ (ℕ0 ↑m 𝐼) → (◡𝑓 “ (ℕ ∩ ran 𝑓)) = (◡𝑓 “ (ran 𝑓 ∖ {0})))
19		fimacnvinrn 7023	. . . . . . 7 ⊢ (Fun 𝑓 → (◡𝑓 “ ℕ) = (◡𝑓 “ (ℕ ∩ ran 𝑓)))
20	5, 19	syl 17	. . . . . 6 ⊢ (𝑓 ∈ (ℕ0 ↑m 𝐼) → (◡𝑓 “ ℕ) = (◡𝑓 “ (ℕ ∩ ran 𝑓)))
21		id 22	. . . . . . 7 ⊢ (𝑓 ∈ (ℕ0 ↑m 𝐼) → 𝑓 ∈ (ℕ0 ↑m 𝐼))
22	2	a1i 11	. . . . . . 7 ⊢ (𝑓 ∈ (ℕ0 ↑m 𝐼) → 0 ∈ ℕ0)
23		supppreima 32764	. . . . . . 7 ⊢ ((Fun 𝑓 ∧ 𝑓 ∈ (ℕ0 ↑m 𝐼) ∧ 0 ∈ ℕ0) → (𝑓 supp 0) = (◡𝑓 “ (ran 𝑓 ∖ {0})))
24	5, 21, 22, 23	syl3anc 1374	. . . . . 6 ⊢ (𝑓 ∈ (ℕ0 ↑m 𝐼) → (𝑓 supp 0) = (◡𝑓 “ (ran 𝑓 ∖ {0})))
25	18, 20, 24	3eqtr4rd 2782	. . . . 5 ⊢ (𝑓 ∈ (ℕ0 ↑m 𝐼) → (𝑓 supp 0) = (◡𝑓 “ ℕ))
26	25	eleq1d 2821	. . . 4 ⊢ (𝑓 ∈ (ℕ0 ↑m 𝐼) → ((𝑓 supp 0) ∈ Fin ↔ (◡𝑓 “ ℕ) ∈ Fin))
27	4, 6, 26	3bitr2d 307	. . 3 ⊢ (𝑓 ∈ (ℕ0 ↑m 𝐼) → (𝑓 finSupp 0 ↔ (◡𝑓 “ ℕ) ∈ Fin))
28	27	rabbiia 3393	. 2 ⊢ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ 𝑓 finSupp 0} = {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}
29	1, 28	eqtri 2759	1 ⊢ 𝐷 = {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114  {crab 3389   ∖ cdif 3886   ∩ cin 3888   ⊆ wss 3889  {csn 4567   class class class wbr 5085  ^◡ccnv 5630  ran crn 5632   “ cima 5634  Fun wfun 6492  (class class class)co 7367   supp csupp 8110   ↑_m cmap 8773  Fincfn 8893   finSupp cfsupp 9274  0cc0 11038  ℕcn 12174  ℕ₀cn0 12437

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689  ax-resscn 11095  ax-1cn 11096  ax-icn 11097  ax-addcl 11098  ax-addrcl 11099  ax-mulcl 11100  ax-mulrcl 11101  ax-mulcom 11102  ax-addass 11103  ax-mulass 11104  ax-distr 11105  ax-i2m1 11106  ax-1ne0 11107  ax-1rid 11108  ax-rnegex 11109  ax-rrecex 11110  ax-cnre 11111  ax-pre-lttri 11112  ax-pre-lttrn 11113  ax-pre-ltadd 11114

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3062  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-pss 3909  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-tr 5193  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6265  df-ord 6326  df-on 6327  df-lim 6328  df-suc 6329  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-ov 7370  df-oprab 7371  df-mpo 7372  df-om 7818  df-1st 7942  df-2nd 7943  df-supp 8111  df-frecs 8231  df-wrecs 8262  df-recs 8311  df-rdg 8349  df-er 8643  df-map 8775  df-en 8894  df-dom 8895  df-sdom 8896  df-fsupp 9275  df-pnf 11181  df-mnf 11182  df-xr 11183  df-ltxr 11184  df-le 11185  df-nn 12175  df-n0 12438

This theorem is referenced by:  extvfvvcl  33679  extvfvcl  33680  mplmulmvr  33683  evlextv  33686  mplvrpmfgalem  33688  mplvrpmga  33689  mplvrpmmhm  33690  mplvrpmrhm  33691  psrgsum  33692  psrmon  33693  psrmonmul  33694  psrmonmul2  33695  psrmonprod  33696  mplgsum  33697  mplmonprod  33698  issply  33705  esplyfval0  33708  esplyfval2  33709  esplympl  33711  esplymhp  33712  esplyfval3  33716  esplyfval1  33717  esplyfvaln  33718  esplyind  33719  vieta  33724