Theorem psrbaglefi 22243

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 3413	. . 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 22235	. . . . . . 7 ⊢ (𝑦 ∈ 𝐷 → 𝑦:𝐼⟶ℕ0)
4	3	a1i 11	. . . . . 6 ⊢ (𝐹 ∈ 𝐷 → (𝑦 ∈ 𝐷 → 𝑦:𝐼⟶ℕ0))
5	4	adantrd 497	. . . . 5 ⊢ (𝐹 ∈ 𝐷 → ((𝑦 ∈ 𝐷 ∧ 𝑦 ∘r ≤ 𝐹) → 𝑦:𝐼⟶ℕ0))
6		ss2ixp 8946	. . . . . . . . 9 ⊢ (∀𝑥 ∈ 𝐼 (0...(𝐹‘𝑥)) ⊆ ℕ0 → X𝑥 ∈ 𝐼 (0...(𝐹‘𝑥)) ⊆ X𝑥 ∈ 𝐼 ℕ0)
7		fz0ssnn0 13763	. . . . . . . . . 10 ⊢ (0...(𝐹‘𝑥)) ⊆ ℕ0
8	7	a1i 11	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐼 → (0...(𝐹‘𝑥)) ⊆ ℕ0)
9	6, 8	mprg 3082	. . . . . . . 8 ⊢ X𝑥 ∈ 𝐼 (0...(𝐹‘𝑥)) ⊆ X𝑥 ∈ 𝐼 ℕ0
10	9	sseli 3905	. . . . . . 7 ⊢ (𝑦 ∈ X𝑥 ∈ 𝐼 (0...(𝐹‘𝑥)) → 𝑦 ∈ X𝑥 ∈ 𝐼 ℕ0)
11		vex 3454	. . . . . . . 8 ⊢ 𝑦 ∈ V
12	11	elixpconst 8941	. . . . . . 7 ⊢ (𝑦 ∈ X𝑥 ∈ 𝐼 ℕ0 ↔ 𝑦:𝐼⟶ℕ0)
13	10, 12	sylib 221	. . . . . 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 6714	. . . . . . . . 9 ⊢ (𝑦:𝐼⟶ℕ0 → 𝑦 Fn 𝐼)
16	15	adantl 487	. . . . . . . 8 ⊢ ((𝐹 ∈ 𝐷 ∧ 𝑦:𝐼⟶ℕ0) → 𝑦 Fn 𝐼)
17	11	elixp 8940	. . . . . . . . 9 ⊢ (𝑦 ∈ X𝑥 ∈ 𝐼 (0...(𝐹‘𝑥)) ↔ (𝑦 Fn 𝐼 ∧ ∀𝑥 ∈ 𝐼 (𝑦‘𝑥) ∈ (0...(𝐹‘𝑥))))
18	17	baib 545	. . . . . . . 8 ⊢ (𝑦 Fn 𝐼 → (𝑦 ∈ X𝑥 ∈ 𝐼 (0...(𝐹‘𝑥)) ↔ ∀𝑥 ∈ 𝐼 (𝑦‘𝑥) ∈ (0...(𝐹‘𝑥))))
19	16, 18	syl 18	. . . . . . 7 ⊢ ((𝐹 ∈ 𝐷 ∧ 𝑦:𝐼⟶ℕ0) → (𝑦 ∈ X𝑥 ∈ 𝐼 (0...(𝐹‘𝑥)) ↔ ∀𝑥 ∈ 𝐼 (𝑦‘𝑥) ∈ (0...(𝐹‘𝑥))))
20		ffvelcdm 7086	. . . . . . . . . . . 12 ⊢ ((𝑦:𝐼⟶ℕ0 ∧ 𝑥 ∈ 𝐼) → (𝑦‘𝑥) ∈ ℕ0)
21	20	adantll 727	. . . . . . . . . . 11 ⊢ (((𝐹 ∈ 𝐷 ∧ 𝑦:𝐼⟶ℕ0) ∧ 𝑥 ∈ 𝐼) → (𝑦‘𝑥) ∈ ℕ0)
22		nn0uz 13010	. . . . . . . . . . 11 ⊢ ℕ0 = (ℤ≥‘0)
23	21, 22	eleqtrdi 2870	. . . . . . . . . 10 ⊢ (((𝐹 ∈ 𝐷 ∧ 𝑦:𝐼⟶ℕ0) ∧ 𝑥 ∈ 𝐼) → (𝑦‘𝑥) ∈ (ℤ≥‘0))
24	2	psrbagf 22235	. . . . . . . . . . . . 13 ⊢ (𝐹 ∈ 𝐷 → 𝐹:𝐼⟶ℕ0)
25	24	adantr 486	. . . . . . . . . . . 12 ⊢ ((𝐹 ∈ 𝐷 ∧ 𝑦:𝐼⟶ℕ0) → 𝐹:𝐼⟶ℕ0)
