Theorem opsrval 22064

Description: The value of the "ordered power series" function. This is the same as mPwSer psrval 21935, but with the addition of a well-order on 𝐼 we can turn a strict order on 𝑅 into a strict order on the power series structure. (Contributed by Mario Carneiro, 8-Feb-2015.)

Hypotheses

Ref	Expression
opsrval.s	⊢ 𝑆 = (𝐼 mPwSer 𝑅)
opsrval.o	⊢ 𝑂 = ((𝐼 ordPwSer 𝑅)‘𝑇)
opsrval.b	⊢ 𝐵 = (Base‘𝑆)
opsrval.q	⊢ < = (lt‘𝑅)
opsrval.c	⊢ 𝐶 = (𝑇 <bag 𝐼)
opsrval.d	⊢ 𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}
opsrval.l	⊢ ≤ = {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝐵 ∧ (∃𝑧 ∈ 𝐷 ((𝑥‘𝑧) < (𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐷 (𝑤𝐶𝑧 → (𝑥‘𝑤) = (𝑦‘𝑤))) ∨ 𝑥 = 𝑦))}
opsrval.i	⊢ (𝜑 → 𝐼 ∈ 𝑉)
opsrval.r	⊢ (𝜑 → 𝑅 ∈ 𝑊)
opsrval.t	⊢ (𝜑 → 𝑇 ⊆ (𝐼 × 𝐼))

Assertion

Ref	Expression
opsrval	⊢ (𝜑 → 𝑂 = (𝑆 sSet ⟨(le‘ndx), ≤ ⟩))

Distinct variable groups:   𝑤,ℎ,𝑥,𝑦,𝑧,𝐼   𝜑,𝑤,𝑥,𝑦,𝑧   𝑤,𝐷,𝑧   𝑤,𝑇,𝑥,𝑦,𝑧   𝑤,𝑅,𝑥,𝑦,𝑧

Allowed substitution hints:   𝜑(ℎ)   𝐵(𝑥,𝑦,𝑧,𝑤,ℎ)   𝐶(𝑥,𝑦,𝑧,𝑤,ℎ)   𝐷(𝑥,𝑦,ℎ)   𝑅(ℎ)   𝑆(𝑥,𝑦,𝑧,𝑤,ℎ)   < (𝑥,𝑦,𝑧,𝑤,ℎ)   𝑇(ℎ)   ≤ (𝑥,𝑦,𝑧,𝑤,ℎ)   𝑂(𝑥,𝑦,𝑧,𝑤,ℎ)   𝑉(𝑥,𝑦,𝑧,𝑤,ℎ)   𝑊(𝑥,𝑦,𝑧,𝑤,ℎ)

