Theorem fsuppmptif 8580

Description: A function mapping an argument to either a value of a finitely supported function or zero is finitely supported. (Contributed by AV, 6-Jun-2019.)

Hypotheses

Ref	Expression
fsuppmptif.f	⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
fsuppmptif.a	⊢ (𝜑 → 𝐴 ∈ 𝑉)
fsuppmptif.z	⊢ (𝜑 → 𝑍 ∈ 𝑊)
fsuppmptif.s	⊢ (𝜑 → 𝐹 finSupp 𝑍)

Assertion

Ref	Expression
fsuppmptif	⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 𝑍)) finSupp 𝑍)

Distinct variable groups:   𝐴,𝑘   𝑘,𝐹   𝑘,𝑍   𝜑,𝑘

Allowed substitution hints:   𝐵(𝑘)   𝐷(𝑘)   𝑉(𝑘)   𝑊(𝑘)

Proof of Theorem fsuppmptif

Step	Hyp	Ref	Expression
1		fvex 6450	. . . . 5 ⊢ (𝐹‘𝑘) ∈ V
2		fsuppmptif.z	. . . . . 6 ⊢ (𝜑 → 𝑍 ∈ 𝑊)
3	2	adantr 474	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝑍 ∈ 𝑊)
4		ifexg 4355	. . . . 5 ⊢ (((𝐹‘𝑘) ∈ V ∧ 𝑍 ∈ 𝑊) → if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 𝑍) ∈ V)
5	1, 3, 4	sylancr 581	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 𝑍) ∈ V)
6	5	fmpttd 6639	. . 3 ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 𝑍)):𝐴⟶V)
7	6	ffund 6286	. 2 ⊢ (𝜑 → Fun (𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 𝑍)))
8		fsuppmptif.s	. . . 4 ⊢ (𝜑 → 𝐹 finSupp 𝑍)
9	8	fsuppimpd 8557	. . 3 ⊢ (𝜑 → (𝐹 supp 𝑍) ∈ Fin)
10		fsuppmptif.f	. . . . . . 7 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
11		ssidd 3849	. . . . . . 7 ⊢ (𝜑 → (𝐹 supp 𝑍) ⊆ (𝐹 supp 𝑍))
12		fsuppmptif.a	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
13	10, 11, 12, 2	suppssr 7596	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐴 ∖ (𝐹 supp 𝑍))) → (𝐹‘𝑘) = 𝑍)
14	13	ifeq1d 4326	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐴 ∖ (𝐹 supp 𝑍))) → if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 𝑍) = if(𝑘 ∈ 𝐷, 𝑍, 𝑍))
15		ifid 4347	. . . . 5 ⊢ if(𝑘 ∈ 𝐷, 𝑍, 𝑍) = 𝑍
16	14, 15	syl6eq 2877	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐴 ∖ (𝐹 supp 𝑍))) → if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 𝑍) = 𝑍)
17	16, 12	suppss2 7599	. . 3 ⊢ (𝜑 → ((𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 𝑍)) supp 𝑍) ⊆ (𝐹 supp 𝑍))
18		ssfi 8455	. . 3 ⊢ (((𝐹 supp 𝑍) ∈ Fin ∧ ((𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 𝑍)) supp 𝑍) ⊆ (𝐹 supp 𝑍)) → ((𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 𝑍)) supp 𝑍) ∈ Fin)
19	9, 17, 18	syl2anc 579	. 2 ⊢ (𝜑 → ((𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 𝑍)) supp 𝑍) ∈ Fin)
20		mptexg 6745	. . . 4 ⊢ (𝐴 ∈ 𝑉 → (𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 𝑍)) ∈ V)
21	12, 20	syl 17	. . 3 ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 𝑍)) ∈ V)
22		isfsupp 8554	. . 3 ⊢ (((𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 𝑍)) ∈ V ∧ 𝑍 ∈ 𝑊) → ((𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 𝑍)) finSupp 𝑍 ↔ (Fun (𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 𝑍)) ∧ ((𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 𝑍)) supp 𝑍) ∈ Fin)))
23	21, 2, 22	syl2anc 579	. 2 ⊢ (𝜑 → ((𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 𝑍)) finSupp 𝑍 ↔ (Fun (𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 𝑍)) ∧ ((𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 𝑍)) supp 𝑍) ∈ Fin)))
24	7, 19, 23	mpbir2and 704	1 ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 𝑍)) finSupp 𝑍)

Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 386   ∈ wcel 2164  Vcvv 3414   ∖ cdif 3795   ⊆ wss 3798  ifcif 4308   class class class wbr 4875   ↦ cmpt 4954  Fun wfun 6121  ⟶wf 6123  ‘cfv 6127  (class class class)co 6910   supp csupp 7564  Fincfn 8228   finSupp cfsupp 8550

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-rep 4996  ax-sep 5007  ax-nul 5015  ax-pow 5067  ax-pr 5129  ax-un 7214

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3or 1112  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-reu 3124  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-pss 3814  df-nul 4147  df-if 4309  df-pw 4382  df-sn 4400  df-pr 4402  df-tp 4404  df-op 4406  df-uni 4661  df-iun 4744  df-br 4876  df-opab 4938  df-mpt 4955  df-tr 4978  df-id 5252  df-eprel 5257  df-po 5265  df-so 5266  df-fr 5305  df-we 5307  df-xp 5352  df-rel 5353  df-cnv 5354  df-co 5355  df-dm 5356  df-rn 5357  df-res 5358  df-ima 5359  df-ord 5970  df-on 5971  df-lim 5972  df-suc 5973  df-iota 6090  df-fun 6129  df-fn 6130  df-f 6131  df-f1 6132  df-fo 6133  df-f1o 6134  df-fv 6135  df-ov 6913  df-oprab 6914  df-mpt2 6915  df-om 7332  df-supp 7565  df-er 8014  df-en 8229  df-fin 8232  df-fsupp 8551

This theorem is referenced by:  cantnflem1d  8869  gsumzsplit  18687