26	25	ffvelcdmda 7089	. . . . . . . . . . 11 ⊢ (((𝐹 ∈ 𝐷 ∧ 𝑦:𝐼⟶ℕ0) ∧ 𝑥 ∈ 𝐼) → (𝐹‘𝑥) ∈ ℕ0)
27	26	nn0zd 12725	. . . . . . . . . 10 ⊢ (((𝐹 ∈ 𝐷 ∧ 𝑦:𝐼⟶ℕ0) ∧ 𝑥 ∈ 𝐼) → (𝐹‘𝑥) ∈ ℤ)
28		elfz5 13655	. . . . . . . . . 10 ⊢ (((𝑦‘𝑥) ∈ (ℤ≥‘0) ∧ (𝐹‘𝑥) ∈ ℤ) → ((𝑦‘𝑥) ∈ (0...(𝐹‘𝑥)) ↔ (𝑦‘𝑥) ≤ (𝐹‘𝑥)))
29	23, 27, 28	syl2anc 596	. . . . . . . . 9 ⊢ (((𝐹 ∈ 𝐷 ∧ 𝑦:𝐼⟶ℕ0) ∧ 𝑥 ∈ 𝐼) → ((𝑦‘𝑥) ∈ (0...(𝐹‘𝑥)) ↔ (𝑦‘𝑥) ≤ (𝐹‘𝑥)))
30	29	ralbidva 3183	. . . . . . . 8 ⊢ ((𝐹 ∈ 𝐷 ∧ 𝑦:𝐼⟶ℕ0) → (∀𝑥 ∈ 𝐼 (𝑦‘𝑥) ∈ (0...(𝐹‘𝑥)) ↔ ∀𝑥 ∈ 𝐼 (𝑦‘𝑥) ≤ (𝐹‘𝑥)))
31	24	ffnd 6715	. . . . . . . . . 10 ⊢ (𝐹 ∈ 𝐷 → 𝐹 Fn 𝐼)
32	31	adantr 486	. . . . . . . . 9 ⊢ ((𝐹 ∈ 𝐷 ∧ 𝑦:𝐼⟶ℕ0) → 𝐹 Fn 𝐼)
33	11	a1i 11	. . . . . . . . 9 ⊢ ((𝐹 ∈ 𝐷 ∧ 𝑦:𝐼⟶ℕ0) → 𝑦 ∈ V)
34		simpl 488	. . . . . . . . 9 ⊢ ((𝐹 ∈ 𝐷 ∧ 𝑦:𝐼⟶ℕ0) → 𝐹 ∈ 𝐷)
35		inidm 4122	. . . . . . . . 9 ⊢ (𝐼 ∩ 𝐼) = 𝐼
36		eqidd 2761	. . . . . . . . 9 ⊢ (((𝐹 ∈ 𝐷 ∧ 𝑦:𝐼⟶ℕ0) ∧ 𝑥 ∈ 𝐼) → (𝑦‘𝑥) = (𝑦‘𝑥))
37		eqidd 2761	. . . . . . . . 9 ⊢ (((𝐹 ∈ 𝐷 ∧ 𝑦:𝐼⟶ℕ0) ∧ 𝑥 ∈ 𝐼) → (𝐹‘𝑥) = (𝐹‘𝑥))
38	16, 32, 33, 34, 35, 36, 37	ofrfvalg 7705	. . . . . . . 8 ⊢ ((𝐹 ∈ 𝐷 ∧ 𝑦:𝐼⟶ℕ0) → (𝑦 ∘r ≤ 𝐹 ↔ ∀𝑥 ∈ 𝐼 (𝑦‘𝑥) ≤ (𝐹‘𝑥)))
39	30, 38	bitr4d 285	. . . . . . 7 ⊢ ((𝐹 ∈ 𝐷 ∧ 𝑦:𝐼⟶ℕ0) → (∀𝑥 ∈ 𝐼 (𝑦‘𝑥) ∈ (0...(𝐹‘𝑥)) ↔ 𝑦 ∘r ≤ 𝐹))
40	2	psrbaglecl 22240	. . . . . . . . 9 ⊢ ((𝐹 ∈ 𝐷 ∧ 𝑦:𝐼⟶ℕ0 ∧ 𝑦 ∘r ≤ 𝐹) → 𝑦 ∈ 𝐷)
41	40	3expia 1139	. . . . . . . 8 ⊢ ((𝐹 ∈ 𝐷 ∧ 𝑦:𝐼⟶ℕ0) → (𝑦 ∘r ≤ 𝐹 → 𝑦 ∈ 𝐷))
42	41	pm4.71rd 572	. . . . . . 7 ⊢ ((𝐹 ∈ 𝐷 ∧ 𝑦:𝐼⟶ℕ0) → (𝑦 ∘r ≤ 𝐹 ↔ (𝑦 ∈ 𝐷 ∧ 𝑦 ∘r ≤ 𝐹)))
43	19, 39, 42	3bitrrd 309	. . . . . 6 ⊢ ((𝐹 ∈ 𝐷 ∧ 𝑦:𝐼⟶ℕ0) → ((𝑦 ∈ 𝐷 ∧ 𝑦 ∘r ≤ 𝐹) ↔ 𝑦 ∈ X𝑥 ∈ 𝐼 (0...(𝐹‘𝑥))))
44	43	ex 418	. . . . 5 ⊢ (𝐹 ∈ 𝐷 → (𝑦:𝐼⟶ℕ0 → ((𝑦 ∈ 𝐷 ∧ 𝑦 ∘r ≤ 𝐹) ↔ 𝑦 ∈ X𝑥 ∈ 𝐼 (0...(𝐹‘𝑥)))))
45	5, 14, 44	pm5.21ndd 382	. . . 4 ⊢ (𝐹 ∈ 𝐷 → ((𝑦 ∈ 𝐷 ∧ 𝑦 ∘r ≤ 𝐹) ↔ 𝑦 ∈ X𝑥 ∈ 𝐼 (0...(𝐹‘𝑥))))
