Theorem psrbaglefi 21919

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 3391	. . 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 21911	. . . . . . 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 8852	. . . . . . . . 9 ⊢ (∀𝑥 ∈ 𝐼 (0...(𝐹‘𝑥)) ⊆ ℕ0 → X𝑥 ∈ 𝐼 (0...(𝐹‘𝑥)) ⊆ X𝑥 ∈ 𝐼 ℕ0)
7		fz0ssnn0 13570	. . . . . . . . . 10 ⊢ (0...(𝐹‘𝑥)) ⊆ ℕ0
8	7	a1i 11	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐼 → (0...(𝐹‘𝑥)) ⊆ ℕ0)
9	6, 8	mprg 3058	. . . . . . . 8 ⊢ X𝑥 ∈ 𝐼 (0...(𝐹‘𝑥)) ⊆ X𝑥 ∈ 𝐼 ℕ0
10	9	sseli 3918	. . . . . . 7 ⊢ (𝑦 ∈ X𝑥 ∈ 𝐼 (0...(𝐹‘𝑥)) → 𝑦 ∈ X𝑥 ∈ 𝐼 ℕ0)
11		vex 3434	. . . . . . . 8 ⊢ 𝑦 ∈ V
12	11	elixpconst 8847	. . . . . . 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 6663	. . . . . . . . 9 ⊢ (𝑦:𝐼⟶ℕ0 → 𝑦 Fn 𝐼)
16	15	adantl 481	. . . . . . . 8 ⊢ ((𝐹 ∈ 𝐷 ∧ 𝑦:𝐼⟶ℕ0) → 𝑦 Fn 𝐼)
17	11	elixp 8846	. . . . . . . . 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 7028	. . . . . . . . . . . 12 ⊢ ((𝑦:𝐼⟶ℕ0 ∧ 𝑥 ∈ 𝐼) → (𝑦‘𝑥) ∈ ℕ0)
21	20	adantll 715	. . . . . . . . . . 11 ⊢ (((𝐹 ∈ 𝐷 ∧ 𝑦:𝐼⟶ℕ0) ∧ 𝑥 ∈ 𝐼) → (𝑦‘𝑥) ∈ ℕ0)
22		nn0uz 12820	. . . . . . . . . . 11 ⊢ ℕ0 = (ℤ≥‘0)
23	21, 22	eleqtrdi 2847	. . . . . . . . . 10 ⊢ (((𝐹 ∈ 𝐷 ∧ 𝑦:𝐼⟶ℕ0) ∧ 𝑥 ∈ 𝐼) → (𝑦‘𝑥) ∈ (ℤ≥‘0))
24	2	psrbagf 21911	. . . . . . . . . . . . 13 ⊢ (𝐹 ∈ 𝐷 → 𝐹:𝐼⟶ℕ0)
25	24	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝐹 ∈ 𝐷 ∧ 𝑦:𝐼⟶ℕ0) → 𝐹:𝐼⟶ℕ0)
26	25	ffvelcdmda 7031	. . . . . . . . . . 11 ⊢ (((𝐹 ∈ 𝐷 ∧ 𝑦:𝐼⟶ℕ0) ∧ 𝑥 ∈ 𝐼) → (𝐹‘𝑥) ∈ ℕ0)
27	26	nn0zd 12543	. . . . . . . . . 10 ⊢ (((𝐹 ∈ 𝐷 ∧ 𝑦:𝐼⟶ℕ0) ∧ 𝑥 ∈ 𝐼) → (𝐹‘𝑥) ∈ ℤ)
28		elfz5 13464	. . . . . . . . . 10 ⊢ (((𝑦‘𝑥) ∈ (ℤ≥‘0) ∧ (𝐹‘𝑥) ∈ ℤ) → ((𝑦‘𝑥) ∈ (0...(𝐹‘𝑥)) ↔ (𝑦‘𝑥) ≤ (𝐹‘𝑥)))
29	23, 27, 28	syl2anc 585	. . . . . . . . 9 ⊢ (((𝐹 ∈ 𝐷 ∧ 𝑦:𝐼⟶ℕ0) ∧ 𝑥 ∈ 𝐼) → ((𝑦‘𝑥) ∈ (0...(𝐹‘𝑥)) ↔ (𝑦‘𝑥) ≤ (𝐹‘𝑥)))
30	29	ralbidva 3159	. . . . . . . 8 ⊢ ((𝐹 ∈ 𝐷 ∧ 𝑦:𝐼⟶ℕ0) → (∀𝑥 ∈ 𝐼 (𝑦‘𝑥) ∈ (0...(𝐹‘𝑥)) ↔ ∀𝑥 ∈ 𝐼 (𝑦‘𝑥) ≤ (𝐹‘𝑥)))
