Theorem psrbaglefi 21135

Description: There are finitely many bags dominated by a given bag. (Contributed by Mario Carneiro, 29-Dec-2014.) (Revised by Mario Carneiro, 25-Jan-2015.) Remove a sethood antecedent. (Revised by SN, 5-Aug-2024.)

Hypothesis

Ref	Expression
psrbag.d	⊢ 𝐷 = {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}

Assertion

Ref	Expression
psrbaglefi	⊢ (𝐹 ∈ 𝐷 → {𝑦 ∈ 𝐷 ∣ 𝑦 ∘r ≤ 𝐹} ∈ Fin)

Distinct variable groups:   𝑦,𝑓,𝐹   𝑓,𝐼,𝑦   𝑦,𝐷

Allowed substitution hint:   𝐷(𝑓)

Proof of Theorem psrbaglefi

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-rab 3073	. . 3 ⊢ {𝑦 ∈ 𝐷 ∣ 𝑦 ∘r ≤ 𝐹} = {𝑦 ∣ (𝑦 ∈ 𝐷 ∧ 𝑦 ∘r ≤ 𝐹)}
2		psrbag.d	. . . . . . . 8 ⊢ 𝐷 = {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}
3	2	psrbagf 21121	. . . . . . 7 ⊢ (𝑦 ∈ 𝐷 → 𝑦:𝐼⟶ℕ0)
4	3	a1i 11	. . . . . 6 ⊢ (𝐹 ∈ 𝐷 → (𝑦 ∈ 𝐷 → 𝑦:𝐼⟶ℕ0))
5	4	adantrd 492	. . . . 5 ⊢ (𝐹 ∈ 𝐷 → ((𝑦 ∈ 𝐷 ∧ 𝑦 ∘r ≤ 𝐹) → 𝑦:𝐼⟶ℕ0))
6		ss2ixp 8698	. . . . . . . . 9 ⊢ (∀𝑥 ∈ 𝐼 (0...(𝐹‘𝑥)) ⊆ ℕ0 → X𝑥 ∈ 𝐼 (0...(𝐹‘𝑥)) ⊆ X𝑥 ∈ 𝐼 ℕ0)
7		fz0ssnn0 13351	. . . . . . . . . 10 ⊢ (0...(𝐹‘𝑥)) ⊆ ℕ0
8	7	a1i 11	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐼 → (0...(𝐹‘𝑥)) ⊆ ℕ0)
9	6, 8	mprg 3078	. . . . . . . 8 ⊢ X𝑥 ∈ 𝐼 (0...(𝐹‘𝑥)) ⊆ X𝑥 ∈ 𝐼 ℕ0
10	9	sseli 3917	. . . . . . 7 ⊢ (𝑦 ∈ X𝑥 ∈ 𝐼 (0...(𝐹‘𝑥)) → 𝑦 ∈ X𝑥 ∈ 𝐼 ℕ0)
11		vex 3436	. . . . . . . 8 ⊢ 𝑦 ∈ V
12	11	elixpconst 8693	. . . . . . 7 ⊢ (𝑦 ∈ X𝑥 ∈ 𝐼 ℕ0 ↔ 𝑦:𝐼⟶ℕ0)
13	10, 12	sylib 217	. . . . . 6 ⊢ (𝑦 ∈ X𝑥 ∈ 𝐼 (0...(𝐹‘𝑥)) → 𝑦:𝐼⟶ℕ0)
14	13	a1i 11	. . . . 5 ⊢ (𝐹 ∈ 𝐷 → (𝑦 ∈ X𝑥 ∈ 𝐼 (0...(𝐹‘𝑥)) → 𝑦:𝐼⟶ℕ0))
