Theorem mptiffisupp 32705

Description: Conditions for a mapping function defined with a conditional to have finite support. (Contributed by Thierry Arnoux, 20-Feb-2025.)

Hypotheses

Ref	Expression
mptiffisupp.f	⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ if(𝑥 ∈ 𝐵, 𝐶, 𝑍))
mptiffisupp.a	⊢ (𝜑 → 𝐴 ∈ 𝑈)
mptiffisupp.b	⊢ (𝜑 → 𝐵 ∈ Fin)
mptiffisupp.c	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝐶 ∈ 𝑉)
mptiffisupp.z	⊢ (𝜑 → 𝑍 ∈ 𝑊)

Assertion

Ref	Expression
mptiffisupp	⊢ (𝜑 → 𝐹 finSupp 𝑍)

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝑉   𝑥,𝑍   𝜑,𝑥

Allowed substitution hints:   𝐶(𝑥)   𝑈(𝑥)   𝐹(𝑥)   𝑊(𝑥)

Proof of Theorem mptiffisupp

Step	Hyp	Ref	Expression
1		mptiffisupp.f	. . 3 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ if(𝑥 ∈ 𝐵, 𝐶, 𝑍))
2		mptiffisupp.a	. . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑈)
3	2	mptexd 7261	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ if(𝑥 ∈ 𝐵, 𝐶, 𝑍)) ∈ V)
4	1, 3	eqeltrid 2848	. 2 ⊢ (𝜑 → 𝐹 ∈ V)
5		mptiffisupp.z	. 2 ⊢ (𝜑 → 𝑍 ∈ 𝑊)
6	1	funmpt2 6617	. . 3 ⊢ Fun 𝐹
7	6	a1i 11	. 2 ⊢ (𝜑 → Fun 𝐹)
8		partfun 6727	. . . . 5 ⊢ (𝑥 ∈ 𝐴 ↦ if(𝑥 ∈ 𝐵, 𝐶, 𝑍)) = ((𝑥 ∈ (𝐴 ∩ 𝐵) ↦ 𝐶) ∪ (𝑥 ∈ (𝐴 ∖ 𝐵) ↦ 𝑍))
9	1, 8	eqtri 2768	. . . 4 ⊢ 𝐹 = ((𝑥 ∈ (𝐴 ∩ 𝐵) ↦ 𝐶) ∪ (𝑥 ∈ (𝐴 ∖ 𝐵) ↦ 𝑍))
10	9	oveq1i 7458	. . 3 ⊢ (𝐹 supp 𝑍) = (((𝑥 ∈ (𝐴 ∩ 𝐵) ↦ 𝐶) ∪ (𝑥 ∈ (𝐴 ∖ 𝐵) ↦ 𝑍)) supp 𝑍)
11		inss2 4259	. . . . . . . . 9 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐵
12	11	a1i 11	. . . . . . . 8 ⊢ (𝜑 → (𝐴 ∩ 𝐵) ⊆ 𝐵)
13	12	sselda 4008	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∩ 𝐵)) → 𝑥 ∈ 𝐵)
14		mptiffisupp.c	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝐶 ∈ 𝑉)
15	13, 14	syldan 590	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∩ 𝐵)) → 𝐶 ∈ 𝑉)
16	15	fmpttd 7149	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ (𝐴 ∩ 𝐵) ↦ 𝐶):(𝐴 ∩ 𝐵)⟶𝑉)
17		incom 4230	. . . . . 6 ⊢ (𝐵 ∩ 𝐴) = (𝐴 ∩ 𝐵)
18		mptiffisupp.b	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ Fin)
19		infi 9330	. . . . . . 7 ⊢ (𝐵 ∈ Fin → (𝐵 ∩ 𝐴) ∈ Fin)
