Theorem psrbaglefi 21969

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:   𝑓,𝐹   𝑓,𝐼   𝑦,𝐷   𝑦,𝐹,𝑓   𝑦,𝐼

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 3444	. . 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 21961	. . . . . . 7 ⊢ (𝑦 ∈ 𝐷 → 𝑦:𝐼⟶ℕ0)
4	3	a1i 11	. . . . . 6 ⊢ (𝐹 ∈ 𝐷 → (𝑦 ∈ 𝐷 → 𝑦:𝐼⟶ℕ0))
5	4	adantrd 491	. . . . 5 ⊢ (𝐹 ∈ 𝐷 → ((𝑦 ∈ 𝐷 ∧ 𝑦 ∘r ≤ 𝐹) → 𝑦:𝐼⟶ℕ0))
6		ss2ixp 8968	. . . . . . . . 9 ⊢ (∀𝑥 ∈ 𝐼 (0...(𝐹‘𝑥)) ⊆ ℕ0 → X𝑥 ∈ 𝐼 (0...(𝐹‘𝑥)) ⊆ X𝑥 ∈ 𝐼 ℕ0)
7		fz0ssnn0 13679	. . . . . . . . . 10 ⊢ (0...(𝐹‘𝑥)) ⊆ ℕ0
8	7	a1i 11	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐼 → (0...(𝐹‘𝑥)) ⊆ ℕ0)
9	6, 8	mprg 3073	. . . . . . . 8 ⊢ X𝑥 ∈ 𝐼 (0...(𝐹‘𝑥)) ⊆ X𝑥 ∈ 𝐼 ℕ0
10	9	sseli 4004	. . . . . . 7 ⊢ (𝑦 ∈ X𝑥 ∈ 𝐼 (0...(𝐹‘𝑥)) → 𝑦 ∈ X𝑥 ∈ 𝐼 ℕ0)
11		vex 3492	. . . . . . . 8 ⊢ 𝑦 ∈ V
12	11	elixpconst 8963	. . . . . . 7 ⊢ (𝑦 ∈ X𝑥 ∈ 𝐼 ℕ0 ↔ 𝑦:𝐼⟶ℕ0)
13	10, 12	sylib 218	. . . . . 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 6747	. . . . . . . . 9 ⊢ (𝑦:𝐼⟶ℕ0 → 𝑦 Fn 𝐼)
16	15	adantl 481	. . . . . . . 8 ⊢ ((𝐹 ∈ 𝐷 ∧ 𝑦:𝐼⟶ℕ0) → 𝑦 Fn 𝐼)
17	11	elixp 8962	. . . . . . . . 9 ⊢ (𝑦 ∈ X𝑥 ∈ 𝐼 (0...(𝐹‘𝑥)) ↔ (𝑦 Fn 𝐼 ∧ ∀𝑥 ∈ 𝐼 (𝑦‘𝑥) ∈ (0...(𝐹‘𝑥))))
18	17	baib 535	. . . . . . . 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		ffvelcdm 7115	. . . . . . . . . . . 12 ⊢ ((𝑦:𝐼⟶ℕ0 ∧ 𝑥 ∈ 𝐼) → (𝑦‘𝑥) ∈ ℕ0)
21	20	adantll 713	. . . . . . . . . . 11 ⊢ (((𝐹 ∈ 𝐷 ∧ 𝑦:𝐼⟶ℕ0) ∧ 𝑥 ∈ 𝐼) → (𝑦‘𝑥) ∈ ℕ0)
22		nn0uz 12945	. . . . . . . . . . 11 ⊢ ℕ0 = (ℤ≥‘0)
23	21, 22	eleqtrdi 2854	. . . . . . . . . 10 ⊢ (((𝐹 ∈ 𝐷 ∧ 𝑦:𝐼⟶ℕ0) ∧ 𝑥 ∈ 𝐼) → (𝑦‘𝑥) ∈ (ℤ≥‘0))
24	2	psrbagf 21961	. . . . . . . . . . . . 13 ⊢ (𝐹 ∈ 𝐷 → 𝐹:𝐼⟶ℕ0)