Proof of Theorem opsrval

Dummy variables 𝑟 𝑖 𝑝 𝑠 𝑑 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		opsrval.o	. 2 ⊢ 𝑂 = ((𝐼 ordPwSer 𝑅)‘𝑇)
2		opsrval.i	. . . . 5 ⊢ (𝜑 → 𝐼 ∈ 𝑉)
3	2	elexd 3504	. . . 4 ⊢ (𝜑 → 𝐼 ∈ V)
4		opsrval.r	. . . . 5 ⊢ (𝜑 → 𝑅 ∈ 𝑊)
5	4	elexd 3504	. . . 4 ⊢ (𝜑 → 𝑅 ∈ V)
6	2, 2	xpexd 7771	. . . . 5 ⊢ (𝜑 → (𝐼 × 𝐼) ∈ V)
7		pwexg 5378	. . . . 5 ⊢ ((𝐼 × 𝐼) ∈ V → 𝒫 (𝐼 × 𝐼) ∈ V)
8		mptexg 7241	. . . . 5 ⊢ (𝒫 (𝐼 × 𝐼) ∈ V → (𝑟 ∈ 𝒫 (𝐼 × 𝐼) ↦ (𝑆 sSet ⟨(le‘ndx), {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝐵 ∧ (∃𝑧 ∈ 𝐷 ((𝑥‘𝑧) < (𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐷 (𝑤(𝑟 <bag 𝐼)𝑧 → (𝑥‘𝑤) = (𝑦‘𝑤))) ∨ 𝑥 = 𝑦))}⟩)) ∈ V)
9	6, 7, 8	3syl 18	. . . 4 ⊢ (𝜑 → (𝑟 ∈ 𝒫 (𝐼 × 𝐼) ↦ (𝑆 sSet ⟨(le‘ndx), {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝐵 ∧ (∃𝑧 ∈ 𝐷 ((𝑥‘𝑧) < (𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐷 (𝑤(𝑟 <bag 𝐼)𝑧 → (𝑥‘𝑤) = (𝑦‘𝑤))) ∨ 𝑥 = 𝑦))}⟩)) ∈ V)
10		simpl 482	. . . . . . . 8 ⊢ ((𝑖 = 𝐼 ∧ 𝑠 = 𝑅) → 𝑖 = 𝐼)
11	10	sqxpeqd 5717	. . . . . . 7 ⊢ ((𝑖 = 𝐼 ∧ 𝑠 = 𝑅) → (𝑖 × 𝑖) = (𝐼 × 𝐼))
12	11	pweqd 4617	. . . . . 6 ⊢ ((𝑖 = 𝐼 ∧ 𝑠 = 𝑅) → 𝒫 (𝑖 × 𝑖) = 𝒫 (𝐼 × 𝐼))
13		ovexd 7466	. . . . . . 7 ⊢ ((𝑖 = 𝐼 ∧ 𝑠 = 𝑅) → (𝑖 mPwSer 𝑠) ∈ V)
14		id 22	. . . . . . . . . 10 ⊢ (𝑝 = (𝑖 mPwSer 𝑠) → 𝑝 = (𝑖 mPwSer 𝑠))
15		oveq12 7440	. . . . . . . . . 10 ⊢ ((𝑖 = 𝐼 ∧ 𝑠 = 𝑅) → (𝑖 mPwSer 𝑠) = (𝐼 mPwSer 𝑅))
16	14, 15	sylan9eqr 2799	. . . . . . . . 9 ⊢ (((𝑖 = 𝐼 ∧ 𝑠 = 𝑅) ∧ 𝑝 = (𝑖 mPwSer 𝑠)) → 𝑝 = (𝐼 mPwSer 𝑅))
17		opsrval.s	. . . . . . . . 9 ⊢ 𝑆 = (𝐼 mPwSer 𝑅)
18	16, 17	eqtr4di 2795	. . . . . . . 8 ⊢ (((𝑖 = 𝐼 ∧ 𝑠 = 𝑅) ∧ 𝑝 = (𝑖 mPwSer 𝑠)) → 𝑝 = 𝑆)
19	18	fveq2d 6910	. . . . . . . . . . . . 13 ⊢ (((𝑖 = 𝐼 ∧ 𝑠 = 𝑅) ∧ 𝑝 = (𝑖 mPwSer 𝑠)) → (Base‘𝑝) = (Base‘𝑆))
20		opsrval.b	. . . . . . . . . . . . 13 ⊢ 𝐵 = (Base‘𝑆)
21	19, 20	eqtr4di 2795	. . . . . . . . . . . 12 ⊢ (((𝑖 = 𝐼 ∧ 𝑠 = 𝑅) ∧ 𝑝 = (𝑖 mPwSer 𝑠)) → (Base‘𝑝) = 𝐵)
22	21	sseq2d 4016	. . . . . . . . . . 11 ⊢ (((𝑖 = 𝐼 ∧ 𝑠 = 𝑅) ∧ 𝑝 = (𝑖 mPwSer 𝑠)) → ({𝑥, 𝑦} ⊆ (Base‘𝑝) ↔ {𝑥, 𝑦} ⊆ 𝐵))