20	18, 19	syl 17	. . . . . 6 ⊢ (𝜑 → (𝐵 ∩ 𝐴) ∈ Fin)
21	17, 20	eqeltrrid 2849	. . . . 5 ⊢ (𝜑 → (𝐴 ∩ 𝐵) ∈ Fin)
22	16, 21, 5	fidmfisupp 9442	. . . 4 ⊢ (𝜑 → (𝑥 ∈ (𝐴 ∩ 𝐵) ↦ 𝐶) finSupp 𝑍)
23		difexg 5347	. . . . . 6 ⊢ (𝐴 ∈ 𝑈 → (𝐴 ∖ 𝐵) ∈ V)
24		mptexg 7258	. . . . . 6 ⊢ ((𝐴 ∖ 𝐵) ∈ V → (𝑥 ∈ (𝐴 ∖ 𝐵) ↦ 𝑍) ∈ V)
25	2, 23, 24	3syl 18	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ (𝐴 ∖ 𝐵) ↦ 𝑍) ∈ V)
26		funmpt 6616	. . . . . 6 ⊢ Fun (𝑥 ∈ (𝐴 ∖ 𝐵) ↦ 𝑍)
27	26	a1i 11	. . . . 5 ⊢ (𝜑 → Fun (𝑥 ∈ (𝐴 ∖ 𝐵) ↦ 𝑍))
28		supppreima 32703	. . . . . . . 8 ⊢ ((Fun (𝑥 ∈ (𝐴 ∖ 𝐵) ↦ 𝑍) ∧ (𝑥 ∈ (𝐴 ∖ 𝐵) ↦ 𝑍) ∈ V ∧ 𝑍 ∈ 𝑊) → ((𝑥 ∈ (𝐴 ∖ 𝐵) ↦ 𝑍) supp 𝑍) = (◡(𝑥 ∈ (𝐴 ∖ 𝐵) ↦ 𝑍) “ (ran (𝑥 ∈ (𝐴 ∖ 𝐵) ↦ 𝑍) ∖ {𝑍})))
29	26, 25, 5, 28	mp3an2i 1466	. . . . . . 7 ⊢ (𝜑 → ((𝑥 ∈ (𝐴 ∖ 𝐵) ↦ 𝑍) supp 𝑍) = (◡(𝑥 ∈ (𝐴 ∖ 𝐵) ↦ 𝑍) “ (ran (𝑥 ∈ (𝐴 ∖ 𝐵) ↦ 𝑍) ∖ {𝑍})))
30		simpr 484	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝐴 ∖ 𝐵) = ∅) → (𝐴 ∖ 𝐵) = ∅)
31	30	mpteq1d 5261	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝐴 ∖ 𝐵) = ∅) → (𝑥 ∈ (𝐴 ∖ 𝐵) ↦ 𝑍) = (𝑥 ∈ ∅ ↦ 𝑍))
32		mpt0 6722	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ∅ ↦ 𝑍) = ∅
33	31, 32	eqtrdi 2796	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝐴 ∖ 𝐵) = ∅) → (𝑥 ∈ (𝐴 ∖ 𝐵) ↦ 𝑍) = ∅)
34	33	cnveqd 5900	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝐴 ∖ 𝐵) = ∅) → ◡(𝑥 ∈ (𝐴 ∖ 𝐵) ↦ 𝑍) = ◡∅)
35		cnv0 6172	. . . . . . . . . . 11 ⊢ ◡∅ = ∅
36	34, 35	eqtrdi 2796	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝐴 ∖ 𝐵) = ∅) → ◡(𝑥 ∈ (𝐴 ∖ 𝐵) ↦ 𝑍) = ∅)
37	36	imaeq1d 6088	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝐴 ∖ 𝐵) = ∅) → (◡(𝑥 ∈ (𝐴 ∖ 𝐵) ↦ 𝑍) “ (ran (𝑥 ∈ (𝐴 ∖ 𝐵) ↦ 𝑍) ∖ {𝑍})) = (∅ “ (ran (𝑥 ∈ (𝐴 ∖ 𝐵) ↦ 𝑍) ∖ {𝑍})))
38		0ima 6107	. . . . . . . . 9 ⊢ (∅ “ (ran (𝑥 ∈ (𝐴 ∖ 𝐵) ↦ 𝑍) ∖ {𝑍})) = ∅
39	37, 38	eqtrdi 2796	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝐴 ∖ 𝐵) = ∅) → (◡(𝑥 ∈ (𝐴 ∖ 𝐵) ↦ 𝑍) “ (ran (𝑥 ∈ (𝐴 ∖ 𝐵) ↦ 𝑍) ∖ {𝑍})) = ∅)
40		eqid 2740	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ (𝐴 ∖ 𝐵) ↦ 𝑍) = (𝑥 ∈ (𝐴 ∖ 𝐵) ↦ 𝑍)