15		ffn 6600	. . . . . . . . 9 ⊢ (𝑦:𝐼⟶ℕ0 → 𝑦 Fn 𝐼)
16	15	adantl 482	. . . . . . . 8 ⊢ ((𝐹 ∈ 𝐷 ∧ 𝑦:𝐼⟶ℕ0) → 𝑦 Fn 𝐼)
17	11	elixp 8692	. . . . . . . . 9 ⊢ (𝑦 ∈ X𝑥 ∈ 𝐼 (0...(𝐹‘𝑥)) ↔ (𝑦 Fn 𝐼 ∧ ∀𝑥 ∈ 𝐼 (𝑦‘𝑥) ∈ (0...(𝐹‘𝑥))))
18	17	baib 536	. . . . . . . 8 ⊢ (𝑦 Fn 𝐼 → (𝑦 ∈ X𝑥 ∈ 𝐼 (0...(𝐹‘𝑥)) ↔ ∀𝑥 ∈ 𝐼 (𝑦‘𝑥) ∈ (0...(𝐹‘𝑥))))
19	16, 18	syl 17	. . . . . . 7 ⊢ ((𝐹 ∈ 𝐷 ∧ 𝑦:𝐼⟶ℕ0) → (𝑦 ∈ X𝑥 ∈ 𝐼 (0...(𝐹‘𝑥)) ↔ ∀𝑥 ∈ 𝐼 (𝑦‘𝑥) ∈ (0...(𝐹‘𝑥))))
20		ffvelrn 6959	. . . . . . . . . . . 12 ⊢ ((𝑦:𝐼⟶ℕ0 ∧ 𝑥 ∈ 𝐼) → (𝑦‘𝑥) ∈ ℕ0)
21	20	adantll 711	. . . . . . . . . . 11 ⊢ (((𝐹 ∈ 𝐷 ∧ 𝑦:𝐼⟶ℕ0) ∧ 𝑥 ∈ 𝐼) → (𝑦‘𝑥) ∈ ℕ0)
22		nn0uz 12620	. . . . . . . . . . 11 ⊢ ℕ0 = (ℤ≥‘0)
23	21, 22	eleqtrdi 2849	. . . . . . . . . 10 ⊢ (((𝐹 ∈ 𝐷 ∧ 𝑦:𝐼⟶ℕ0) ∧ 𝑥 ∈ 𝐼) → (𝑦‘𝑥) ∈ (ℤ≥‘0))
24	2	psrbagf 21121	. . . . . . . . . . . . 13 ⊢ (𝐹 ∈ 𝐷 → 𝐹:𝐼⟶ℕ0)
25	24	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝐹 ∈ 𝐷 ∧ 𝑦:𝐼⟶ℕ0) → 𝐹:𝐼⟶ℕ0)
26	25	ffvelrnda 6961	. . . . . . . . . . 11 ⊢ (((𝐹 ∈ 𝐷 ∧ 𝑦:𝐼⟶ℕ0) ∧ 𝑥 ∈ 𝐼) → (𝐹‘𝑥) ∈ ℕ0)
27	26	nn0zd 12424	. . . . . . . . . 10 ⊢ (((𝐹 ∈ 𝐷 ∧ 𝑦:𝐼⟶ℕ0) ∧ 𝑥 ∈ 𝐼) → (𝐹‘𝑥) ∈ ℤ)
28		elfz5 13248	. . . . . . . . . 10 ⊢ (((𝑦‘𝑥) ∈ (ℤ≥‘0) ∧ (𝐹‘𝑥) ∈ ℤ) → ((𝑦‘𝑥) ∈ (0...(𝐹‘𝑥)) ↔ (𝑦‘𝑥) ≤ (𝐹‘𝑥)))
29	23, 27, 28	syl2anc 584	. . . . . . . . 9 ⊢ (((𝐹 ∈ 𝐷 ∧ 𝑦:𝐼⟶ℕ0) ∧ 𝑥 ∈ 𝐼) → ((𝑦‘𝑥) ∈ (0...(𝐹‘𝑥)) ↔ (𝑦‘𝑥) ≤ (𝐹‘𝑥)))
30	29	ralbidva 3111	. . . . . . . 8 ⊢ ((𝐹 ∈ 𝐷 ∧ 𝑦:𝐼⟶ℕ0) → (∀𝑥 ∈ 𝐼 (𝑦‘𝑥) ∈ (0...(𝐹‘𝑥)) ↔ ∀𝑥 ∈ 𝐼 (𝑦‘𝑥) ≤ (𝐹‘𝑥)))
31	24	ffnd 6601	. . . . . . . . . 10 ⊢ (𝐹 ∈ 𝐷 → 𝐹 Fn 𝐼)
32	31	adantr 481	. . . . . . . . 9 ⊢ ((𝐹 ∈ 𝐷 ∧ 𝑦:𝐼⟶ℕ0) → 𝐹 Fn 𝐼)