23		ovex 7464	. . . . . . . . . . . . . . 15 ⊢ (ℕ0 ↑m 𝑖) ∈ V
24	23	rabex 5339	. . . . . . . . . . . . . 14 ⊢ {ℎ ∈ (ℕ0 ↑m 𝑖) ∣ (◡ℎ “ ℕ) ∈ Fin} ∈ V
25	24	a1i 11	. . . . . . . . . . . . 13 ⊢ (((𝑖 = 𝐼 ∧ 𝑠 = 𝑅) ∧ 𝑝 = (𝑖 mPwSer 𝑠)) → {ℎ ∈ (ℕ0 ↑m 𝑖) ∣ (◡ℎ “ ℕ) ∈ Fin} ∈ V)
26	10	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ (((𝑖 = 𝐼 ∧ 𝑠 = 𝑅) ∧ 𝑝 = (𝑖 mPwSer 𝑠)) → 𝑖 = 𝐼)
27	26	oveq2d 7447	. . . . . . . . . . . . . . 15 ⊢ (((𝑖 = 𝐼 ∧ 𝑠 = 𝑅) ∧ 𝑝 = (𝑖 mPwSer 𝑠)) → (ℕ0 ↑m 𝑖) = (ℕ0 ↑m 𝐼))
28		rabeq 3451	. . . . . . . . . . . . . . 15 ⊢ ((ℕ0 ↑m 𝑖) = (ℕ0 ↑m 𝐼) → {ℎ ∈ (ℕ0 ↑m 𝑖) ∣ (◡ℎ “ ℕ) ∈ Fin} = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin})
29	27, 28	syl 17	. . . . . . . . . . . . . 14 ⊢ (((𝑖 = 𝐼 ∧ 𝑠 = 𝑅) ∧ 𝑝 = (𝑖 mPwSer 𝑠)) → {ℎ ∈ (ℕ0 ↑m 𝑖) ∣ (◡ℎ “ ℕ) ∈ Fin} = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin})
30		opsrval.d	. . . . . . . . . . . . . 14 ⊢ 𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}
31	29, 30	eqtr4di 2795	. . . . . . . . . . . . 13 ⊢ (((𝑖 = 𝐼 ∧ 𝑠 = 𝑅) ∧ 𝑝 = (𝑖 mPwSer 𝑠)) → {ℎ ∈ (ℕ0 ↑m 𝑖) ∣ (◡ℎ “ ℕ) ∈ Fin} = 𝐷)
32		simpr 484	. . . . . . . . . . . . . 14 ⊢ ((((𝑖 = 𝐼 ∧ 𝑠 = 𝑅) ∧ 𝑝 = (𝑖 mPwSer 𝑠)) ∧ 𝑑 = 𝐷) → 𝑑 = 𝐷)
33		simpllr 776	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑖 = 𝐼 ∧ 𝑠 = 𝑅) ∧ 𝑝 = (𝑖 mPwSer 𝑠)) ∧ 𝑑 = 𝐷) → 𝑠 = 𝑅)
34	33	fveq2d 6910	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑖 = 𝐼 ∧ 𝑠 = 𝑅) ∧ 𝑝 = (𝑖 mPwSer 𝑠)) ∧ 𝑑 = 𝐷) → (lt‘𝑠) = (lt‘𝑅))
35		opsrval.q	. . . . . . . . . . . . . . . . 17 ⊢ < = (lt‘𝑅)
36	34, 35	eqtr4di 2795	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑖 = 𝐼 ∧ 𝑠 = 𝑅) ∧ 𝑝 = (𝑖 mPwSer 𝑠)) ∧ 𝑑 = 𝐷) → (lt‘𝑠) = < )
37	36	breqd 5154	. . . . . . . . . . . . . . 15 ⊢ ((((𝑖 = 𝐼 ∧ 𝑠 = 𝑅) ∧ 𝑝 = (𝑖 mPwSer 𝑠)) ∧ 𝑑 = 𝐷) → ((𝑥‘𝑧)(lt‘𝑠)(𝑦‘𝑧) ↔ (𝑥‘𝑧) < (𝑦‘𝑧)))
38	26	adantr 480	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝑖 = 𝐼 ∧ 𝑠 = 𝑅) ∧ 𝑝 = (𝑖 mPwSer 𝑠)) ∧ 𝑑 = 𝐷) → 𝑖 = 𝐼)
