Theorem psrbaglefiOLD 20695

Description: Obsolete version of psrbaglefi 20694 as of 6-Aug-2024. (Contributed by Mario Carneiro, 29-Dec-2014.) (Revised by Mario Carneiro, 25-Jan-2015.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypothesis

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

Assertion

Ref	Expression
psrbaglefiOLD	⊢ ((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 psrbaglefiOLD

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-rab 3079	. . 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 20679	. . . . . . . 8 ⊢ (𝐼 ∈ 𝑉 → (𝑦 ∈ 𝐷 ↔ (𝑦:𝐼⟶ℕ0 ∧ (◡𝑦 “ ℕ) ∈ Fin)))
4	3	adantr 484	. . . . . . 7 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷) → (𝑦 ∈ 𝐷 ↔ (𝑦:𝐼⟶ℕ0 ∧ (◡𝑦 “ ℕ) ∈ Fin)))
5		simpl 486	. . . . . . 7 ⊢ ((𝑦:𝐼⟶ℕ0 ∧ (◡𝑦 “ ℕ) ∈ Fin) → 𝑦:𝐼⟶ℕ0)
6	4, 5	syl6bi 256	. . . . . 6 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷) → (𝑦 ∈ 𝐷 → 𝑦:𝐼⟶ℕ0))
7	6	adantrd 495	. . . . 5 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷) → ((𝑦 ∈ 𝐷 ∧ 𝑦 ∘r ≤ 𝐹) → 𝑦:𝐼⟶ℕ0))
8		ss2ixp 8492	. . . . . . . . 9 ⊢ (∀𝑥 ∈ 𝐼 (0...(𝐹‘𝑥)) ⊆ ℕ0 → X𝑥 ∈ 𝐼 (0...(𝐹‘𝑥)) ⊆ X𝑥 ∈ 𝐼 ℕ0)
9		fz0ssnn0 13051	. . . . . . . . . 10 ⊢ (0...(𝐹‘𝑥)) ⊆ ℕ0
10	9	a1i 11	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐼 → (0...(𝐹‘𝑥)) ⊆ ℕ0)
11	8, 10	mprg 3084	. . . . . . . 8 ⊢ X𝑥 ∈ 𝐼 (0...(𝐹‘𝑥)) ⊆ X𝑥 ∈ 𝐼 ℕ0
12	11	sseli 3888	. . . . . . 7 ⊢ (𝑦 ∈ X𝑥 ∈ 𝐼 (0...(𝐹‘𝑥)) → 𝑦 ∈ X𝑥 ∈ 𝐼 ℕ0)
13		vex 3413	. . . . . . . 8 ⊢ 𝑦 ∈ V
14	13	elixpconst 8487	. . . . . . 7 ⊢ (𝑦 ∈ X𝑥 ∈ 𝐼 ℕ0 ↔ 𝑦:𝐼⟶ℕ0)
15	12, 14	sylib 221	. . . . . 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 6498	. . . . . . . . 9 ⊢ (𝑦:𝐼⟶ℕ0 → 𝑦 Fn 𝐼)
18	17	adantl 485	. . . . . . . 8 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷) ∧ 𝑦:𝐼⟶ℕ0) → 𝑦 Fn 𝐼)
19	13	elixp 8486	. . . . . . . . 9 ⊢ (𝑦 ∈ X𝑥 ∈ 𝐼 (0...(𝐹‘𝑥)) ↔ (𝑦 Fn 𝐼 ∧ ∀𝑥 ∈ 𝐼 (𝑦‘𝑥) ∈ (0...(𝐹‘𝑥))))
20	19	baib 539	. . . . . . . 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 6840	. . . . . . . . . . . 12 ⊢ ((𝑦:𝐼⟶ℕ0 ∧ 𝑥 ∈ 𝐼) → (𝑦‘𝑥) ∈ ℕ0)
23	22	adantll 713	. . . . . . . . . . 11 ⊢ ((((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷) ∧ 𝑦:𝐼⟶ℕ0) ∧ 𝑥 ∈ 𝐼) → (𝑦‘𝑥) ∈ ℕ0)
24		nn0uz 12320	. . . . . . . . . . 11 ⊢ ℕ0 = (ℤ≥‘0)
25	23, 24	eleqtrdi 2862	. . . . . . . . . 10 ⊢ ((((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷) ∧ 𝑦:𝐼⟶ℕ0) ∧ 𝑥 ∈ 𝐼) → (𝑦‘𝑥) ∈ (ℤ≥‘0))
