Theorem psrbaglefi 19372

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.)

Hypothesis

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

Assertion

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

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

Allowed substitution hints:   𝐷(𝑓)   𝑉(𝑓)

Proof of Theorem psrbaglefi

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-rab 2921	. . 3 ⊢ {𝑦 ∈ 𝐷 ∣ 𝑦 ∘𝑟 ≤ 𝐹} = {𝑦 ∣ (𝑦 ∈ 𝐷 ∧ 𝑦 ∘𝑟 ≤ 𝐹)}
2		psrbag.d	. . . . . . . . 9 ⊢ 𝐷 = {𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}
3	2	psrbag 19364	. . . . . . . 8 ⊢ (𝐼 ∈ 𝑉 → (𝑦 ∈ 𝐷 ↔ (𝑦:𝐼⟶ℕ0 ∧ (◡𝑦 “ ℕ) ∈ Fin)))
4	3	adantr 481	. . . . . . 7 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷) → (𝑦 ∈ 𝐷 ↔ (𝑦:𝐼⟶ℕ0 ∧ (◡𝑦 “ ℕ) ∈ Fin)))
5		simpl 473	. . . . . . 7 ⊢ ((𝑦:𝐼⟶ℕ0 ∧ (◡𝑦 “ ℕ) ∈ Fin) → 𝑦:𝐼⟶ℕ0)
6	4, 5	syl6bi 243	. . . . . 6 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷) → (𝑦 ∈ 𝐷 → 𝑦:𝐼⟶ℕ0))
7	6	adantrd 484	. . . . 5 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷) → ((𝑦 ∈ 𝐷 ∧ 𝑦 ∘𝑟 ≤ 𝐹) → 𝑦:𝐼⟶ℕ0))
8		ss2ixp 7921	. . . . . . . . 9 ⊢ (∀𝑥 ∈ 𝐼 (0...(𝐹‘𝑥)) ⊆ ℕ0 → X𝑥 ∈ 𝐼 (0...(𝐹‘𝑥)) ⊆ X𝑥 ∈ 𝐼 ℕ0)
9		fz0ssnn0 12435	. . . . . . . . . 10 ⊢ (0...(𝐹‘𝑥)) ⊆ ℕ0
10	9	a1i 11	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐼 → (0...(𝐹‘𝑥)) ⊆ ℕ0)
11	8, 10	mprg 2926	. . . . . . . 8 ⊢ X𝑥 ∈ 𝐼 (0...(𝐹‘𝑥)) ⊆ X𝑥 ∈ 𝐼 ℕ0
12	11	sseli 3599	. . . . . . 7 ⊢ (𝑦 ∈ X𝑥 ∈ 𝐼 (0...(𝐹‘𝑥)) → 𝑦 ∈ X𝑥 ∈ 𝐼 ℕ0)
13		vex 3203	. . . . . . . 8 ⊢ 𝑦 ∈ V
14	13	elixpconst 7916	. . . . . . 7 ⊢ (𝑦 ∈ X𝑥 ∈ 𝐼 ℕ0 ↔ 𝑦:𝐼⟶ℕ0)
15	12, 14	sylib 208	. . . . . 6 ⊢ (𝑦 ∈ X𝑥 ∈ 𝐼 (0...(𝐹‘𝑥)) → 𝑦:𝐼⟶ℕ0)
16	15	a1i 11	. . . . 5 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷) → (𝑦 ∈ X𝑥 ∈ 𝐼 (0...(𝐹‘𝑥)) → 𝑦:𝐼⟶ℕ0))
17		ffn 6045	. . . . . . . . 9 ⊢ (𝑦:𝐼⟶ℕ0 → 𝑦 Fn 𝐼)
18	17	adantl 482	. . . . . . . 8 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷) ∧ 𝑦:𝐼⟶ℕ0) → 𝑦 Fn 𝐼)
19	13	elixp 7915	. . . . . . . . 9 ⊢ (𝑦 ∈ X𝑥 ∈ 𝐼 (0...(𝐹‘𝑥)) ↔ (𝑦 Fn 𝐼 ∧ ∀𝑥 ∈ 𝐼 (𝑦‘𝑥) ∈ (0...(𝐹‘𝑥))))
20	19	baib 944	. . . . . . . 8 ⊢ (𝑦 Fn 𝐼 → (𝑦 ∈ X𝑥 ∈ 𝐼 (0...(𝐹‘𝑥)) ↔ ∀𝑥 ∈ 𝐼 (𝑦‘𝑥) ∈ (0...(𝐹‘𝑥))))
21	18, 20	syl 17	. . . . . . 7 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷) ∧ 𝑦:𝐼⟶ℕ0) → (𝑦 ∈ X𝑥 ∈ 𝐼 (0...(𝐹‘𝑥)) ↔ ∀𝑥 ∈ 𝐼 (𝑦‘𝑥) ∈ (0...(𝐹‘𝑥))))