33	11	a1i 11	. . . . . . . . 9 ⊢ ((𝐹 ∈ 𝐷 ∧ 𝑦:𝐼⟶ℕ0) → 𝑦 ∈ V)
34		simpl 483	. . . . . . . . 9 ⊢ ((𝐹 ∈ 𝐷 ∧ 𝑦:𝐼⟶ℕ0) → 𝐹 ∈ 𝐷)
35		inidm 4152	. . . . . . . . 9 ⊢ (𝐼 ∩ 𝐼) = 𝐼
36		eqidd 2739	. . . . . . . . 9 ⊢ (((𝐹 ∈ 𝐷 ∧ 𝑦:𝐼⟶ℕ0) ∧ 𝑥 ∈ 𝐼) → (𝑦‘𝑥) = (𝑦‘𝑥))
37		eqidd 2739	. . . . . . . . 9 ⊢ (((𝐹 ∈ 𝐷 ∧ 𝑦:𝐼⟶ℕ0) ∧ 𝑥 ∈ 𝐼) → (𝐹‘𝑥) = (𝐹‘𝑥))
38	16, 32, 33, 34, 35, 36, 37	ofrfvalg 7541	. . . . . . . 8 ⊢ ((𝐹 ∈ 𝐷 ∧ 𝑦:𝐼⟶ℕ0) → (𝑦 ∘r ≤ 𝐹 ↔ ∀𝑥 ∈ 𝐼 (𝑦‘𝑥) ≤ (𝐹‘𝑥)))
39	30, 38	bitr4d 281	. . . . . . 7 ⊢ ((𝐹 ∈ 𝐷 ∧ 𝑦:𝐼⟶ℕ0) → (∀𝑥 ∈ 𝐼 (𝑦‘𝑥) ∈ (0...(𝐹‘𝑥)) ↔ 𝑦 ∘r ≤ 𝐹))
40	2	psrbaglecl 21129	. . . . . . . . 9 ⊢ ((𝐹 ∈ 𝐷 ∧ 𝑦:𝐼⟶ℕ0 ∧ 𝑦 ∘r ≤ 𝐹) → 𝑦 ∈ 𝐷)
41	40	3expia 1120	. . . . . . . 8 ⊢ ((𝐹 ∈ 𝐷 ∧ 𝑦:𝐼⟶ℕ0) → (𝑦 ∘r ≤ 𝐹 → 𝑦 ∈ 𝐷))
42	41	pm4.71rd 563	. . . . . . 7 ⊢ ((𝐹 ∈ 𝐷 ∧ 𝑦:𝐼⟶ℕ0) → (𝑦 ∘r ≤ 𝐹 ↔ (𝑦 ∈ 𝐷 ∧ 𝑦 ∘r ≤ 𝐹)))
43	19, 39, 42	3bitrrd 306	. . . . . 6 ⊢ ((𝐹 ∈ 𝐷 ∧ 𝑦:𝐼⟶ℕ0) → ((𝑦 ∈ 𝐷 ∧ 𝑦 ∘r ≤ 𝐹) ↔ 𝑦 ∈ X𝑥 ∈ 𝐼 (0...(𝐹‘𝑥))))
44	43	ex 413	. . . . 5 ⊢ (𝐹 ∈ 𝐷 → (𝑦:𝐼⟶ℕ0 → ((𝑦 ∈ 𝐷 ∧ 𝑦 ∘r ≤ 𝐹) ↔ 𝑦 ∈ X𝑥 ∈ 𝐼 (0...(𝐹‘𝑥)))))
45	5, 14, 44	pm5.21ndd 381	. . . 4 ⊢ (𝐹 ∈ 𝐷 → ((𝑦 ∈ 𝐷 ∧ 𝑦 ∘r ≤ 𝐹) ↔ 𝑦 ∈ X𝑥 ∈ 𝐼 (0...(𝐹‘𝑥))))
46	45	abbi1dv 2878	. . 3 ⊢ (𝐹 ∈ 𝐷 → {𝑦 ∣ (𝑦 ∈ 𝐷 ∧ 𝑦 ∘r ≤ 𝐹)} = X𝑥 ∈ 𝐼 (0...(𝐹‘𝑥)))
47	1, 46	eqtrid 2790	. 2 ⊢ (𝐹 ∈ 𝐷 → {𝑦 ∈ 𝐷 ∣ 𝑦 ∘r ≤ 𝐹} = X𝑥 ∈ 𝐼 (0...(𝐹‘𝑥)))
48		cnveq 5782	. . . . . . 7 ⊢ (𝑓 = 𝐹 → ◡𝑓 = ◡𝐹)
49	48	imaeq1d 5968	. . . . . 6 ⊢ (𝑓 = 𝐹 → (◡𝑓 “ ℕ) = (◡𝐹 “ ℕ))
50	49	eleq1d 2823	. . . . 5 ⊢ (𝑓 = 𝐹 → ((◡𝑓 “ ℕ) ∈ Fin ↔ (◡𝐹 “ ℕ) ∈ Fin))
51	50, 2	elrab2 3627	. . . 4 ⊢ (𝐹 ∈ 𝐷 ↔ (𝐹 ∈ (ℕ0 ↑m 𝐼) ∧ (◡𝐹 “ ℕ) ∈ Fin))
52	51	simprbi 497	. . 3 ⊢ (𝐹 ∈ 𝐷 → (◡𝐹 “ ℕ) ∈ Fin)