46	45	eqabcdv 2894	. . 3 ⊢ (𝐹 ∈ 𝐷 → {𝑦 ∣ (𝑦 ∈ 𝐷 ∧ 𝑦 ∘r ≤ 𝐹)} = X𝑥 ∈ 𝐼 (0...(𝐹‘𝑥)))
47	1, 46	eqtrid 2807	. 2 ⊢ (𝐹 ∈ 𝐷 → {𝑦 ∈ 𝐷 ∣ 𝑦 ∘r ≤ 𝐹} = X𝑥 ∈ 𝐼 (0...(𝐹‘𝑥)))
48		cnveq 5852	. . . . . . 7 ⊢ (𝑓 = 𝐹 → ◡𝑓 = ◡𝐹)
49	48	imaeq1d 6053	. . . . . 6 ⊢ (𝑓 = 𝐹 → (◡𝑓 “ ℕ) = (◡𝐹 “ ℕ))
50	49	eleq1d 2845	. . . . 5 ⊢ (𝑓 = 𝐹 → ((◡𝑓 “ ℕ) ∈ Fin ↔ (◡𝐹 “ ℕ) ∈ Fin))
51	50, 2	elrab2 3648	. . . 4 ⊢ (𝐹 ∈ 𝐷 ↔ (𝐹 ∈ (ℕ0 ↑m 𝐼) ∧ (◡𝐹 “ ℕ) ∈ Fin))
52	51	simprbi 503	. . 3 ⊢ (𝐹 ∈ 𝐷 → (◡𝐹 “ ℕ) ∈ Fin)
53		fzfid 14123	. . 3 ⊢ ((𝐹 ∈ 𝐷 ∧ 𝑥 ∈ 𝐼) → (0...(𝐹‘𝑥)) ∈ Fin)
54		fcdmnn0suppg 12672	. . . . . . . . 9 ⊢ ((𝐹 ∈ 𝐷 ∧ 𝐹:𝐼⟶ℕ0) → (𝐹 supp 0) = (◡𝐹 “ ℕ))
55	24, 54	mpdan 700	. . . . . . . 8 ⊢ (𝐹 ∈ 𝐷 → (𝐹 supp 0) = (◡𝐹 “ ℕ))
56		eqimss 3967	. . . . . . . 8 ⊢ ((𝐹 supp 0) = (◡𝐹 “ ℕ) → (𝐹 supp 0) ⊆ (◡𝐹 “ ℕ))
57	55, 56	syl 18	. . . . . . 7 ⊢ (𝐹 ∈ 𝐷 → (𝐹 supp 0) ⊆ (◡𝐹 “ ℕ))
58		id 23	. . . . . . 7 ⊢ (𝐹 ∈ 𝐷 → 𝐹 ∈ 𝐷)
59		c0ex 11307	. . . . . . . 8 ⊢ 0 ∈ V
60	59	a1i 11	. . . . . . 7 ⊢ (𝐹 ∈ 𝐷 → 0 ∈ V)
61	24, 57, 58, 60	suppssrg 8221	. . . . . 6 ⊢ ((𝐹 ∈ 𝐷 ∧ 𝑥 ∈ (𝐼 ∖ (◡𝐹 “ ℕ))) → (𝐹‘𝑥) = 0)
62	61	oveq2d 7440	. . . . 5 ⊢ ((𝐹 ∈ 𝐷 ∧ 𝑥 ∈ (𝐼 ∖ (◡𝐹 “ ℕ))) → (0...(𝐹‘𝑥)) = (0...0))
63		fz0sn 13768	. . . . 5 ⊢ (0...0) = {0}
64	62, 63	eqtrdi 2811	. . . 4 ⊢ ((𝐹 ∈ 𝐷 ∧ 𝑥 ∈ (𝐼 ∖ (◡𝐹 “ ℕ))) → (0...(𝐹‘𝑥)) = {0})
65		eqimss 3967	. . . 4 ⊢ ((0...(𝐹‘𝑥)) = {0} → (0...(𝐹‘𝑥)) ⊆ {0})
66	64, 65	syl 18	. . 3 ⊢ ((𝐹 ∈ 𝐷 ∧ 𝑥 ∈ (𝐼 ∖ (◡𝐹 “ ℕ))) → (0...(𝐹‘𝑥)) ⊆ {0})
67	52, 53, 66	ixpfi2 9347	. 2 ⊢ (𝐹 ∈ 𝐷 → X𝑥 ∈ 𝐼 (0...(𝐹‘𝑥)) ∈ Fin)