22		ffvelrn 6357	. . . . . . . . . . . 12 ⊢ ((𝑦:𝐼⟶ℕ0 ∧ 𝑥 ∈ 𝐼) → (𝑦‘𝑥) ∈ ℕ0)
23	22	adantll 750	. . . . . . . . . . 11 ⊢ ((((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷) ∧ 𝑦:𝐼⟶ℕ0) ∧ 𝑥 ∈ 𝐼) → (𝑦‘𝑥) ∈ ℕ0)
24		nn0uz 11722	. . . . . . . . . . 11 ⊢ ℕ0 = (ℤ≥‘0)
25	23, 24	syl6eleq 2711	. . . . . . . . . 10 ⊢ ((((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷) ∧ 𝑦:𝐼⟶ℕ0) ∧ 𝑥 ∈ 𝐼) → (𝑦‘𝑥) ∈ (ℤ≥‘0))
26	2	psrbagf 19365	. . . . . . . . . . . . 13 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷) → 𝐹:𝐼⟶ℕ0)
27	26	adantr 481	. . . . . . . . . . . 12 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷) ∧ 𝑦:𝐼⟶ℕ0) → 𝐹:𝐼⟶ℕ0)
28	27	ffvelrnda 6359	. . . . . . . . . . 11 ⊢ ((((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷) ∧ 𝑦:𝐼⟶ℕ0) ∧ 𝑥 ∈ 𝐼) → (𝐹‘𝑥) ∈ ℕ0)
29	28	nn0zd 11480	. . . . . . . . . 10 ⊢ ((((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷) ∧ 𝑦:𝐼⟶ℕ0) ∧ 𝑥 ∈ 𝐼) → (𝐹‘𝑥) ∈ ℤ)
30		elfz5 12334	. . . . . . . . . 10 ⊢ (((𝑦‘𝑥) ∈ (ℤ≥‘0) ∧ (𝐹‘𝑥) ∈ ℤ) → ((𝑦‘𝑥) ∈ (0...(𝐹‘𝑥)) ↔ (𝑦‘𝑥) ≤ (𝐹‘𝑥)))
31	25, 29, 30	syl2anc 693	. . . . . . . . 9 ⊢ ((((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷) ∧ 𝑦:𝐼⟶ℕ0) ∧ 𝑥 ∈ 𝐼) → ((𝑦‘𝑥) ∈ (0...(𝐹‘𝑥)) ↔ (𝑦‘𝑥) ≤ (𝐹‘𝑥)))
32	31	ralbidva 2985	. . . . . . . 8 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷) ∧ 𝑦:𝐼⟶ℕ0) → (∀𝑥 ∈ 𝐼 (𝑦‘𝑥) ∈ (0...(𝐹‘𝑥)) ↔ ∀𝑥 ∈ 𝐼 (𝑦‘𝑥) ≤ (𝐹‘𝑥)))
33	27	ffnd 6046	. . . . . . . . 9 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷) ∧ 𝑦:𝐼⟶ℕ0) → 𝐹 Fn 𝐼)
34		simpll 790	. . . . . . . . 9 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷) ∧ 𝑦:𝐼⟶ℕ0) → 𝐼 ∈ 𝑉)
35		inidm 3822	. . . . . . . . 9 ⊢ (𝐼 ∩ 𝐼) = 𝐼
36		eqidd 2623	. . . . . . . . 9 ⊢ ((((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷) ∧ 𝑦:𝐼⟶ℕ0) ∧ 𝑥 ∈ 𝐼) → (𝑦‘𝑥) = (𝑦‘𝑥))
37		eqidd 2623	. . . . . . . . 9 ⊢ ((((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷) ∧ 𝑦:𝐼⟶ℕ0) ∧ 𝑥 ∈ 𝐼) → (𝐹‘𝑥) = (𝐹‘𝑥))
38	18, 33, 34, 34, 35, 36, 37	ofrfval 6905	. . . . . . . 8 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷) ∧ 𝑦:𝐼⟶ℕ0) → (𝑦 ∘𝑟 ≤ 𝐹 ↔ ∀𝑥 ∈ 𝐼 (𝑦‘𝑥) ≤ (𝐹‘𝑥)))
39	32, 38	bitr4d 271	. . . . . . 7 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷) ∧ 𝑦:𝐼⟶ℕ0) → (∀𝑥 ∈ 𝐼 (𝑦‘𝑥) ∈ (0...(𝐹‘𝑥)) ↔ 𝑦 ∘𝑟 ≤ 𝐹))
