Theorem mptiffisupp 32783

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 7180	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ if(𝑥 ∈ 𝐵, 𝐶, 𝑍)) ∈ V)
4	1, 3	eqeltrid 2841	. 2 ⊢ (𝜑 → 𝐹 ∈ V)
5		mptiffisupp.z	. 2 ⊢ (𝜑 → 𝑍 ∈ 𝑊)
6	1	funmpt2 6539	. . 3 ⊢ Fun 𝐹
7	6	a1i 11	. 2 ⊢ (𝜑 → Fun 𝐹)
8		partfun 6647	. . . . 5 ⊢ (𝑥 ∈ 𝐴 ↦ if(𝑥 ∈ 𝐵, 𝐶, 𝑍)) = ((𝑥 ∈ (𝐴 ∩ 𝐵) ↦ 𝐶) ∪ (𝑥 ∈ (𝐴 ∖ 𝐵) ↦ 𝑍))
9	1, 8	eqtri 2760	. . . 4 ⊢ 𝐹 = ((𝑥 ∈ (𝐴 ∩ 𝐵) ↦ 𝐶) ∪ (𝑥 ∈ (𝐴 ∖ 𝐵) ↦ 𝑍))
10	9	oveq1i 7378	. . 3 ⊢ (𝐹 supp 𝑍) = (((𝑥 ∈ (𝐴 ∩ 𝐵) ↦ 𝐶) ∪ (𝑥 ∈ (𝐴 ∖ 𝐵) ↦ 𝑍)) supp 𝑍)
11		inss2 4192	. . . . . . . . 9 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐵
12	11	a1i 11	. . . . . . . 8 ⊢ (𝜑 → (𝐴 ∩ 𝐵) ⊆ 𝐵)
13	12	sselda 3935	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∩ 𝐵)) → 𝑥 ∈ 𝐵)
14		mptiffisupp.c	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝐶 ∈ 𝑉)
15	13, 14	syldan 592	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∩ 𝐵)) → 𝐶 ∈ 𝑉)
16	15	fmpttd 7069	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ (𝐴 ∩ 𝐵) ↦ 𝐶):(𝐴 ∩ 𝐵)⟶𝑉)
17		incom 4163	. . . . . 6 ⊢ (𝐵 ∩ 𝐴) = (𝐴 ∩ 𝐵)
18		mptiffisupp.b	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ Fin)
19		infi 9182	. . . . . . 7 ⊢ (𝐵 ∈ Fin → (𝐵 ∩ 𝐴) ∈ Fin)
20	18, 19	syl 17	. . . . . 6 ⊢ (𝜑 → (𝐵 ∩ 𝐴) ∈ Fin)
21	17, 20	eqeltrrid 2842	. . . . 5 ⊢ (𝜑 → (𝐴 ∩ 𝐵) ∈ Fin)
22	16, 21, 5	fidmfisupp 9287	. . . 4 ⊢ (𝜑 → (𝑥 ∈ (𝐴 ∩ 𝐵) ↦ 𝐶) finSupp 𝑍)
23		difexg 5276	. . . . . 6 ⊢ (𝐴 ∈ 𝑈 → (𝐴 ∖ 𝐵) ∈ V)
24		mptexg 7177	. . . . . 6 ⊢ ((𝐴 ∖ 𝐵) ∈ V → (𝑥 ∈ (𝐴 ∖ 𝐵) ↦ 𝑍) ∈ V)
25	2, 23, 24	3syl 18	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ (𝐴 ∖ 𝐵) ↦ 𝑍) ∈ V)
26		funmpt 6538	. . . . . 6 ⊢ Fun (𝑥 ∈ (𝐴 ∖ 𝐵) ↦ 𝑍)
27	26	a1i 11	. . . . 5 ⊢ (𝜑 → Fun (𝑥 ∈ (𝐴 ∖ 𝐵) ↦ 𝑍))
28		supppreima 32781	. . . . . . . 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 1469	. . . . . . 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 5190	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝐴 ∖ 𝐵) = ∅) → (𝑥 ∈ (𝐴 ∖ 𝐵) ↦ 𝑍) = (𝑥 ∈ ∅ ↦ 𝑍))
32		mpt0 6642	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ∅ ↦ 𝑍) = ∅
33	31, 32	eqtrdi 2788	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝐴 ∖ 𝐵) = ∅) → (𝑥 ∈ (𝐴 ∖ 𝐵) ↦ 𝑍) = ∅)
34	33	cnveqd 5832	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝐴 ∖ 𝐵) = ∅) → ◡(𝑥 ∈ (𝐴 ∖ 𝐵) ↦ 𝑍) = ◡∅)
35		cnv0 6105	. . . . . . . . . . 11 ⊢ ◡∅ = ∅
36	34, 35	eqtrdi 2788	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝐴 ∖ 𝐵) = ∅) → ◡(𝑥 ∈ (𝐴 ∖ 𝐵) ↦ 𝑍) = ∅)
37	36	imaeq1d 6026	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝐴 ∖ 𝐵) = ∅) → (◡(𝑥 ∈ (𝐴 ∖ 𝐵) ↦ 𝑍) “ (ran (𝑥 ∈ (𝐴 ∖ 𝐵) ↦ 𝑍) ∖ {𝑍})) = (∅ “ (ran (𝑥 ∈ (𝐴 ∖ 𝐵) ↦ 𝑍) ∖ {𝑍})))
38		0ima 6045	. . . . . . . . 9 ⊢ (∅ “ (ran (𝑥 ∈ (𝐴 ∖ 𝐵) ↦ 𝑍) ∖ {𝑍})) = ∅
