Theorem mptiffisupp 33383

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 2868	. 2 ⊢ (𝜑 → 𝐹 ∈ V)
5		mptiffisupp.z	. 2 ⊢ (𝜑 → 𝑍 ∈ 𝑊)
6	1	funmpt2 6607	. . 3 ⊢ Fun 𝐹
7	6	a1i 11	. 2 ⊢ (𝜑 → Fun 𝐹)
8		partfun 6716	. . . . 5 ⊢ (𝑥 ∈ 𝐴 ↦ if(𝑥 ∈ 𝐵, 𝐶, 𝑍)) = ((𝑥 ∈ (𝐴 ∩ 𝐵) ↦ 𝐶) ∪ (𝑥 ∈ (𝐴 ∖ 𝐵) ↦ 𝑍))
9	1, 8	eqtri 2786	. . . 4 ⊢ 𝐹 = ((𝑥 ∈ (𝐴 ∩ 𝐵) ↦ 𝐶) ∪ (𝑥 ∈ (𝐴 ∖ 𝐵) ↦ 𝑍))
10	9	oveq1i 7461	. . 3 ⊢ (𝐹 supp 𝑍) = (((𝑥 ∈ (𝐴 ∩ 𝐵) ↦ 𝐶) ∪ (𝑥 ∈ (𝐴 ∖ 𝐵) ↦ 𝑍)) supp 𝑍)
11		inss2 4137	. . . . . . . . 9 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐵
12	11	a1i 11	. . . . . . . 8 ⊢ (𝜑 → (𝐴 ∩ 𝐵) ⊆ 𝐵)
13	12	sselda 3913	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∩ 𝐵)) → 𝑥 ∈ 𝐵)
14		mptiffisupp.c	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝐶 ∈ 𝑉)
15	13, 14	syldan 603	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∩ 𝐵)) → 𝐶 ∈ 𝑉)
16	15	fmpttd 7147	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ (𝐴 ∩ 𝐵) ↦ 𝐶):(𝐴 ∩ 𝐵)⟶𝑉)
17		incom 4109	. . . . . 6 ⊢ (𝐵 ∩ 𝐴) = (𝐴 ∩ 𝐵)
18		mptiffisupp.b	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ Fin)
19		infi 9299	. . . . . . 7 ⊢ (𝐵 ∈ Fin → (𝐵 ∩ 𝐴) ∈ Fin)
20	18, 19	syl 18	. . . . . 6 ⊢ (𝜑 → (𝐵 ∩ 𝐴) ∈ Fin)
21	17, 20	eqeltrrid 2869	. . . . 5 ⊢ (𝜑 → (𝐴 ∩ 𝐵) ∈ Fin)
22	16, 21, 5	fdmfifsupp 9404	. . . 4 ⊢ (𝜑 → (𝑥 ∈ (𝐴 ∩ 𝐵) ↦ 𝐶) finSupp 𝑍)
23		difexg 5298	. . . . . 6 ⊢ (𝐴 ∈ 𝑈 → (𝐴 ∖ 𝐵) ∈ V)
24		mptexg 7258	. . . . . 6 ⊢ ((𝐴 ∖ 𝐵) ∈ V → (𝑥 ∈ (𝐴 ∖ 𝐵) ↦ 𝑍) ∈ V)
25	2, 23, 24	3syl 19	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ (𝐴 ∖ 𝐵) ↦ 𝑍) ∈ V)
26		funmpt 6606	. . . . . 6 ⊢ Fun (𝑥 ∈ (𝐴 ∖ 𝐵) ↦ 𝑍)
27	26	a1i 11	. . . . 5 ⊢ (𝜑 → Fun (𝑥 ∈ (𝐴 ∖ 𝐵) ↦ 𝑍))
28		supppreima 33381	. . . . . . . 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 1496	. . . . . . 7 ⊢ (𝜑 → ((𝑥 ∈ (𝐴 ∖ 𝐵) ↦ 𝑍) supp 𝑍) = (◡(𝑥 ∈ (𝐴 ∖ 𝐵) ↦ 𝑍) “ (ran (𝑥 ∈ (𝐴 ∖ 𝐵) ↦ 𝑍) ∖ {𝑍})))
30		simpr 490	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝐴 ∖ 𝐵) = ∅) → (𝐴 ∖ 𝐵) = ∅)
31	30	mpteq1d 5201	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝐴 ∖ 𝐵) = ∅) → (𝑥 ∈ (𝐴 ∖ 𝐵) ↦ 𝑍) = (𝑥 ∈ ∅ ↦ 𝑍))
32		mpt0 6711	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ∅ ↦ 𝑍) = ∅
33	31, 32	eqtrdi 2814	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝐴 ∖ 𝐵) = ∅) → (𝑥 ∈ (𝐴 ∖ 𝐵) ↦ 𝑍) = ∅)
34	33	cnveqd 5865	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝐴 ∖ 𝐵) = ∅) → ◡(𝑥 ∈ (𝐴 ∖ 𝐵) ↦ 𝑍) = ◡∅)
35		cnv0 5873	. . . . . . . . . . 11 ⊢ ◡∅ = ∅
36	34, 35	eqtrdi 2814	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝐴 ∖ 𝐵) = ∅) → ◡(𝑥 ∈ (𝐴 ∖ 𝐵) ↦ 𝑍) = ∅)
37	36	imaeq1d 6069	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝐴 ∖ 𝐵) = ∅) → (◡(𝑥 ∈ (𝐴 ∖ 𝐵) ↦ 𝑍) “ (ran (𝑥 ∈ (𝐴 ∖ 𝐵) ↦ 𝑍) ∖ {𝑍})) = (∅ “ (ran (𝑥 ∈ (𝐴 ∖ 𝐵) ↦ 𝑍) ∖ {𝑍})))
38		0ima 6219	. . . . . . . . 9 ⊢ (∅ “ (ran (𝑥 ∈ (𝐴 ∖ 𝐵) ↦ 𝑍) ∖ {𝑍})) = ∅
