Theorem mptiffisupp 32892

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 7208	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ if(𝑥 ∈ 𝐵, 𝐶, 𝑍)) ∈ V)
4	1, 3	eqeltrid 2866	. 2 ⊢ (𝜑 → 𝐹 ∈ V)
5		mptiffisupp.z	. 2 ⊢ (𝜑 → 𝑍 ∈ 𝑊)
6	1	funmpt2 6560	. . 3 ⊢ Fun 𝐹
7	6	a1i 11	. 2 ⊢ (𝜑 → Fun 𝐹)
8		partfun 6668	. . . . 5 ⊢ (𝑥 ∈ 𝐴 ↦ if(𝑥 ∈ 𝐵, 𝐶, 𝑍)) = ((𝑥 ∈ (𝐴 ∩ 𝐵) ↦ 𝐶) ∪ (𝑥 ∈ (𝐴 ∖ 𝐵) ↦ 𝑍))
9	1, 8	eqtri 2785	. . . 4 ⊢ 𝐹 = ((𝑥 ∈ (𝐴 ∩ 𝐵) ↦ 𝐶) ∪ (𝑥 ∈ (𝐴 ∖ 𝐵) ↦ 𝑍))
10	9	oveq1i 7406	. . 3 ⊢ (𝐹 supp 𝑍) = (((𝑥 ∈ (𝐴 ∩ 𝐵) ↦ 𝐶) ∪ (𝑥 ∈ (𝐴 ∖ 𝐵) ↦ 𝑍)) supp 𝑍)
11		inss2 4189	. . . . . . . . 9 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐵
12	11	a1i 11	. . . . . . . 8 ⊢ (𝜑 → (𝐴 ∩ 𝐵) ⊆ 𝐵)
13	12	sselda 3936	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∩ 𝐵)) → 𝑥 ∈ 𝐵)
14		mptiffisupp.c	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝐶 ∈ 𝑉)
15	13, 14	syldan 600	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∩ 𝐵)) → 𝐶 ∈ 𝑉)
16	15	fmpttd 7096	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ (𝐴 ∩ 𝐵) ↦ 𝐶):(𝐴 ∩ 𝐵)⟶𝑉)
17		incom 4161	. . . . . 6 ⊢ (𝐵 ∩ 𝐴) = (𝐴 ∩ 𝐵)
18		mptiffisupp.b	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ Fin)
19		infi 9214	. . . . . . 7 ⊢ (𝐵 ∈ Fin → (𝐵 ∩ 𝐴) ∈ Fin)
20	18, 19	syl 17	. . . . . 6 ⊢ (𝜑 → (𝐵 ∩ 𝐴) ∈ Fin)
21	17, 20	eqeltrrid 2867	. . . . 5 ⊢ (𝜑 → (𝐴 ∩ 𝐵) ∈ Fin)
22	16, 21, 5	fidmfisupp 9318	. . . 4 ⊢ (𝜑 → (𝑥 ∈ (𝐴 ∩ 𝐵) ↦ 𝐶) finSupp 𝑍)
23		difexg 5285	. . . . . 6 ⊢ (𝐴 ∈ 𝑈 → (𝐴 ∖ 𝐵) ∈ V)
24		mptexg 7205	. . . . . 6 ⊢ ((𝐴 ∖ 𝐵) ∈ V → (𝑥 ∈ (𝐴 ∖ 𝐵) ↦ 𝑍) ∈ V)
25	2, 23, 24	3syl 18	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ (𝐴 ∖ 𝐵) ↦ 𝑍) ∈ V)
26		funmpt 6559	. . . . . 6 ⊢ Fun (𝑥 ∈ (𝐴 ∖ 𝐵) ↦ 𝑍)
27	26	a1i 11	. . . . 5 ⊢ (𝜑 → Fun (𝑥 ∈ (𝐴 ∖ 𝐵) ↦ 𝑍))
28		supppreima 32890	. . . . . . . 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 1487	. . . . . . 7 ⊢ (𝜑 → ((𝑥 ∈ (𝐴 ∖ 𝐵) ↦ 𝑍) supp 𝑍) = (◡(𝑥 ∈ (𝐴 ∖ 𝐵) ↦ 𝑍) “ (ran (𝑥 ∈ (𝐴 ∖ 𝐵) ↦ 𝑍) ∖ {𝑍})))
30		simpr 488	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝐴 ∖ 𝐵) = ∅) → (𝐴 ∖ 𝐵) = ∅)
31	30	mpteq1d 5190	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝐴 ∖ 𝐵) = ∅) → (𝑥 ∈ (𝐴 ∖ 𝐵) ↦ 𝑍) = (𝑥 ∈ ∅ ↦ 𝑍))
32		mpt0 6663	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ∅ ↦ 𝑍) = ∅
33	31, 32	eqtrdi 2813	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝐴 ∖ 𝐵) = ∅) → (𝑥 ∈ (𝐴 ∖ 𝐵) ↦ 𝑍) = ∅)
34	33	cnveqd 5847	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝐴 ∖ 𝐵) = ∅) → ◡(𝑥 ∈ (𝐴 ∖ 𝐵) ↦ 𝑍) = ◡∅)
35		cnv0 5855	. . . . . . . . . . 11 ⊢ ◡∅ = ∅
36	34, 35	eqtrdi 2813	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝐴 ∖ 𝐵) = ∅) → ◡(𝑥 ∈ (𝐴 ∖ 𝐵) ↦ 𝑍) = ∅)
37	36	imaeq1d 6048	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝐴 ∖ 𝐵) = ∅) → (◡(𝑥 ∈ (𝐴 ∖ 𝐵) ↦ 𝑍) “ (ran (𝑥 ∈ (𝐴 ∖ 𝐵) ↦ 𝑍) ∖ {𝑍})) = (∅ “ (ran (𝑥 ∈ (𝐴 ∖ 𝐵) ↦ 𝑍) ∖ {𝑍})))
38		0ima 6067	. . . . . . . . 9 ⊢ (∅ “ (ran (𝑥 ∈ (𝐴 ∖ 𝐵) ↦ 𝑍) ∖ {𝑍})) = ∅
39	37, 38	eqtrdi 2813	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝐴 ∖ 𝐵) = ∅) → (◡(𝑥 ∈ (𝐴 ∖ 𝐵) ↦ 𝑍) “ (ran (𝑥 ∈ (𝐴 ∖ 𝐵) ↦ 𝑍) ∖ {𝑍})) = ∅)
40		eqid 2762	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ (𝐴 ∖ 𝐵) ↦ 𝑍) = (𝑥 ∈ (𝐴 ∖ 𝐵) ↦ 𝑍)
41		simpr 488	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝐴 ∖ 𝐵) ≠ ∅) → (𝐴 ∖ 𝐵) ≠ ∅)
42	40, 41	rnmptc 7191	. . . . . . . . . . . 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 4329	. . . . . . . . . . 11 ⊢ ({𝑍} ∖ {𝑍}) = ∅
45	43, 44	eqtrdi 2813	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝐴 ∖ 𝐵) ≠ ∅) → (ran (𝑥 ∈ (𝐴 ∖ 𝐵) ↦ 𝑍) ∖ {𝑍}) = ∅)
46	45	imaeq2d 6049	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝐴 ∖ 𝐵) ≠ ∅) → (◡(𝑥 ∈ (𝐴 ∖ 𝐵) ↦ 𝑍) “ (ran (𝑥 ∈ (𝐴 ∖ 𝐵) ↦ 𝑍) ∖ {𝑍})) = (◡(𝑥 ∈ (𝐴 ∖ 𝐵) ↦ 𝑍) “ ∅))
47		ima0 6066	. . . . . . . . 9 ⊢ (◡(𝑥 ∈ (𝐴 ∖ 𝐵) ↦ 𝑍) “ ∅) = ∅
48	46, 47	eqtrdi 2813	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝐴 ∖ 𝐵) ≠ ∅) → (◡(𝑥 ∈ (𝐴 ∖ 𝐵) ↦ 𝑍) “ (ran (𝑥 ∈ (𝐴 ∖ 𝐵) ↦ 𝑍) ∖ {𝑍})) = ∅)
49	39, 48	pm2.61dane 3044	. . . . . . 7 ⊢ (𝜑 → (◡(𝑥 ∈ (𝐴 ∖ 𝐵) ↦ 𝑍) “ (ran (𝑥 ∈ (𝐴 ∖ 𝐵) ↦ 𝑍) ∖ {𝑍})) = ∅)
50	29, 49	eqtrd 2797	. . . . . 6 ⊢ (𝜑 → ((𝑥 ∈ (𝐴 ∖ 𝐵) ↦ 𝑍) supp 𝑍) = ∅)
51		0fi 9023	. . . . . 6 ⊢ ∅ ∈ Fin
52	50, 51	eqeltrdi 2870	. . . . 5 ⊢ (𝜑 → ((𝑥 ∈ (𝐴 ∖ 𝐵) ↦ 𝑍) supp 𝑍) ∈ Fin)
53	25, 5, 27, 52	isfsuppd 9312	. . . 4 ⊢ (𝜑 → (𝑥 ∈ (𝐴 ∖ 𝐵) ↦ 𝑍) finSupp 𝑍)
54	22, 53	fsuppun 9333	. . 3 ⊢ (𝜑 → (((𝑥 ∈ (𝐴 ∩ 𝐵) ↦ 𝐶) ∪ (𝑥 ∈ (𝐴 ∖ 𝐵) ↦ 𝑍)) supp 𝑍) ∈ Fin)
55	10, 54	eqeltrid 2866	. 2 ⊢ (𝜑 → (𝐹 supp 𝑍) ∈ Fin)
56	4, 5, 7, 55	isfsuppd 9312	1 ⊢ (𝜑 → 𝐹 finSupp 𝑍)