41		simpr 484	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝐴 ∖ 𝐵) ≠ ∅) → (𝐴 ∖ 𝐵) ≠ ∅)
42	40, 41	rnmptc 7244	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝐴 ∖ 𝐵) ≠ ∅) → ran (𝑥 ∈ (𝐴 ∖ 𝐵) ↦ 𝑍) = {𝑍})
43	42	difeq1d 4148	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝐴 ∖ 𝐵) ≠ ∅) → (ran (𝑥 ∈ (𝐴 ∖ 𝐵) ↦ 𝑍) ∖ {𝑍}) = ({𝑍} ∖ {𝑍}))
44		difid 4398	. . . . . . . . . . 11 ⊢ ({𝑍} ∖ {𝑍}) = ∅
45	43, 44	eqtrdi 2796	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝐴 ∖ 𝐵) ≠ ∅) → (ran (𝑥 ∈ (𝐴 ∖ 𝐵) ↦ 𝑍) ∖ {𝑍}) = ∅)
46	45	imaeq2d 6089	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝐴 ∖ 𝐵) ≠ ∅) → (◡(𝑥 ∈ (𝐴 ∖ 𝐵) ↦ 𝑍) “ (ran (𝑥 ∈ (𝐴 ∖ 𝐵) ↦ 𝑍) ∖ {𝑍})) = (◡(𝑥 ∈ (𝐴 ∖ 𝐵) ↦ 𝑍) “ ∅))
47		ima0 6106	. . . . . . . . 9 ⊢ (◡(𝑥 ∈ (𝐴 ∖ 𝐵) ↦ 𝑍) “ ∅) = ∅
48	46, 47	eqtrdi 2796	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝐴 ∖ 𝐵) ≠ ∅) → (◡(𝑥 ∈ (𝐴 ∖ 𝐵) ↦ 𝑍) “ (ran (𝑥 ∈ (𝐴 ∖ 𝐵) ↦ 𝑍) ∖ {𝑍})) = ∅)
49	39, 48	pm2.61dane 3035	. . . . . . 7 ⊢ (𝜑 → (◡(𝑥 ∈ (𝐴 ∖ 𝐵) ↦ 𝑍) “ (ran (𝑥 ∈ (𝐴 ∖ 𝐵) ↦ 𝑍) ∖ {𝑍})) = ∅)
50	29, 49	eqtrd 2780	. . . . . 6 ⊢ (𝜑 → ((𝑥 ∈ (𝐴 ∖ 𝐵) ↦ 𝑍) supp 𝑍) = ∅)
51		0fi 9108	. . . . . 6 ⊢ ∅ ∈ Fin
52	50, 51	eqeltrdi 2852	. . . . 5 ⊢ (𝜑 → ((𝑥 ∈ (𝐴 ∖ 𝐵) ↦ 𝑍) supp 𝑍) ∈ Fin)
53	25, 5, 27, 52	isfsuppd 9436	. . . 4 ⊢ (𝜑 → (𝑥 ∈ (𝐴 ∖ 𝐵) ↦ 𝑍) finSupp 𝑍)
54	22, 53	fsuppun 9456	. . 3 ⊢ (𝜑 → (((𝑥 ∈ (𝐴 ∩ 𝐵) ↦ 𝐶) ∪ (𝑥 ∈ (𝐴 ∖ 𝐵) ↦ 𝑍)) supp 𝑍) ∈ Fin)
55	10, 54	eqeltrid 2848	. 2 ⊢ (𝜑 → (𝐹 supp 𝑍) ∈ Fin)
56	4, 5, 7, 55	isfsuppd 9436	1 ⊢ (𝜑 → 𝐹 finSupp 𝑍)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1537   ∈ wcel 2108   ≠ wne 2946  Vcvv 3488   ∖ cdif 3973   ∪ cun 3974   ∩ cin 3975   ⊆ wss 3976  ∅c0 4352  ifcif 4548  {csn 4648   class class class wbr 5166   ↦ cmpt 5249  ^◡ccnv 5699  ran crn 5701   “ cima 5703  Fun wfun 6567  (class class class)co 7448   supp csupp 8201  Fincfn 9003   finSupp cfsupp 9431

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-ov 7451  df-oprab 7452  df-mpo 7453  df-om 7904  df-supp 8202  df-1o 8522  df-en 9004  df-fin 9007  df-fsupp 9432

This theorem is referenced by:  elrspunsn  33422  gsummoncoe1fzo  33583