68	47, 67	eqeltrd 2860	1 ⊢ (𝐹 ∈ 𝐷 → {𝑦 ∈ 𝐷 ∣ 𝑦 ∘r ≤ 𝐹} ∈ Fin)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1569   ∈ wcel 2144  {cab 2738  ∀wral 3076  {crab 3412  Vcvv 3450   ⊆ wss 3896   ∖ cdif 4044  {csn 4584   class class class wbr 5102  ^◡ccnv 5650   “ cima 5654   Fn wfn 6538  ⟶wf 6539  ‘cfv 6543  (class class class)co 7424   ∘_r cofr 7696   supp csupp 8184   ↑_m cmap 8855  Xcixp 8933  Fincfn 8981  0cc0 11207   ≤ cle 11351  ℕcn 12342  ℕ₀cn0 12613  ℤcz 12700  ℤ_≥cuz 12972  ...cfz 13646

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pow 5327  ax-pr 5391  ax-un 7755  ax-cnex 11263  ax-resscn 11264  ax-1cn 11265  ax-icn 11266  ax-addcl 11267  ax-addrcl 11268  ax-mulcl 11269  ax-mulrcl 11270  ax-mulcom 11271  ax-addass 11272  ax-mulass 11273  ax-distr 11274  ax-i2m1 11275  ax-1ne0 11276  ax-1rid 11277  ax-rnegex 11278  ax-rrecex 11279  ax-cnre 11280  ax-pre-lttri 11281  ax-pre-lttrn 11282  ax-pre-ltadd 11283  ax-pre-mulgt0 11284

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3848  df-ss 3897  df-pss 4013  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6307  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6497  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7381  df-ov 7427  df-oprab 7428  df-mpo 7429  df-ofr 7698  df-om 7882  df-1st 8005  df-2nd 8006  df-supp 8185  df-frecs 8307  df-wrecs 8338  df-recs 8387  df-rdg 8426  df-1o 8484  df-er 8725  df-map 8857  df-pm 8858  df-ixp 8934  df-en 8982  df-dom 8983  df-sdom 8984  df-fin 8985  df-pnf 11352  df-mnf 11353  df-xr 11354  df-ltxr 11355  df-le 11356  df-sub 11550  df-neg 11551  df-nn 12343  df-n0 12614  df-z 12701  df-uz 12973  df-fz 13647

This theorem is used by:  gsumbagdiag  22249  psrass1lem  22250  rhmpsrlem1  22257  rhmpsrlem2  22258  psrass1  22280  psrdi  22281  psrdir  22282  psrass23l  22283  psrcom  22284  psrass23  22285  resspsrmul  22292  mplsubrglem  22320  mplmonmul  22354  psdmul  22496  psropprmul  22564  psrmonmul  34232