26	2	psrbagfOLD 20681	. . . . . . . . . . . . 13 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷) → 𝐹:𝐼⟶ℕ0)
27	26	adantr 484	. . . . . . . . . . . 12 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷) ∧ 𝑦:𝐼⟶ℕ0) → 𝐹:𝐼⟶ℕ0)
28	27	ffvelrnda 6842	. . . . . . . . . . 11 ⊢ ((((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷) ∧ 𝑦:𝐼⟶ℕ0) ∧ 𝑥 ∈ 𝐼) → (𝐹‘𝑥) ∈ ℕ0)
29	28	nn0zd 12124	. . . . . . . . . 10 ⊢ ((((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷) ∧ 𝑦:𝐼⟶ℕ0) ∧ 𝑥 ∈ 𝐼) → (𝐹‘𝑥) ∈ ℤ)
30		elfz5 12948	. . . . . . . . . 10 ⊢ (((𝑦‘𝑥) ∈ (ℤ≥‘0) ∧ (𝐹‘𝑥) ∈ ℤ) → ((𝑦‘𝑥) ∈ (0...(𝐹‘𝑥)) ↔ (𝑦‘𝑥) ≤ (𝐹‘𝑥)))
31	25, 29, 30	syl2anc 587	. . . . . . . . 9 ⊢ ((((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷) ∧ 𝑦:𝐼⟶ℕ0) ∧ 𝑥 ∈ 𝐼) → ((𝑦‘𝑥) ∈ (0...(𝐹‘𝑥)) ↔ (𝑦‘𝑥) ≤ (𝐹‘𝑥)))
32	31	ralbidva 3125	. . . . . . . 8 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷) ∧ 𝑦:𝐼⟶ℕ0) → (∀𝑥 ∈ 𝐼 (𝑦‘𝑥) ∈ (0...(𝐹‘𝑥)) ↔ ∀𝑥 ∈ 𝐼 (𝑦‘𝑥) ≤ (𝐹‘𝑥)))
33	27	ffnd 6499	. . . . . . . . 9 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷) ∧ 𝑦:𝐼⟶ℕ0) → 𝐹 Fn 𝐼)
34		simpll 766	. . . . . . . . 9 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷) ∧ 𝑦:𝐼⟶ℕ0) → 𝐼 ∈ 𝑉)
35		inidm 4123	. . . . . . . . 9 ⊢ (𝐼 ∩ 𝐼) = 𝐼
36		eqidd 2759	. . . . . . . . 9 ⊢ ((((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷) ∧ 𝑦:𝐼⟶ℕ0) ∧ 𝑥 ∈ 𝐼) → (𝑦‘𝑥) = (𝑦‘𝑥))
37		eqidd 2759	. . . . . . . . 9 ⊢ ((((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷) ∧ 𝑦:𝐼⟶ℕ0) ∧ 𝑥 ∈ 𝐼) → (𝐹‘𝑥) = (𝐹‘𝑥))
38	18, 33, 34, 34, 35, 36, 37	ofrfval 7414	. . . . . . . 8 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷) ∧ 𝑦:𝐼⟶ℕ0) → (𝑦 ∘r ≤ 𝐹 ↔ ∀𝑥 ∈ 𝐼 (𝑦‘𝑥) ≤ (𝐹‘𝑥)))
39	32, 38	bitr4d 285	. . . . . . 7 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷) ∧ 𝑦:𝐼⟶ℕ0) → (∀𝑥 ∈ 𝐼 (𝑦‘𝑥) ∈ (0...(𝐹‘𝑥)) ↔ 𝑦 ∘r ≤ 𝐹))
40	2	psrbagleclOLD 20689	. . . . . . . . . 10 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝐹 ∈ 𝐷 ∧ 𝑦:𝐼⟶ℕ0 ∧ 𝑦 ∘r ≤ 𝐹)) → 𝑦 ∈ 𝐷)
41	40	3exp2 1351	. . . . . . . . 9 ⊢ (𝐼 ∈ 𝑉 → (𝐹 ∈ 𝐷 → (𝑦:𝐼⟶ℕ0 → (𝑦 ∘r ≤ 𝐹 → 𝑦 ∈ 𝐷))))
42	41	imp31 421	. . . . . . . 8 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷) ∧ 𝑦:𝐼⟶ℕ0) → (𝑦 ∘r ≤ 𝐹 → 𝑦 ∈ 𝐷))
43	42	pm4.71rd 566	. . . . . . 7 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷) ∧ 𝑦:𝐼⟶ℕ0) → (𝑦 ∘r ≤ 𝐹 ↔ (𝑦 ∈ 𝐷 ∧ 𝑦 ∘r ≤ 𝐹)))
44	21, 39, 43	3bitrrd 309	. . . . . 6 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷) ∧ 𝑦:𝐼⟶ℕ0) → ((𝑦 ∈ 𝐷 ∧ 𝑦 ∘r ≤ 𝐹) ↔ 𝑦 ∈ X𝑥 ∈ 𝐼 (0...(𝐹‘𝑥))))