39	38	oveq2d 7447	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑖 = 𝐼 ∧ 𝑠 = 𝑅) ∧ 𝑝 = (𝑖 mPwSer 𝑠)) ∧ 𝑑 = 𝐷) → (𝑟 <bag 𝑖) = (𝑟 <bag 𝐼))
40	39	breqd 5154	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑖 = 𝐼 ∧ 𝑠 = 𝑅) ∧ 𝑝 = (𝑖 mPwSer 𝑠)) ∧ 𝑑 = 𝐷) → (𝑤(𝑟 <bag 𝑖)𝑧 ↔ 𝑤(𝑟 <bag 𝐼)𝑧))
41	40	imbi1d 341	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑖 = 𝐼 ∧ 𝑠 = 𝑅) ∧ 𝑝 = (𝑖 mPwSer 𝑠)) ∧ 𝑑 = 𝐷) → ((𝑤(𝑟 <bag 𝑖)𝑧 → (𝑥‘𝑤) = (𝑦‘𝑤)) ↔ (𝑤(𝑟 <bag 𝐼)𝑧 → (𝑥‘𝑤) = (𝑦‘𝑤))))
42	32, 41	raleqbidv 3346	. . . . . . . . . . . . . . 15 ⊢ ((((𝑖 = 𝐼 ∧ 𝑠 = 𝑅) ∧ 𝑝 = (𝑖 mPwSer 𝑠)) ∧ 𝑑 = 𝐷) → (∀𝑤 ∈ 𝑑 (𝑤(𝑟 <bag 𝑖)𝑧 → (𝑥‘𝑤) = (𝑦‘𝑤)) ↔ ∀𝑤 ∈ 𝐷 (𝑤(𝑟 <bag 𝐼)𝑧 → (𝑥‘𝑤) = (𝑦‘𝑤))))
43	37, 42	anbi12d 632	. . . . . . . . . . . . . 14 ⊢ ((((𝑖 = 𝐼 ∧ 𝑠 = 𝑅) ∧ 𝑝 = (𝑖 mPwSer 𝑠)) ∧ 𝑑 = 𝐷) → (((𝑥‘𝑧)(lt‘𝑠)(𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝑑 (𝑤(𝑟 <bag 𝑖)𝑧 → (𝑥‘𝑤) = (𝑦‘𝑤))) ↔ ((𝑥‘𝑧) < (𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐷 (𝑤(𝑟 <bag 𝐼)𝑧 → (𝑥‘𝑤) = (𝑦‘𝑤)))))
44	32, 43	rexeqbidv 3347	. . . . . . . . . . . . 13 ⊢ ((((𝑖 = 𝐼 ∧ 𝑠 = 𝑅) ∧ 𝑝 = (𝑖 mPwSer 𝑠)) ∧ 𝑑 = 𝐷) → (∃𝑧 ∈ 𝑑 ((𝑥‘𝑧)(lt‘𝑠)(𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝑑 (𝑤(𝑟 <bag 𝑖)𝑧 → (𝑥‘𝑤) = (𝑦‘𝑤))) ↔ ∃𝑧 ∈ 𝐷 ((𝑥‘𝑧) < (𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐷 (𝑤(𝑟 <bag 𝐼)𝑧 → (𝑥‘𝑤) = (𝑦‘𝑤)))))
45	25, 31, 44	sbcied2 3833	. . . . . . . . . . . 12 ⊢ (((𝑖 = 𝐼 ∧ 𝑠 = 𝑅) ∧ 𝑝 = (𝑖 mPwSer 𝑠)) → ([{ℎ ∈ (ℕ0 ↑m 𝑖) ∣ (◡ℎ “ ℕ) ∈ Fin} / 𝑑]∃𝑧 ∈ 𝑑 ((𝑥‘𝑧)(lt‘𝑠)(𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝑑 (𝑤(𝑟 <bag 𝑖)𝑧 → (𝑥‘𝑤) = (𝑦‘𝑤))) ↔ ∃𝑧 ∈ 𝐷 ((𝑥‘𝑧) < (𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐷 (𝑤(𝑟 <bag 𝐼)𝑧 → (𝑥‘𝑤) = (𝑦‘𝑤)))))
46	45	orbi1d 917	. . . . . . . . . . 11 ⊢ (((𝑖 = 𝐼 ∧ 𝑠 = 𝑅) ∧ 𝑝 = (𝑖 mPwSer 𝑠)) → (([{ℎ ∈ (ℕ0 ↑m 𝑖) ∣ (◡ℎ “ ℕ) ∈ Fin} / 𝑑]∃𝑧 ∈ 𝑑 ((𝑥‘𝑧)(lt‘𝑠)(𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝑑 (𝑤(𝑟 <bag 𝑖)𝑧 → (𝑥‘𝑤) = (𝑦‘𝑤))) ∨ 𝑥 = 𝑦) ↔ (∃𝑧 ∈ 𝐷 ((𝑥‘𝑧) < (𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐷 (𝑤(𝑟 <bag 𝐼)𝑧 → (𝑥‘𝑤) = (𝑦‘𝑤))) ∨ 𝑥 = 𝑦)))