25	24	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝐹 ∈ 𝐷 ∧ 𝑦:𝐼⟶ℕ0) → 𝐹:𝐼⟶ℕ0)
26	25	ffvelcdmda 7118	. . . . . . . . . . 11 ⊢ (((𝐹 ∈ 𝐷 ∧ 𝑦:𝐼⟶ℕ0) ∧ 𝑥 ∈ 𝐼) → (𝐹‘𝑥) ∈ ℕ0)
27	26	nn0zd 12665	. . . . . . . . . 10 ⊢ (((𝐹 ∈ 𝐷 ∧ 𝑦:𝐼⟶ℕ0) ∧ 𝑥 ∈ 𝐼) → (𝐹‘𝑥) ∈ ℤ)
28		elfz5 13576	. . . . . . . . . 10 ⊢ (((𝑦‘𝑥) ∈ (ℤ≥‘0) ∧ (𝐹‘𝑥) ∈ ℤ) → ((𝑦‘𝑥) ∈ (0...(𝐹‘𝑥)) ↔ (𝑦‘𝑥) ≤ (𝐹‘𝑥)))
29	23, 27, 28	syl2anc 583	. . . . . . . . 9 ⊢ (((𝐹 ∈ 𝐷 ∧ 𝑦:𝐼⟶ℕ0) ∧ 𝑥 ∈ 𝐼) → ((𝑦‘𝑥) ∈ (0...(𝐹‘𝑥)) ↔ (𝑦‘𝑥) ≤ (𝐹‘𝑥)))
30	29	ralbidva 3182	. . . . . . . 8 ⊢ ((𝐹 ∈ 𝐷 ∧ 𝑦:𝐼⟶ℕ0) → (∀𝑥 ∈ 𝐼 (𝑦‘𝑥) ∈ (0...(𝐹‘𝑥)) ↔ ∀𝑥 ∈ 𝐼 (𝑦‘𝑥) ≤ (𝐹‘𝑥)))
31	24	ffnd 6748	. . . . . . . . . 10 ⊢ (𝐹 ∈ 𝐷 → 𝐹 Fn 𝐼)
32	31	adantr 480	. . . . . . . . 9 ⊢ ((𝐹 ∈ 𝐷 ∧ 𝑦:𝐼⟶ℕ0) → 𝐹 Fn 𝐼)
33	11	a1i 11	. . . . . . . . 9 ⊢ ((𝐹 ∈ 𝐷 ∧ 𝑦:𝐼⟶ℕ0) → 𝑦 ∈ V)
34		simpl 482	. . . . . . . . 9 ⊢ ((𝐹 ∈ 𝐷 ∧ 𝑦:𝐼⟶ℕ0) → 𝐹 ∈ 𝐷)
35		inidm 4248	. . . . . . . . 9 ⊢ (𝐼 ∩ 𝐼) = 𝐼
36		eqidd 2741	. . . . . . . . 9 ⊢ (((𝐹 ∈ 𝐷 ∧ 𝑦:𝐼⟶ℕ0) ∧ 𝑥 ∈ 𝐼) → (𝑦‘𝑥) = (𝑦‘𝑥))
37		eqidd 2741	. . . . . . . . 9 ⊢ (((𝐹 ∈ 𝐷 ∧ 𝑦:𝐼⟶ℕ0) ∧ 𝑥 ∈ 𝐼) → (𝐹‘𝑥) = (𝐹‘𝑥))
38	16, 32, 33, 34, 35, 36, 37	ofrfvalg 7722	. . . . . . . 8 ⊢ ((𝐹 ∈ 𝐷 ∧ 𝑦:𝐼⟶ℕ0) → (𝑦 ∘r ≤ 𝐹 ↔ ∀𝑥 ∈ 𝐼 (𝑦‘𝑥) ≤ (𝐹‘𝑥)))
39	30, 38	bitr4d 282	. . . . . . 7 ⊢ ((𝐹 ∈ 𝐷 ∧ 𝑦:𝐼⟶ℕ0) → (∀𝑥 ∈ 𝐼 (𝑦‘𝑥) ∈ (0...(𝐹‘𝑥)) ↔ 𝑦 ∘r ≤ 𝐹))
40	2	psrbaglecl 21966	. . . . . . . . 9 ⊢ ((𝐹 ∈ 𝐷 ∧ 𝑦:𝐼⟶ℕ0 ∧ 𝑦 ∘r ≤ 𝐹) → 𝑦 ∈ 𝐷)
41	40	3expia 1121	. . . . . . . 8 ⊢ ((𝐹 ∈ 𝐷 ∧ 𝑦:𝐼⟶ℕ0) → (𝑦 ∘r ≤ 𝐹 → 𝑦 ∈ 𝐷))
42	41	pm4.71rd 562	. . . . . . 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 412	. . . . 5 ⊢ (𝐹 ∈ 𝐷 → (𝑦:𝐼⟶ℕ0 → ((𝑦 ∈ 𝐷 ∧ 𝑦 ∘r ≤ 𝐹) ↔ 𝑦 ∈ X𝑥 ∈ 𝐼 (0...(𝐹‘𝑥)))))
