Theorem psrbasfsupp 33563

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 21859, 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 12418	. . . . 5 ⊢ 0 ∈ ℕ0
3		isfsupp 9274	. . . . 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 8800	. . . . 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 12416	. . . . . . . . . 10 ⊢ ℕ = (ℕ0 ∖ {0})
8	7	ineq2i 4170	. . . . . . . . 9 ⊢ (ran 𝑓 ∩ ℕ) = (ran 𝑓 ∩ (ℕ0 ∖ {0}))
9		incom 4162	. . . . . . . . 9 ⊢ (ran 𝑓 ∩ ℕ) = (ℕ ∩ ran 𝑓)
10		indif2 4234	. . . . . . . . 9 ⊢ (ran 𝑓 ∩ (ℕ0 ∖ {0})) = ((ran 𝑓 ∩ ℕ0) ∖ {0})
11	8, 9, 10	3eqtr3i 2760	. . . . . . . 8 ⊢ (ℕ ∩ ran 𝑓) = ((ran 𝑓 ∩ ℕ0) ∖ {0})
12		elmapi 8783	. . . . . . . . . . 11 ⊢ (𝑓 ∈ (ℕ0 ↑m 𝐼) → 𝑓:𝐼⟶ℕ0)
13	12	frnd 6664	. . . . . . . . . 10 ⊢ (𝑓 ∈ (ℕ0 ↑m 𝐼) → ran 𝑓 ⊆ ℕ0)
14		dfss2 3923	. . . . . . . . . 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 4078	. . . . . . . 8 ⊢ (𝑓 ∈ (ℕ0 ↑m 𝐼) → ((ran 𝑓 ∩ ℕ0) ∖ {0}) = (ran 𝑓 ∖ {0}))
17	11, 16	eqtrid 2776	. . . . . . 7 ⊢ (𝑓 ∈ (ℕ0 ↑m 𝐼) → (ℕ ∩ ran 𝑓) = (ran 𝑓 ∖ {0}))
18	17	imaeq2d 6015	. . . . . 6 ⊢ (𝑓 ∈ (ℕ0 ↑m 𝐼) → (◡𝑓 “ (ℕ ∩ ran 𝑓)) = (◡𝑓 “ (ran 𝑓 ∖ {0})))
19		fimacnvinrn 7009	. . . . . . 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 32652	. . . . . . 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 2775	. . . . 5 ⊢ (𝑓 ∈ (ℕ0 ↑m 𝐼) → (𝑓 supp 0) = (◡𝑓 “ ℕ))
26	25	eleq1d 2813	. . . 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 2752	1 ⊢ 𝐷 = {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  {crab 3396   ∖ cdif 3902   ∩ cin 3904   ⊆ wss 3905  {csn 4579   class class class wbr 5095  ^◡ccnv 5622  ran crn 5624   “ cima 5626  Fun wfun 6480  (class class class)co 7353   supp csupp 8100   ↑_m cmap 8760  Fincfn 8879   finSupp cfsupp 9270  0cc0 11028  ℕcn 12147  ℕ₀cn0 12403

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7675  ax-resscn 11085  ax-1cn 11086  ax-icn 11087  ax-addcl 11088  ax-addrcl 11089  ax-mulcl 11090  ax-mulrcl 11091  ax-mulcom 11092  ax-addass 11093  ax-mulass 11094  ax-distr 11095  ax-i2m1 11096  ax-1ne0 11097  ax-1rid 11098  ax-rnegex 11099  ax-rrecex 11100  ax-cnre 11101  ax-pre-lttri 11102  ax-pre-lttrn 11103  ax-pre-ltadd 11104

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-reu 3346  df-rab 3397  df-v 3440  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4862  df-iun 4946  df-br 5096  df-opab 5158  df-mpt 5177  df-tr 5203  df-id 5518  df-eprel 5523  df-po 5531  df-so 5532  df-fr 5576  df-we 5578  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  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 7356  df-oprab 7357  df-mpo 7358  df-om 7807  df-1st 7931  df-2nd 7932  df-supp 8101  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-rdg 8339  df-er 8632  df-map 8762  df-en 8880  df-dom 8881  df-sdom 8882  df-fsupp 9271  df-pnf 11170  df-mnf 11171  df-xr 11172  df-ltxr 11173  df-le 11174  df-nn 12148  df-n0 12404

This theorem is referenced by:  mplvrpmfgalem  33564  mplvrpmga  33565