53		fzfid 13693	. . 3 ⊢ ((𝐹 ∈ 𝐷 ∧ 𝑥 ∈ 𝐼) → (0...(𝐹‘𝑥)) ∈ Fin)
54		frnnn0suppg 12291	. . . . . . . . 9 ⊢ ((𝐹 ∈ 𝐷 ∧ 𝐹:𝐼⟶ℕ0) → (𝐹 supp 0) = (◡𝐹 “ ℕ))
55	24, 54	mpdan 684	. . . . . . . 8 ⊢ (𝐹 ∈ 𝐷 → (𝐹 supp 0) = (◡𝐹 “ ℕ))
56		eqimss 3977	. . . . . . . 8 ⊢ ((𝐹 supp 0) = (◡𝐹 “ ℕ) → (𝐹 supp 0) ⊆ (◡𝐹 “ ℕ))
57	55, 56	syl 17	. . . . . . 7 ⊢ (𝐹 ∈ 𝐷 → (𝐹 supp 0) ⊆ (◡𝐹 “ ℕ))
58		id 22	. . . . . . 7 ⊢ (𝐹 ∈ 𝐷 → 𝐹 ∈ 𝐷)
59		c0ex 10969	. . . . . . . 8 ⊢ 0 ∈ V
60	59	a1i 11	. . . . . . 7 ⊢ (𝐹 ∈ 𝐷 → 0 ∈ V)
61	24, 57, 58, 60	suppssrg 8013	. . . . . 6 ⊢ ((𝐹 ∈ 𝐷 ∧ 𝑥 ∈ (𝐼 ∖ (◡𝐹 “ ℕ))) → (𝐹‘𝑥) = 0)
62	61	oveq2d 7291	. . . . 5 ⊢ ((𝐹 ∈ 𝐷 ∧ 𝑥 ∈ (𝐼 ∖ (◡𝐹 “ ℕ))) → (0...(𝐹‘𝑥)) = (0...0))
63		fz0sn 13356	. . . . 5 ⊢ (0...0) = {0}
64	62, 63	eqtrdi 2794	. . . 4 ⊢ ((𝐹 ∈ 𝐷 ∧ 𝑥 ∈ (𝐼 ∖ (◡𝐹 “ ℕ))) → (0...(𝐹‘𝑥)) = {0})
65		eqimss 3977	. . . 4 ⊢ ((0...(𝐹‘𝑥)) = {0} → (0...(𝐹‘𝑥)) ⊆ {0})
66	64, 65	syl 17	. . 3 ⊢ ((𝐹 ∈ 𝐷 ∧ 𝑥 ∈ (𝐼 ∖ (◡𝐹 “ ℕ))) → (0...(𝐹‘𝑥)) ⊆ {0})
67	52, 53, 66	ixpfi2 9117	. 2 ⊢ (𝐹 ∈ 𝐷 → X𝑥 ∈ 𝐼 (0...(𝐹‘𝑥)) ∈ Fin)
68	47, 67	eqeltrd 2839	1 ⊢ (𝐹 ∈ 𝐷 → {𝑦 ∈ 𝐷 ∣ 𝑦 ∘r ≤ 𝐹} ∈ Fin)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1539   ∈ wcel 2106  {cab 2715  ∀wral 3064  {crab 3068  Vcvv 3432   ∖ cdif 3884   ⊆ wss 3887  {csn 4561   class class class wbr 5074  ^◡ccnv 5588   “ cima 5592   Fn wfn 6428  ⟶wf 6429  ‘cfv 6433  (class class class)co 7275   ∘_r cofr 7532   supp csupp 7977   ↑_m cmap 8615  Xcixp 8685  Fincfn 8733  0cc0 10871   ≤ cle 11010  ℕcn 11973  ℕ₀cn0 12233  ℤcz 12319  ℤ_≥cuz 12582  ...cfz 13239