39	37, 38	eqtrdi 2788	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝐴 ∖ 𝐵) = ∅) → (◡(𝑥 ∈ (𝐴 ∖ 𝐵) ↦ 𝑍) “ (ran (𝑥 ∈ (𝐴 ∖ 𝐵) ↦ 𝑍) ∖ {𝑍})) = ∅)
40		eqid 2737	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ (𝐴 ∖ 𝐵) ↦ 𝑍) = (𝑥 ∈ (𝐴 ∖ 𝐵) ↦ 𝑍)
41		simpr 484	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝐴 ∖ 𝐵) ≠ ∅) → (𝐴 ∖ 𝐵) ≠ ∅)
42	40, 41	rnmptc 7163	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝐴 ∖ 𝐵) ≠ ∅) → ran (𝑥 ∈ (𝐴 ∖ 𝐵) ↦ 𝑍) = {𝑍})
43	42	difeq1d 4079	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝐴 ∖ 𝐵) ≠ ∅) → (ran (𝑥 ∈ (𝐴 ∖ 𝐵) ↦ 𝑍) ∖ {𝑍}) = ({𝑍} ∖ {𝑍}))
44		difid 4330	. . . . . . . . . . 11 ⊢ ({𝑍} ∖ {𝑍}) = ∅
45	43, 44	eqtrdi 2788	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝐴 ∖ 𝐵) ≠ ∅) → (ran (𝑥 ∈ (𝐴 ∖ 𝐵) ↦ 𝑍) ∖ {𝑍}) = ∅)
46	45	imaeq2d 6027	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝐴 ∖ 𝐵) ≠ ∅) → (◡(𝑥 ∈ (𝐴 ∖ 𝐵) ↦ 𝑍) “ (ran (𝑥 ∈ (𝐴 ∖ 𝐵) ↦ 𝑍) ∖ {𝑍})) = (◡(𝑥 ∈ (𝐴 ∖ 𝐵) ↦ 𝑍) “ ∅))
47		ima0 6044	. . . . . . . . 9 ⊢ (◡(𝑥 ∈ (𝐴 ∖ 𝐵) ↦ 𝑍) “ ∅) = ∅
48	46, 47	eqtrdi 2788	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝐴 ∖ 𝐵) ≠ ∅) → (◡(𝑥 ∈ (𝐴 ∖ 𝐵) ↦ 𝑍) “ (ran (𝑥 ∈ (𝐴 ∖ 𝐵) ↦ 𝑍) ∖ {𝑍})) = ∅)
49	39, 48	pm2.61dane 3020	. . . . . . 7 ⊢ (𝜑 → (◡(𝑥 ∈ (𝐴 ∖ 𝐵) ↦ 𝑍) “ (ran (𝑥 ∈ (𝐴 ∖ 𝐵) ↦ 𝑍) ∖ {𝑍})) = ∅)
50	29, 49	eqtrd 2772	. . . . . 6 ⊢ (𝜑 → ((𝑥 ∈ (𝐴 ∖ 𝐵) ↦ 𝑍) supp 𝑍) = ∅)
51		0fi 8991	. . . . . 6 ⊢ ∅ ∈ Fin
52	50, 51	eqeltrdi 2845	. . . . 5 ⊢ (𝜑 → ((𝑥 ∈ (𝐴 ∖ 𝐵) ↦ 𝑍) supp 𝑍) ∈ Fin)
53	25, 5, 27, 52	isfsuppd 9281	. . . 4 ⊢ (𝜑 → (𝑥 ∈ (𝐴 ∖ 𝐵) ↦ 𝑍) finSupp 𝑍)
54	22, 53	fsuppun 9302	. . 3 ⊢ (𝜑 → (((𝑥 ∈ (𝐴 ∩ 𝐵) ↦ 𝐶) ∪ (𝑥 ∈ (𝐴 ∖ 𝐵) ↦ 𝑍)) supp 𝑍) ∈ Fin)
55	10, 54	eqeltrid 2841	. 2 ⊢ (𝜑 → (𝐹 supp 𝑍) ∈ Fin)
56	4, 5, 7, 55	isfsuppd 9281	1 ⊢ (𝜑 → 𝐹 finSupp 𝑍)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114   ≠ wne 2933  Vcvv 3442   ∖ cdif 3900   ∪ cun 3901   ∩ cin 3902   ⊆ wss 3903  ∅c0 4287  ifcif 4481  {csn 4582   class class class wbr 5100   ↦ cmpt 5181  ^◡ccnv 5631  ran crn 5633   “ cima 5635  Fun wfun 6494  (class class class)co 7368   supp csupp 8112  Fincfn 8895   finSupp cfsupp 9276

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 2709  ax-rep 5226  ax-sep 5243  ax-nul 5253  ax-pr 5379  ax-un 7690

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 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5527  df-eprel 5532  df-po 5540  df-so 5541  df-fr 5585  df-we 5587  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-ord 6328  df-on 6329  df-lim 6330  df-suc 6331  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-ov 7371  df-oprab 7372  df-mpo 7373  df-om 7819  df-supp 8113  df-1o 8407  df-en 8896  df-fin 8899  df-fsupp 9277

This theorem is referenced by:  elrspunsn  33522  gsummoncoe1fzo  33690  mplmonprod  33731  extdgfialglem2  33871