45	44	ex 416	. . . . 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 384	. . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷) → ((𝑦 ∈ 𝐷 ∧ 𝑦 ∘r ≤ 𝐹) ↔ 𝑦 ∈ X𝑥 ∈ 𝐼 (0...(𝐹‘𝑥))))
47	46	abbi1dv 2890	. . 3 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷) → {𝑦 ∣ (𝑦 ∈ 𝐷 ∧ 𝑦 ∘r ≤ 𝐹)} = X𝑥 ∈ 𝐼 (0...(𝐹‘𝑥)))
48	1, 47	syl5eq 2805	. 2 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷) → {𝑦 ∈ 𝐷 ∣ 𝑦 ∘r ≤ 𝐹} = X𝑥 ∈ 𝐼 (0...(𝐹‘𝑥)))
49		simpr 488	. . . . 5 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷) → 𝐹 ∈ 𝐷)
50		cnveq 5713	. . . . . . . 8 ⊢ (𝑓 = 𝐹 → ◡𝑓 = ◡𝐹)
51	50	imaeq1d 5900	. . . . . . 7 ⊢ (𝑓 = 𝐹 → (◡𝑓 “ ℕ) = (◡𝐹 “ ℕ))
52	51	eleq1d 2836	. . . . . 6 ⊢ (𝑓 = 𝐹 → ((◡𝑓 “ ℕ) ∈ Fin ↔ (◡𝐹 “ ℕ) ∈ Fin))
53	52, 2	elrab2 3605	. . . . 5 ⊢ (𝐹 ∈ 𝐷 ↔ (𝐹 ∈ (ℕ0 ↑m 𝐼) ∧ (◡𝐹 “ ℕ) ∈ Fin))
54	49, 53	sylib 221	. . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷) → (𝐹 ∈ (ℕ0 ↑m 𝐼) ∧ (◡𝐹 “ ℕ) ∈ Fin))
55	54	simprd 499	. . 3 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷) → (◡𝐹 “ ℕ) ∈ Fin)
56		fzfid 13390	. . 3 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷) ∧ 𝑥 ∈ 𝐼) → (0...(𝐹‘𝑥)) ∈ Fin)
57		simpl 486	. . . . . . . . 9 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷) → 𝐼 ∈ 𝑉)
58	57, 26	jca 515	. . . . . . . 8 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷) → (𝐼 ∈ 𝑉 ∧ 𝐹:𝐼⟶ℕ0))
59		frnnn0supp 11990	. . . . . . . 8 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐹:𝐼⟶ℕ0) → (𝐹 supp 0) = (◡𝐹 “ ℕ))
60		eqimss 3948	. . . . . . . 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 10673	. . . . . . . 8 ⊢ 0 ∈ V
63	62	a1i 11	. . . . . . 7 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷) → 0 ∈ V)
64	26, 61, 57, 63	suppssr 7870	. . . . . 6 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷) ∧ 𝑥 ∈ (𝐼 ∖ (◡𝐹 “ ℕ))) → (𝐹‘𝑥) = 0)
65	64	oveq2d 7166	. . . . 5 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷) ∧ 𝑥 ∈ (𝐼 ∖ (◡𝐹 “ ℕ))) → (0...(𝐹‘𝑥)) = (0...0))
66		fz0sn 13056	. . . . 5 ⊢ (0...0) = {0}
67	65, 66	eqtrdi 2809	. . . 4 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷) ∧ 𝑥 ∈ (𝐼 ∖ (◡𝐹 “ ℕ))) → (0...(𝐹‘𝑥)) = {0})
68		eqimss 3948	. . . 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 8855	. 2 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷) → X𝑥 ∈ 𝐼 (0...(𝐹‘𝑥)) ∈ Fin)
71	48, 70	eqeltrd 2852	1 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷) → {𝑦 ∈ 𝐷 ∣ 𝑦 ∘r ≤ 𝐹} ∈ Fin)