31	24	ffnd 6664	. . . . . . . . . 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 4168	. . . . . . . . 9 ⊢ (𝐼 ∩ 𝐼) = 𝐼
36		eqidd 2738	. . . . . . . . 9 ⊢ (((𝐹 ∈ 𝐷 ∧ 𝑦:𝐼⟶ℕ0) ∧ 𝑥 ∈ 𝐼) → (𝑦‘𝑥) = (𝑦‘𝑥))
37		eqidd 2738	. . . . . . . . 9 ⊢ (((𝐹 ∈ 𝐷 ∧ 𝑦:𝐼⟶ℕ0) ∧ 𝑥 ∈ 𝐼) → (𝐹‘𝑥) = (𝐹‘𝑥))
38	16, 32, 33, 34, 35, 36, 37	ofrfvalg 7633	. . . . . . . 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 21916	. . . . . . . . 9 ⊢ ((𝐹 ∈ 𝐷 ∧ 𝑦:𝐼⟶ℕ0 ∧ 𝑦 ∘r ≤ 𝐹) → 𝑦 ∈ 𝐷)
41	40	3expia 1122	. . . . . . . 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 2871	. . 3 ⊢ (𝐹 ∈ 𝐷 → {𝑦 ∣ (𝑦 ∈ 𝐷 ∧ 𝑦 ∘r ≤ 𝐹)} = X𝑥 ∈ 𝐼 (0...(𝐹‘𝑥)))
47	1, 46	eqtrid 2784	. 2 ⊢ (𝐹 ∈ 𝐷 → {𝑦 ∈ 𝐷 ∣ 𝑦 ∘r ≤ 𝐹} = X𝑥 ∈ 𝐼 (0...(𝐹‘𝑥)))
48		cnveq 5823	. . . . . . 7 ⊢ (𝑓 = 𝐹 → ◡𝑓 = ◡𝐹)
49	48	imaeq1d 6019	. . . . . 6 ⊢ (𝑓 = 𝐹 → (◡𝑓 “ ℕ) = (◡𝐹 “ ℕ))
50	49	eleq1d 2822	. . . . 5 ⊢ (𝑓 = 𝐹 → ((◡𝑓 “ ℕ) ∈ Fin ↔ (◡𝐹 “ ℕ) ∈ Fin))
51	50, 2	elrab2 3638	. . . 4 ⊢ (𝐹 ∈ 𝐷 ↔ (𝐹 ∈ (ℕ0 ↑m 𝐼) ∧ (◡𝐹 “ ℕ) ∈ Fin))
52	51	simprbi 497	. . 3 ⊢ (𝐹 ∈ 𝐷 → (◡𝐹 “ ℕ) ∈ Fin)
53		fzfid 13929	. . 3 ⊢ ((𝐹 ∈ 𝐷 ∧ 𝑥 ∈ 𝐼) → (0...(𝐹‘𝑥)) ∈ Fin)
54		fcdmnn0suppg 12490	. . . . . . . . 9 ⊢ ((𝐹 ∈ 𝐷 ∧ 𝐹:𝐼⟶ℕ0) → (𝐹 supp 0) = (◡𝐹 “ ℕ))
55	24, 54	mpdan 688	. . . . . . . 8 ⊢ (𝐹 ∈ 𝐷 → (𝐹 supp 0) = (◡𝐹 “ ℕ))
56		eqimss 3981	. . . . . . . 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 11132	. . . . . . . 8 ⊢ 0 ∈ V
60	59	a1i 11	. . . . . . 7 ⊢ (𝐹 ∈ 𝐷 → 0 ∈ V)
61	24, 57, 58, 60	suppssrg 8140	. . . . . 6 ⊢ ((𝐹 ∈ 𝐷 ∧ 𝑥 ∈ (𝐼 ∖ (◡𝐹 “ ℕ))) → (𝐹‘𝑥) = 0)
62	61	oveq2d 7377	. . . . 5 ⊢ ((𝐹 ∈ 𝐷 ∧ 𝑥 ∈ (𝐼 ∖ (◡𝐹 “ ℕ))) → (0...(𝐹‘𝑥)) = (0...0))
63		fz0sn 13575	. . . . 5 ⊢ (0...0) = {0}
64	62, 63	eqtrdi 2788	. . . 4 ⊢ ((𝐹 ∈ 𝐷 ∧ 𝑥 ∈ (𝐼 ∖ (◡𝐹 “ ℕ))) → (0...(𝐹‘𝑥)) = {0})
65		eqimss 3981	. . . 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 9254	. 2 ⊢ (𝐹 ∈ 𝐷 → X𝑥 ∈ 𝐼 (0...(𝐹‘𝑥)) ∈ Fin)