45	5, 14, 44	pm5.21ndd 379	. . . 4 ⊢ (𝐹 ∈ 𝐷 → ((𝑦 ∈ 𝐷 ∧ 𝑦 ∘r ≤ 𝐹) ↔ 𝑦 ∈ X𝑥 ∈ 𝐼 (0...(𝐹‘𝑥))))
46	45	eqabcdv 2879	. . 3 ⊢ (𝐹 ∈ 𝐷 → {𝑦 ∣ (𝑦 ∈ 𝐷 ∧ 𝑦 ∘r ≤ 𝐹)} = X𝑥 ∈ 𝐼 (0...(𝐹‘𝑥)))
47	1, 46	eqtrid 2792	. 2 ⊢ (𝐹 ∈ 𝐷 → {𝑦 ∈ 𝐷 ∣ 𝑦 ∘r ≤ 𝐹} = X𝑥 ∈ 𝐼 (0...(𝐹‘𝑥)))
48		cnveq 5898	. . . . . . 7 ⊢ (𝑓 = 𝐹 → ◡𝑓 = ◡𝐹)
49	48	imaeq1d 6088	. . . . . 6 ⊢ (𝑓 = 𝐹 → (◡𝑓 “ ℕ) = (◡𝐹 “ ℕ))
50	49	eleq1d 2829	. . . . 5 ⊢ (𝑓 = 𝐹 → ((◡𝑓 “ ℕ) ∈ Fin ↔ (◡𝐹 “ ℕ) ∈ Fin))
51	50, 2	elrab2 3711	. . . 4 ⊢ (𝐹 ∈ 𝐷 ↔ (𝐹 ∈ (ℕ0 ↑m 𝐼) ∧ (◡𝐹 “ ℕ) ∈ Fin))
52	51	simprbi 496	. . 3 ⊢ (𝐹 ∈ 𝐷 → (◡𝐹 “ ℕ) ∈ Fin)
53		fzfid 14024	. . 3 ⊢ ((𝐹 ∈ 𝐷 ∧ 𝑥 ∈ 𝐼) → (0...(𝐹‘𝑥)) ∈ Fin)
54		fcdmnn0suppg 12611	. . . . . . . . 9 ⊢ ((𝐹 ∈ 𝐷 ∧ 𝐹:𝐼⟶ℕ0) → (𝐹 supp 0) = (◡𝐹 “ ℕ))
55	24, 54	mpdan 686	. . . . . . . 8 ⊢ (𝐹 ∈ 𝐷 → (𝐹 supp 0) = (◡𝐹 “ ℕ))
56		eqimss 4067	. . . . . . . 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 11284	. . . . . . . 8 ⊢ 0 ∈ V
60	59	a1i 11	. . . . . . 7 ⊢ (𝐹 ∈ 𝐷 → 0 ∈ V)
61	24, 57, 58, 60	suppssrg 8237	. . . . . 6 ⊢ ((𝐹 ∈ 𝐷 ∧ 𝑥 ∈ (𝐼 ∖ (◡𝐹 “ ℕ))) → (𝐹‘𝑥) = 0)
62	61	oveq2d 7464	. . . . 5 ⊢ ((𝐹 ∈ 𝐷 ∧ 𝑥 ∈ (𝐼 ∖ (◡𝐹 “ ℕ))) → (0...(𝐹‘𝑥)) = (0...0))
63		fz0sn 13684	. . . . 5 ⊢ (0...0) = {0}
64	62, 63	eqtrdi 2796	. . . 4 ⊢ ((𝐹 ∈ 𝐷 ∧ 𝑥 ∈ (𝐼 ∖ (◡𝐹 “ ℕ))) → (0...(𝐹‘𝑥)) = {0})
65		eqimss 4067	. . . 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 9420	. 2 ⊢ (𝐹 ∈ 𝐷 → X𝑥 ∈ 𝐼 (0...(𝐹‘𝑥)) ∈ Fin)
68	47, 67	eqeltrd 2844	1 ⊢ (𝐹 ∈ 𝐷 → {𝑦 ∈ 𝐷 ∣ 𝑦 ∘r ≤ 𝐹} ∈ Fin)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1537   ∈ wcel 2108  {cab 2717  ∀wral 3067  {crab 3443  Vcvv 3488   ∖ cdif 3973   ⊆ wss 3976  {csn 4648   class class class wbr 5166  ^◡ccnv 5699   “ cima 5703   Fn wfn 6568  ⟶wf 6569  ‘cfv 6573  (class class class)co 7448   ∘_r cofr 7713   supp csupp 8201   ↑_m cmap 8884  Xcixp 8955  Fincfn 9003  0cc0 11184   ≤ cle 11325  ℕcn 12293  ℕ₀cn0 12553  ℤcz 12639  ℤ_≥cuz 12903  ...cfz 13567