40	2	psrbaglecl 19369	. . . . . . . . . 10 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝐹 ∈ 𝐷 ∧ 𝑦:𝐼⟶ℕ0 ∧ 𝑦 ∘𝑟 ≤ 𝐹)) → 𝑦 ∈ 𝐷)
41	40	3exp2 1285	. . . . . . . . 9 ⊢ (𝐼 ∈ 𝑉 → (𝐹 ∈ 𝐷 → (𝑦:𝐼⟶ℕ0 → (𝑦 ∘𝑟 ≤ 𝐹 → 𝑦 ∈ 𝐷))))
42	41	imp31 448	. . . . . . . 8 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷) ∧ 𝑦:𝐼⟶ℕ0) → (𝑦 ∘𝑟 ≤ 𝐹 → 𝑦 ∈ 𝐷))
43	42	pm4.71rd 667	. . . . . . 7 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷) ∧ 𝑦:𝐼⟶ℕ0) → (𝑦 ∘𝑟 ≤ 𝐹 ↔ (𝑦 ∈ 𝐷 ∧ 𝑦 ∘𝑟 ≤ 𝐹)))
44	21, 39, 43	3bitrrd 295	. . . . . 6 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷) ∧ 𝑦:𝐼⟶ℕ0) → ((𝑦 ∈ 𝐷 ∧ 𝑦 ∘𝑟 ≤ 𝐹) ↔ 𝑦 ∈ X𝑥 ∈ 𝐼 (0...(𝐹‘𝑥))))
45	44	ex 450	. . . . 5 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷) → (𝑦:𝐼⟶ℕ0 → ((𝑦 ∈ 𝐷 ∧ 𝑦 ∘𝑟 ≤ 𝐹) ↔ 𝑦 ∈ X𝑥 ∈ 𝐼 (0...(𝐹‘𝑥)))))
46	7, 16, 45	pm5.21ndd 369	. . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷) → ((𝑦 ∈ 𝐷 ∧ 𝑦 ∘𝑟 ≤ 𝐹) ↔ 𝑦 ∈ X𝑥 ∈ 𝐼 (0...(𝐹‘𝑥))))
47	46	abbi1dv 2743	. . 3 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷) → {𝑦 ∣ (𝑦 ∈ 𝐷 ∧ 𝑦 ∘𝑟 ≤ 𝐹)} = X𝑥 ∈ 𝐼 (0...(𝐹‘𝑥)))
48	1, 47	syl5eq 2668	. 2 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷) → {𝑦 ∈ 𝐷 ∣ 𝑦 ∘𝑟 ≤ 𝐹} = X𝑥 ∈ 𝐼 (0...(𝐹‘𝑥)))
49		simpr 477	. . . . 5 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷) → 𝐹 ∈ 𝐷)
50		cnveq 5296	. . . . . . . 8 ⊢ (𝑓 = 𝐹 → ◡𝑓 = ◡𝐹)
51	50	imaeq1d 5465	. . . . . . 7 ⊢ (𝑓 = 𝐹 → (◡𝑓 “ ℕ) = (◡𝐹 “ ℕ))
52	51	eleq1d 2686	. . . . . 6 ⊢ (𝑓 = 𝐹 → ((◡𝑓 “ ℕ) ∈ Fin ↔ (◡𝐹 “ ℕ) ∈ Fin))
53	52, 2	elrab2 3366	. . . . 5 ⊢ (𝐹 ∈ 𝐷 ↔ (𝐹 ∈ (ℕ0 ↑𝑚 𝐼) ∧ (◡𝐹 “ ℕ) ∈ Fin))
54	49, 53	sylib 208	. . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷) → (𝐹 ∈ (ℕ0 ↑𝑚 𝐼) ∧ (◡𝐹 “ ℕ) ∈ Fin))
55	54	simprd 479	. . 3 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷) → (◡𝐹 “ ℕ) ∈ Fin)
56		fzfid 12772	. . 3 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷) ∧ 𝑥 ∈ 𝐼) → (0...(𝐹‘𝑥)) ∈ Fin)
57		simpl 473	. . . . . . . . 9 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷) → 𝐼 ∈ 𝑉)
58	57, 26	jca 554	. . . . . . . 8 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷) → (𝐼 ∈ 𝑉 ∧ 𝐹:𝐼⟶ℕ0))
59		frnnn0supp 11349	. . . . . . . 8 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐹:𝐼⟶ℕ0) → (𝐹 supp 0) = (◡𝐹 “ ℕ))
