Theorem psrbaglefi 21882

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 3400	. . 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 21874	. . . . . . 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 8848	. . . . . . . . 9 ⊢ (∀𝑥 ∈ 𝐼 (0...(𝐹‘𝑥)) ⊆ ℕ0 → X𝑥 ∈ 𝐼 (0...(𝐹‘𝑥)) ⊆ X𝑥 ∈ 𝐼 ℕ0)
7		fz0ssnn0 13538	. . . . . . . . . 10 ⊢ (0...(𝐹‘𝑥)) ⊆ ℕ0
8	7	a1i 11	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐼 → (0...(𝐹‘𝑥)) ⊆ ℕ0)
9	6, 8	mprg 3057	. . . . . . . 8 ⊢ X𝑥 ∈ 𝐼 (0...(𝐹‘𝑥)) ⊆ X𝑥 ∈ 𝐼 ℕ0
10	9	sseli 3929	. . . . . . 7 ⊢ (𝑦 ∈ X𝑥 ∈ 𝐼 (0...(𝐹‘𝑥)) → 𝑦 ∈ X𝑥 ∈ 𝐼 ℕ0)
11		vex 3444	. . . . . . . 8 ⊢ 𝑦 ∈ V
12	11	elixpconst 8843	. . . . . . 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 6662	. . . . . . . . 9 ⊢ (𝑦:𝐼⟶ℕ0 → 𝑦 Fn 𝐼)
16	15	adantl 481	. . . . . . . 8 ⊢ ((𝐹 ∈ 𝐷 ∧ 𝑦:𝐼⟶ℕ0) → 𝑦 Fn 𝐼)
17	11	elixp 8842	. . . . . . . . 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 7026	. . . . . . . . . . . 12 ⊢ ((𝑦:𝐼⟶ℕ0 ∧ 𝑥 ∈ 𝐼) → (𝑦‘𝑥) ∈ ℕ0)
21	20	adantll 714	. . . . . . . . . . 11 ⊢ (((𝐹 ∈ 𝐷 ∧ 𝑦:𝐼⟶ℕ0) ∧ 𝑥 ∈ 𝐼) → (𝑦‘𝑥) ∈ ℕ0)
22		nn0uz 12789	. . . . . . . . . . 11 ⊢ ℕ0 = (ℤ≥‘0)
23	21, 22	eleqtrdi 2846	. . . . . . . . . 10 ⊢ (((𝐹 ∈ 𝐷 ∧ 𝑦:𝐼⟶ℕ0) ∧ 𝑥 ∈ 𝐼) → (𝑦‘𝑥) ∈ (ℤ≥‘0))
24	2	psrbagf 21874	. . . . . . . . . . . . 13 ⊢ (𝐹 ∈ 𝐷 → 𝐹:𝐼⟶ℕ0)
25	24	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝐹 ∈ 𝐷 ∧ 𝑦:𝐼⟶ℕ0) → 𝐹:𝐼⟶ℕ0)
26	25	ffvelcdmda 7029	. . . . . . . . . . 11 ⊢ (((𝐹 ∈ 𝐷 ∧ 𝑦:𝐼⟶ℕ0) ∧ 𝑥 ∈ 𝐼) → (𝐹‘𝑥) ∈ ℕ0)
27	26	nn0zd 12513	. . . . . . . . . 10 ⊢ (((𝐹 ∈ 𝐷 ∧ 𝑦:𝐼⟶ℕ0) ∧ 𝑥 ∈ 𝐼) → (𝐹‘𝑥) ∈ ℤ)
28		elfz5 13432	. . . . . . . . . 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 3157	. . . . . . . 8 ⊢ ((𝐹 ∈ 𝐷 ∧ 𝑦:𝐼⟶ℕ0) → (∀𝑥 ∈ 𝐼 (𝑦‘𝑥) ∈ (0...(𝐹‘𝑥)) ↔ ∀𝑥 ∈ 𝐼 (𝑦‘𝑥) ≤ (𝐹‘𝑥)))
31	24	ffnd 6663	. . . . . . . . . 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 4179	. . . . . . . . 9 ⊢ (𝐼 ∩ 𝐼) = 𝐼
36		eqidd 2737	. . . . . . . . 9 ⊢ (((𝐹 ∈ 𝐷 ∧ 𝑦:𝐼⟶ℕ0) ∧ 𝑥 ∈ 𝐼) → (𝑦‘𝑥) = (𝑦‘𝑥))
37		eqidd 2737	. . . . . . . . 9 ⊢ (((𝐹 ∈ 𝐷 ∧ 𝑦:𝐼⟶ℕ0) ∧ 𝑥 ∈ 𝐼) → (𝐹‘𝑥) = (𝐹‘𝑥))
38	16, 32, 33, 34, 35, 36, 37	ofrfvalg 7630	. . . . . . . 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 21879	. . . . . . . . 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 2870	. . 3 ⊢ (𝐹 ∈ 𝐷 → {𝑦 ∣ (𝑦 ∈ 𝐷 ∧ 𝑦 ∘r ≤ 𝐹)} = X𝑥 ∈ 𝐼 (0...(𝐹‘𝑥)))
47	1, 46	eqtrid 2783	. 2 ⊢ (𝐹 ∈ 𝐷 → {𝑦 ∈ 𝐷 ∣ 𝑦 ∘r ≤ 𝐹} = X𝑥 ∈ 𝐼 (0...(𝐹‘𝑥)))
48		cnveq 5822	. . . . . . 7 ⊢ (𝑓 = 𝐹 → ◡𝑓 = ◡𝐹)
49	48	imaeq1d 6018	. . . . . 6 ⊢ (𝑓 = 𝐹 → (◡𝑓 “ ℕ) = (◡𝐹 “ ℕ))
50	49	eleq1d 2821	. . . . 5 ⊢ (𝑓 = 𝐹 → ((◡𝑓 “ ℕ) ∈ Fin ↔ (◡𝐹 “ ℕ) ∈ Fin))
51	50, 2	elrab2 3649	. . . 4 ⊢ (𝐹 ∈ 𝐷 ↔ (𝐹 ∈ (ℕ0 ↑m 𝐼) ∧ (◡𝐹 “ ℕ) ∈ Fin))
52	51	simprbi 496	. . 3 ⊢ (𝐹 ∈ 𝐷 → (◡𝐹 “ ℕ) ∈ Fin)
53		fzfid 13896	. . 3 ⊢ ((𝐹 ∈ 𝐷 ∧ 𝑥 ∈ 𝐼) → (0...(𝐹‘𝑥)) ∈ Fin)
54		fcdmnn0suppg 12460	. . . . . . . . 9 ⊢ ((𝐹 ∈ 𝐷 ∧ 𝐹:𝐼⟶ℕ0) → (𝐹 supp 0) = (◡𝐹 “ ℕ))
55	24, 54	mpdan 687	. . . . . . . 8 ⊢ (𝐹 ∈ 𝐷 → (𝐹 supp 0) = (◡𝐹 “ ℕ))
56		eqimss 3992	. . . . . . . 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 11126	. . . . . . . 8 ⊢ 0 ∈ V
60	59	a1i 11	. . . . . . 7 ⊢ (𝐹 ∈ 𝐷 → 0 ∈ V)
61	24, 57, 58, 60	suppssrg 8138	. . . . . 6 ⊢ ((𝐹 ∈ 𝐷 ∧ 𝑥 ∈ (𝐼 ∖ (◡𝐹 “ ℕ))) → (𝐹‘𝑥) = 0)
62	61	oveq2d 7374	. . . . 5 ⊢ ((𝐹 ∈ 𝐷 ∧ 𝑥 ∈ (𝐼 ∖ (◡𝐹 “ ℕ))) → (0...(𝐹‘𝑥)) = (0...0))
63		fz0sn 13543	. . . . 5 ⊢ (0...0) = {0}
64	62, 63	eqtrdi 2787	. . . 4 ⊢ ((𝐹 ∈ 𝐷 ∧ 𝑥 ∈ (𝐼 ∖ (◡𝐹 “ ℕ))) → (0...(𝐹‘𝑥)) = {0})
65		eqimss 3992	. . . 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 9250	. 2 ⊢ (𝐹 ∈ 𝐷 → X𝑥 ∈ 𝐼 (0...(𝐹‘𝑥)) ∈ Fin)
68	47, 67	eqeltrd 2836	1 ⊢ (𝐹 ∈ 𝐷 → {𝑦 ∈ 𝐷 ∣ 𝑦 ∘r ≤ 𝐹} ∈ Fin)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541   ∈ wcel 2113  {cab 2714  ∀wral 3051  {crab 3399  Vcvv 3440   ∖ cdif 3898   ⊆ wss 3901  {csn 4580   class class class wbr 5098  ^◡ccnv 5623   “ cima 5627   Fn wfn 6487  ⟶wf 6488  ‘cfv 6492  (class class class)co 7358   ∘_r cofr 7621   supp csupp 8102   ↑_m cmap 8763  Xcixp 8835  Fincfn 8883  0cc0 11026   ≤ cle 11167  ℕcn 12145  ℕ₀cn0 12401  ℤcz 12488  ℤ_≥cuz 12751  ...cfz 13423