68	47, 67	eqeltrd 2837	1 ⊢ (𝐹 ∈ 𝐷 → {𝑦 ∈ 𝐷 ∣ 𝑦 ∘r ≤ 𝐹} ∈ Fin)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114  {cab 2715  ∀wral 3052  {crab 3390  Vcvv 3430   ∖ cdif 3887   ⊆ wss 3890  {csn 4568   class class class wbr 5086  ^◡ccnv 5624   “ cima 5628   Fn wfn 6488  ⟶wf 6489  ‘cfv 6493  (class class class)co 7361   ∘_r cofr 7624   supp csupp 8104   ↑_m cmap 8767  Xcixp 8839  Fincfn 8887  0cc0 11032   ≤ cle 11174  ℕcn 12168  ℕ₀cn0 12431  ℤcz 12518  ℤ_≥cuz 12782  ...cfz 13455

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-sep 5232  ax-nul 5242  ax-pow 5303  ax-pr 5371  ax-un 7683  ax-cnex 11088  ax-resscn 11089  ax-1cn 11090  ax-icn 11091  ax-addcl 11092  ax-addrcl 11093  ax-mulcl 11094  ax-mulrcl 11095  ax-mulcom 11096  ax-addass 11097  ax-mulass 11098  ax-distr 11099  ax-i2m1 11100  ax-1ne0 11101  ax-1rid 11102  ax-rnegex 11103  ax-rrecex 11104  ax-cnre 11105  ax-pre-lttri 11106  ax-pre-lttrn 11107  ax-pre-ltadd 11108  ax-pre-mulgt0 11109

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-nel 3038  df-ral 3053  df-rex 3063  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-pred 6260  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-riota 7318  df-ov 7364  df-oprab 7365  df-mpo 7366  df-ofr 7626  df-om 7812  df-1st 7936  df-2nd 7937  df-supp 8105  df-frecs 8225  df-wrecs 8256  df-recs 8305  df-rdg 8343  df-1o 8399  df-er 8637  df-map 8769  df-pm 8770  df-ixp 8840  df-en 8888  df-dom 8889  df-sdom 8890  df-fin 8891  df-pnf 11175  df-mnf 11176  df-xr 11177  df-ltxr 11178  df-le 11179  df-sub 11373  df-neg 11374  df-nn 12169  df-n0 12432  df-z 12519  df-uz 12783  df-fz 13456

This theorem is referenced by:  gsumbagdiag  21924  psrass1lem  21925  rhmpsrlem1  21932  rhmpsrlem2  21933  psrass1  21955  psrdi  21956  psrdir  21957  psrass23l  21958  psrcom  21959  psrass23  21960  resspsrmul  21967  mplsubrglem  21995  mplmonmul  22027  psdmul  22145  psropprmul  22214  psrmonmul  33712