60		eqimss 3657	. . . . . . . 8 ⊢ ((𝐹 supp 0) = (◡𝐹 “ ℕ) → (𝐹 supp 0) ⊆ (◡𝐹 “ ℕ))
61	58, 59, 60	3syl 18	. . . . . . 7 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷) → (𝐹 supp 0) ⊆ (◡𝐹 “ ℕ))
62		c0ex 10034	. . . . . . . 8 ⊢ 0 ∈ V
63	62	a1i 11	. . . . . . 7 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷) → 0 ∈ V)
64	26, 61, 57, 63	suppssr 7326	. . . . . 6 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷) ∧ 𝑥 ∈ (𝐼 ∖ (◡𝐹 “ ℕ))) → (𝐹‘𝑥) = 0)
65	64	oveq2d 6666	. . . . 5 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷) ∧ 𝑥 ∈ (𝐼 ∖ (◡𝐹 “ ℕ))) → (0...(𝐹‘𝑥)) = (0...0))
66		fz0sn 12439	. . . . 5 ⊢ (0...0) = {0}
67	65, 66	syl6eq 2672	. . . 4 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷) ∧ 𝑥 ∈ (𝐼 ∖ (◡𝐹 “ ℕ))) → (0...(𝐹‘𝑥)) = {0})
68		eqimss 3657	. . . 4 ⊢ ((0...(𝐹‘𝑥)) = {0} → (0...(𝐹‘𝑥)) ⊆ {0})
69	67, 68	syl 17	. . 3 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷) ∧ 𝑥 ∈ (𝐼 ∖ (◡𝐹 “ ℕ))) → (0...(𝐹‘𝑥)) ⊆ {0})
70	55, 56, 69	ixpfi2 8264	. 2 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷) → X𝑥 ∈ 𝐼 (0...(𝐹‘𝑥)) ∈ Fin)
71	48, 70	eqeltrd 2701	1 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷) → {𝑦 ∈ 𝐷 ∣ 𝑦 ∘𝑟 ≤ 𝐹} ∈ Fin)

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1483   ∈ wcel 1990  {cab 2608  ∀wral 2912  {crab 2916  Vcvv 3200   ∖ cdif 3571   ⊆ wss 3574  {csn 4177   class class class wbr 4653  ^◡ccnv 5113   “ cima 5117   Fn wfn 5883  ⟶wf 5884  ‘cfv 5888  (class class class)co 6650   ∘_𝑟 cofr 6896   supp csupp 7295   ↑_𝑚 cmap 7857  Xcixp 7908  Fincfn 7955  0cc0 9936   ≤ cle 10075  ℕcn 11020  ℕ₀cn0 11292  ℤcz 11377  ℤ_≥cuz 11687  ...cfz 12326

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-ofr 6898  df-om 7066  df-1st 7168  df-2nd 7169  df-supp 7296  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-map 7859  df-pm 7860  df-ixp 7909  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-n0 11293  df-z 11378  df-uz 11688  df-fz 12327

This theorem is referenced by:  gsumbagdiag  19376  psrass1lem  19377  psrmulcllem  19387  psrass1  19405  psrdi  19406  psrdir  19407  psrass23l  19408  psrcom  19409  psrass23  19410  resspsrmul  19417  mplsubrglem  19439  mplmonmul  19464  psropprmul  19608