Theorem psrbasfsupp 33579

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 21872, 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 12403	. . . . 5 ⊢ 0 ∈ ℕ0
3		isfsupp 9256	. . . . 5 ⊢ ((𝑓 ∈ (ℕ0 ↑m 𝐼) ∧ 0 ∈ ℕ0) → (𝑓 finSupp 0 ↔ (Fun 𝑓 ∧ (𝑓 supp 0) ∈ Fin)))
4	2, 3	mpan2 691	. . . 4 ⊢ (𝑓 ∈ (ℕ0 ↑m 𝐼) → (𝑓 finSupp 0 ↔ (Fun 𝑓 ∧ (𝑓 supp 0) ∈ Fin)))
5		elmapfun 8796	. . . . 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 12401	. . . . . . . . . 10 ⊢ ℕ = (ℕ0 ∖ {0})
8	7	ineq2i 4166	. . . . . . . . 9 ⊢ (ran 𝑓 ∩ ℕ) = (ran 𝑓 ∩ (ℕ0 ∖ {0}))
9		incom 4158	. . . . . . . . 9 ⊢ (ran 𝑓 ∩ ℕ) = (ℕ ∩ ran 𝑓)
10		indif2 4230	. . . . . . . . 9 ⊢ (ran 𝑓 ∩ (ℕ0 ∖ {0})) = ((ran 𝑓 ∩ ℕ0) ∖ {0})
11	8, 9, 10	3eqtr3i 2764	. . . . . . . 8 ⊢ (ℕ ∩ ran 𝑓) = ((ran 𝑓 ∩ ℕ0) ∖ {0})
12		elmapi 8779	. . . . . . . . . . 11 ⊢ (𝑓 ∈ (ℕ0 ↑m 𝐼) → 𝑓:𝐼⟶ℕ0)
13	12	frnd 6664	. . . . . . . . . 10 ⊢ (𝑓 ∈ (ℕ0 ↑m 𝐼) → ran 𝑓 ⊆ ℕ0)
14		dfss2 3916	. . . . . . . . . 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 4074	. . . . . . . 8 ⊢ (𝑓 ∈ (ℕ0 ↑m 𝐼) → ((ran 𝑓 ∩ ℕ0) ∖ {0}) = (ran 𝑓 ∖ {0}))
17	11, 16	eqtrid 2780	. . . . . . 7 ⊢ (𝑓 ∈ (ℕ0 ↑m 𝐼) → (ℕ ∩ ran 𝑓) = (ran 𝑓 ∖ {0}))
18	17	imaeq2d 6013	. . . . . 6 ⊢ (𝑓 ∈ (ℕ0 ↑m 𝐼) → (◡𝑓 “ (ℕ ∩ ran 𝑓)) = (◡𝑓 “ (ran 𝑓 ∖ {0})))
19		fimacnvinrn 7010	. . . . . . 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 32676	. . . . . . 7 ⊢ ((Fun 𝑓 ∧ 𝑓 ∈ (ℕ0 ↑m 𝐼) ∧ 0 ∈ ℕ0) → (𝑓 supp 0) = (◡𝑓 “ (ran 𝑓 ∖ {0})))
24	5, 21, 22, 23	syl3anc 1373	. . . . . 6 ⊢ (𝑓 ∈ (ℕ0 ↑m 𝐼) → (𝑓 supp 0) = (◡𝑓 “ (ran 𝑓 ∖ {0})))
25	18, 20, 24	3eqtr4rd 2779	. . . . 5 ⊢ (𝑓 ∈ (ℕ0 ↑m 𝐼) → (𝑓 supp 0) = (◡𝑓 “ ℕ))
26	25	eleq1d 2818	. . . 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 3400	. 2 ⊢ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ 𝑓 finSupp 0} = {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}
29	1, 28	eqtri 2756	1 ⊢ 𝐷 = {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1541   ∈ wcel 2113  {crab 3396   ∖ cdif 3895   ∩ cin 3897   ⊆ wss 3898  {csn 4575   class class class wbr 5093  ^◡ccnv 5618  ran crn 5620   “ cima 5622  Fun wfun 6480  (class class class)co 7352   supp csupp 8096   ↑_m cmap 8756  Fincfn 8875   finSupp cfsupp 9252  0cc0 11013  ℕcn 12132  ℕ₀cn0 12388

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 5236  ax-nul 5246  ax-pow 5305  ax-pr 5372  ax-un 7674  ax-resscn 11070  ax-1cn 11071  ax-icn 11072  ax-addcl 11073  ax-addrcl 11074  ax-mulcl 11075  ax-mulrcl 11076  ax-mulcom 11077  ax-addass 11078  ax-mulass 11079  ax-distr 11080  ax-i2m1 11081  ax-1ne0 11082  ax-1rid 11083  ax-rnegex 11084  ax-rrecex 11085  ax-cnre 11086  ax-pre-lttri 11087  ax-pre-lttrn 11088  ax-pre-ltadd 11089

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 4475  df-pw 4551  df-sn 4576  df-pr 4578  df-op 4582  df-uni 4859  df-iun 4943  df-br 5094  df-opab 5156  df-mpt 5175  df-tr 5201  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-pred 6253  df-ord 6314  df-on 6315  df-lim 6316  df-suc 6317  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-ov 7355  df-oprab 7356  df-mpo 7357  df-om 7803  df-1st 7927  df-2nd 7928  df-supp 8097  df-frecs 8217  df-wrecs 8248  df-recs 8297  df-rdg 8335  df-er 8628  df-map 8758  df-en 8876  df-dom 8877  df-sdom 8878  df-fsupp 9253  df-pnf 11155  df-mnf 11156  df-xr 11157  df-ltxr 11158  df-le 11159  df-nn 12133  df-n0 12389

This theorem is referenced by:  extvfvvcl  33586  extvfvcl  33587  mplmulmvr  33590  mplvrpmfgalem  33592  mplvrpmga  33593  mplvrpmmhm  33594  mplvrpmrhm  33595  issply  33602  esplyfval2  33605  esplympl  33607  esplymhp  33608  esplyfval3  33612  esplyind  33613