39	37, 38	eqtrdi 2814	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝐴 ∖ 𝐵) = ∅) → (◡(𝑥 ∈ (𝐴 ∖ 𝐵) ↦ 𝑍) “ (ran (𝑥 ∈ (𝐴 ∖ 𝐵) ↦ 𝑍) ∖ {𝑍})) = ∅)
40		eqid 2763	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ (𝐴 ∖ 𝐵) ↦ 𝑍) = (𝑥 ∈ (𝐴 ∖ 𝐵) ↦ 𝑍)
41		simpr 490	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝐴 ∖ 𝐵) ≠ ∅) → (𝐴 ∖ 𝐵) ≠ ∅)
42	40, 41	rnmptc 7244	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝐴 ∖ 𝐵) ≠ ∅) → ran (𝑥 ∈ (𝐴 ∖ 𝐵) ↦ 𝑍) = {𝑍})
43	42	difeq1d 4065	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝐴 ∖ 𝐵) ≠ ∅) → (ran (𝑥 ∈ (𝐴 ∖ 𝐵) ↦ 𝑍) ∖ {𝑍}) = ({𝑍} ∖ {𝑍}))
44		difid 4328	. . . . . . . . . . 11 ⊢ ({𝑍} ∖ {𝑍}) = ∅
45	43, 44	eqtrdi 2814	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝐴 ∖ 𝐵) ≠ ∅) → (ran (𝑥 ∈ (𝐴 ∖ 𝐵) ↦ 𝑍) ∖ {𝑍}) = ∅)
46	45	imaeq2d 6070	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝐴 ∖ 𝐵) ≠ ∅) → (◡(𝑥 ∈ (𝐴 ∖ 𝐵) ↦ 𝑍) “ (ran (𝑥 ∈ (𝐴 ∖ 𝐵) ↦ 𝑍) ∖ {𝑍})) = (◡(𝑥 ∈ (𝐴 ∖ 𝐵) ↦ 𝑍) “ ∅))
47		ima0 6093	. . . . . . . . 9 ⊢ (◡(𝑥 ∈ (𝐴 ∖ 𝐵) ↦ 𝑍) “ ∅) = ∅
48	46, 47	eqtrdi 2814	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝐴 ∖ 𝐵) ≠ ∅) → (◡(𝑥 ∈ (𝐴 ∖ 𝐵) ↦ 𝑍) “ (ran (𝑥 ∈ (𝐴 ∖ 𝐵) ↦ 𝑍) ∖ {𝑍})) = ∅)
49	39, 48	pm2.61dane 3046	. . . . . . 7 ⊢ (𝜑 → (◡(𝑥 ∈ (𝐴 ∖ 𝐵) ↦ 𝑍) “ (ran (𝑥 ∈ (𝐴 ∖ 𝐵) ↦ 𝑍) ∖ {𝑍})) = ∅)
50	29, 49	eqtrd 2798	. . . . . 6 ⊢ (𝜑 → ((𝑥 ∈ (𝐴 ∖ 𝐵) ↦ 𝑍) supp 𝑍) = ∅)
51		0fi 9108	. . . . . 6 ⊢ ∅ ∈ Fin
52	50, 51	eqeltrdi 2872	. . . . 5 ⊢ (𝜑 → ((𝑥 ∈ (𝐴 ∖ 𝐵) ↦ 𝑍) supp 𝑍) ∈ Fin)
53	25, 5, 27, 52	isfsuppd 9396	. . . 4 ⊢ (𝜑 → (𝑥 ∈ (𝐴 ∖ 𝐵) ↦ 𝑍) finSupp 𝑍)
54	22, 53	fsuppun 9416	. . 3 ⊢ (𝜑 → (((𝑥 ∈ (𝐴 ∩ 𝐵) ↦ 𝐶) ∪ (𝑥 ∈ (𝐴 ∖ 𝐵) ↦ 𝑍)) supp 𝑍) ∈ Fin)
55	10, 54	eqeltrid 2868	. 2 ⊢ (𝜑 → (𝐹 supp 𝑍) ∈ Fin)
56	4, 5, 7, 55	isfsuppd 9396	1 ⊢ (𝜑 → 𝐹 finSupp 𝑍)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2147   ≠ wne 2959  Vcvv 3454   ⊆ wss 3900   ∖ cdif 4048   ∩ cin 4092   ∪ cun 4152  ∅c0 4281  ifcif 4485  {csn 4588   class class class wbr 5108   ↦ cmpt 5192  ^◡ccnv 5657  ran crn 5659   “ cima 5661  Fun wfun 6560  (class class class)co 7451   supp csupp 8214  Fincfn 9011   finSupp cfsupp 9391

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5238  ax-sep 5255  ax-nul 5266  ax-pr 5398  ax-un 7783

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-iun 4958  df-br 5109  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-we 5613  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-ord 6391  df-on 6392  df-lim 6393  df-suc 6394  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fo 6572  df-f1o 6573  df-fv 6574  df-ov 7454  df-oprab 7455  df-mpo 7456  df-om 7912  df-supp 8215  df-1o 8514  df-en 9012  df-fin 9015  df-fsupp 9392

This theorem is used by:  elrspunsn  34083  gsummoncoe1fzo  34232  mplmonprod  34297  extdgfialglem2  34436