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1560   ∈ wcel 2142   ≠ wne 2957  Vcvv 3454   ∖ cdif 3901   ∪ cun 3902   ∩ cin 3903   ⊆ wss 3904  ∅c0 4285  ifcif 4480  {csn 4582   class class class wbr 5100   ↦ cmpt 5181  ^◡ccnv 5646  ran crn 5648   “ cima 5650  Fun wfun 6515  (class class class)co 7396   supp csupp 8140  Fincfn 8927   finSupp cfsupp 9307

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-rep 5227  ax-sep 5246  ax-nul 5256  ax-pr 5390  ax-un 7718

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1099  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-nf 1804  df-sb 2091  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3077  df-rex 3087  df-reu 3368  df-rab 3415  df-v 3456  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4481  df-pw 4557  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4951  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5542  df-eprel 5547  df-po 5555  df-so 5556  df-fr 5600  df-we 5602  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-ord 6349  df-on 6350  df-lim 6351  df-suc 6352  df-iota 6477  df-fun 6523  df-fn 6524  df-f 6525  df-f1 6526  df-fo 6527  df-f1o 6528  df-fv 6529  df-ov 7399  df-oprab 7400  df-mpo 7401  df-om 7847  df-supp 8141  df-1o 8437  df-en 8928  df-fin 8931  df-fsupp 9308

This theorem is referenced by:  elrspunsn  33612  gsummoncoe1fzo  33790  mplmonprod  33848  extdgfialglem2  33987