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-cnex 10927  ax-resscn 10928  ax-1cn 10929  ax-icn 10930  ax-addcl 10931  ax-addrcl 10932  ax-mulcl 10933  ax-mulrcl 10934  ax-mulcom 10935  ax-addass 10936  ax-mulass 10937  ax-distr 10938  ax-i2m1 10939  ax-1ne0 10940  ax-1rid 10941  ax-rnegex 10942  ax-rrecex 10943  ax-cnre 10944  ax-pre-lttri 10945  ax-pre-lttrn 10946  ax-pre-ltadd 10947  ax-pre-mulgt0 10948

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-riota 7232  df-ov 7278  df-oprab 7279  df-mpo 7280  df-ofr 7534  df-om 7713  df-1st 7831  df-2nd 7832  df-supp 7978  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-rdg 8241  df-1o 8297  df-er 8498  df-map 8617  df-pm 8618  df-ixp 8686  df-en 8734  df-dom 8735  df-sdom 8736  df-fin 8737  df-pnf 11011  df-mnf 11012  df-xr 11013  df-ltxr 11014  df-le 11015  df-sub 11207  df-neg 11208  df-nn 11974  df-n0 12234  df-z 12320  df-uz 12583  df-fz 13240

This theorem is referenced by:  gsumbagdiag  21145  psrass1lem  21146  psrmulcllem  21156  psrass1  21174  psrdi  21175  psrdir  21176  psrass23l  21177  psrcom  21178  psrass23  21179  resspsrmul  21186  mplsubrglem  21210  mplmonmul  21237  psropprmul  21409