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-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770  ax-cnex 11240  ax-resscn 11241  ax-1cn 11242  ax-icn 11243  ax-addcl 11244  ax-addrcl 11245  ax-mulcl 11246  ax-mulrcl 11247  ax-mulcom 11248  ax-addass 11249  ax-mulass 11250  ax-distr 11251  ax-i2m1 11252  ax-1ne0 11253  ax-1rid 11254  ax-rnegex 11255  ax-rrecex 11256  ax-cnre 11257  ax-pre-lttri 11258  ax-pre-lttrn 11259  ax-pre-ltadd 11260  ax-pre-mulgt0 11261

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-nel 3053  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-pred 6332  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-riota 7404  df-ov 7451  df-oprab 7452  df-mpo 7453  df-ofr 7715  df-om 7904  df-1st 8030  df-2nd 8031  df-supp 8202  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-rdg 8466  df-1o 8522  df-er 8763  df-map 8886  df-pm 8887  df-ixp 8956  df-en 9004  df-dom 9005  df-sdom 9006  df-fin 9007  df-pnf 11326  df-mnf 11327  df-xr 11328  df-ltxr 11329  df-le 11330  df-sub 11522  df-neg 11523  df-nn 12294  df-n0 12554  df-z 12640  df-uz 12904  df-fz 13568

This theorem is referenced by:  gsumbagdiag  21974  psrass1lem  21975  rhmpsrlem1  21983  rhmpsrlem2  21984  psrass1  22007  psrdi  22008  psrdir  22009  psrass23l  22010  psrcom  22011  psrass23  22012  resspsrmul  22019  mplsubrglem  22047  mplmonmul  22077  psdmul  22193  psropprmul  22260