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680  ax-cnex 11082  ax-resscn 11083  ax-1cn 11084  ax-icn 11085  ax-addcl 11086  ax-addrcl 11087  ax-mulcl 11088  ax-mulrcl 11089  ax-mulcom 11090  ax-addass 11091  ax-mulass 11092  ax-distr 11093  ax-i2m1 11094  ax-1ne0 11095  ax-1rid 11096  ax-rnegex 11097  ax-rrecex 11098  ax-cnre 11099  ax-pre-lttri 11100  ax-pre-lttrn 11101  ax-pre-ltadd 11102  ax-pre-mulgt0 11103

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3061  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-pss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-tr 5206  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-riota 7315  df-ov 7361  df-oprab 7362  df-mpo 7363  df-ofr 7623  df-om 7809  df-1st 7933  df-2nd 7934  df-supp 8103  df-frecs 8223  df-wrecs 8254  df-recs 8303  df-rdg 8341  df-1o 8397  df-er 8635  df-map 8765  df-pm 8766  df-ixp 8836  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-pnf 11168  df-mnf 11169  df-xr 11170  df-ltxr 11171  df-le 11172  df-sub 11366  df-neg 11367  df-nn 12146  df-n0 12402  df-z 12489  df-uz 12752  df-fz 13424

This theorem is referenced by:  gsumbagdiag  21887  psrass1lem  21888  rhmpsrlem1  21896  rhmpsrlem2  21897  psrass1  21919  psrdi  21920  psrdir  21921  psrass23l  21922  psrcom  21923  psrass23  21924  resspsrmul  21931  mplsubrglem  21959  mplmonmul  21991  psdmul  22109  psropprmul  22178