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2111  {cab 2735  ∀wral 3070  {crab 3074  Vcvv 3409   ∖ cdif 3855   ⊆ wss 3858  {csn 4522   class class class wbr 5032  ^◡ccnv 5523   “ cima 5527   Fn wfn 6330  ⟶wf 6331  ‘cfv 6335  (class class class)co 7150   ∘_r cofr 7404   supp csupp 7835   ↑_m cmap 8416  Xcixp 8479  Fincfn 8527  0cc0 10575   ≤ cle 10714  ℕcn 11674  ℕ₀cn0 11934  ℤcz 12020  ℤ_≥cuz 12282  ...cfz 12939

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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-rep 5156  ax-sep 5169  ax-nul 5176  ax-pow 5234  ax-pr 5298  ax-un 7459  ax-cnex 10631  ax-resscn 10632  ax-1cn 10633  ax-icn 10634  ax-addcl 10635  ax-addrcl 10636  ax-mulcl 10637  ax-mulrcl 10638  ax-mulcom 10639  ax-addass 10640  ax-mulass 10641  ax-distr 10642  ax-i2m1 10643  ax-1ne0 10644  ax-1rid 10645  ax-rnegex 10646  ax-rrecex 10647  ax-cnre 10648  ax-pre-lttri 10649  ax-pre-lttrn 10650  ax-pre-ltadd 10651  ax-pre-mulgt0 10652

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-nel 3056  df-ral 3075  df-rex 3076  df-reu 3077  df-rab 3079  df-v 3411  df-sbc 3697  df-csb 3806  df-dif 3861  df-un 3863  df-in 3865  df-ss 3875  df-pss 3877  df-nul 4226  df-if 4421  df-pw 4496  df-sn 4523  df-pr 4525  df-tp 4527  df-op 4529  df-uni 4799  df-iun 4885  df-br 5033  df-opab 5095  df-mpt 5113  df-tr 5139  df-id 5430  df-eprel 5435  df-po 5443  df-so 5444  df-fr 5483  df-we 5485  df-xp 5530  df-rel 5531  df-cnv 5532  df-co 5533  df-dm 5534  df-rn 5535  df-res 5536  df-ima 5537  df-pred 6126  df-ord 6172  df-on 6173  df-lim 6174  df-suc 6175  df-iota 6294  df-fun 6337  df-fn 6338  df-f 6339  df-f1 6340  df-fo 6341  df-f1o 6342  df-fv 6343  df-riota 7108  df-ov 7153  df-oprab 7154  df-mpo 7155  df-ofr 7406  df-om 7580  df-1st 7693  df-2nd 7694  df-supp 7836  df-wrecs 7957  df-recs 8018  df-rdg 8056  df-1o 8112  df-er 8299  df-map 8418  df-pm 8419  df-ixp 8480  df-en 8528  df-dom 8529  df-sdom 8530  df-fin 8531  df-pnf 10715  df-mnf 10716  df-xr 10717  df-ltxr 10718  df-le 10719  df-sub 10910  df-neg 10911  df-nn 11675  df-n0 11935  df-z 12021  df-uz 12283  df-fz 12940

This theorem is referenced by:  gsumbagdiagOLD  20701  psrass1lemOLD  20702