47	22, 46	anbi12d 632	. . . . . . . . . 10 ⊢ (((𝑖 = 𝐼 ∧ 𝑠 = 𝑅) ∧ 𝑝 = (𝑖 mPwSer 𝑠)) → (({𝑥, 𝑦} ⊆ (Base‘𝑝) ∧ ([{ℎ ∈ (ℕ0 ↑m 𝑖) ∣ (◡ℎ “ ℕ) ∈ Fin} / 𝑑]∃𝑧 ∈ 𝑑 ((𝑥‘𝑧)(lt‘𝑠)(𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝑑 (𝑤(𝑟 <bag 𝑖)𝑧 → (𝑥‘𝑤) = (𝑦‘𝑤))) ∨ 𝑥 = 𝑦)) ↔ ({𝑥, 𝑦} ⊆ 𝐵 ∧ (∃𝑧 ∈ 𝐷 ((𝑥‘𝑧) < (𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐷 (𝑤(𝑟 <bag 𝐼)𝑧 → (𝑥‘𝑤) = (𝑦‘𝑤))) ∨ 𝑥 = 𝑦))))
48	47	opabbidv 5209	. . . . . . . . 9 ⊢ (((𝑖 = 𝐼 ∧ 𝑠 = 𝑅) ∧ 𝑝 = (𝑖 mPwSer 𝑠)) → {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ (Base‘𝑝) ∧ ([{ℎ ∈ (ℕ0 ↑m 𝑖) ∣ (◡ℎ “ ℕ) ∈ Fin} / 𝑑]∃𝑧 ∈ 𝑑 ((𝑥‘𝑧)(lt‘𝑠)(𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝑑 (𝑤(𝑟 <bag 𝑖)𝑧 → (𝑥‘𝑤) = (𝑦‘𝑤))) ∨ 𝑥 = 𝑦))} = {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝐵 ∧ (∃𝑧 ∈ 𝐷 ((𝑥‘𝑧) < (𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐷 (𝑤(𝑟 <bag 𝐼)𝑧 → (𝑥‘𝑤) = (𝑦‘𝑤))) ∨ 𝑥 = 𝑦))})
49	48	opeq2d 4880	. . . . . . . 8 ⊢ (((𝑖 = 𝐼 ∧ 𝑠 = 𝑅) ∧ 𝑝 = (𝑖 mPwSer 𝑠)) → ⟨(le‘ndx), {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ (Base‘𝑝) ∧ ([{ℎ ∈ (ℕ0 ↑m 𝑖) ∣ (◡ℎ “ ℕ) ∈ Fin} / 𝑑]∃𝑧 ∈ 𝑑 ((𝑥‘𝑧)(lt‘𝑠)(𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝑑 (𝑤(𝑟 <bag 𝑖)𝑧 → (𝑥‘𝑤) = (𝑦‘𝑤))) ∨ 𝑥 = 𝑦))}⟩ = ⟨(le‘ndx), {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝐵 ∧ (∃𝑧 ∈ 𝐷 ((𝑥‘𝑧) < (𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐷 (𝑤(𝑟 <bag 𝐼)𝑧 → (𝑥‘𝑤) = (𝑦‘𝑤))) ∨ 𝑥 = 𝑦))}⟩)
50	18, 49	oveq12d 7449	. . . . . . 7 ⊢ (((𝑖 = 𝐼 ∧ 𝑠 = 𝑅) ∧ 𝑝 = (𝑖 mPwSer 𝑠)) → (𝑝 sSet ⟨(le‘ndx), {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ (Base‘𝑝) ∧ ([{ℎ ∈ (ℕ0 ↑m 𝑖) ∣ (◡ℎ “ ℕ) ∈ Fin} / 𝑑]∃𝑧 ∈ 𝑑 ((𝑥‘𝑧)(lt‘𝑠)(𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝑑 (𝑤(𝑟 <bag 𝑖)𝑧 → (𝑥‘𝑤) = (𝑦‘𝑤))) ∨ 𝑥 = 𝑦))}⟩) = (𝑆 sSet ⟨(le‘ndx), {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝐵 ∧ (∃𝑧 ∈ 𝐷 ((𝑥‘𝑧) < (𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐷 (𝑤(𝑟 <bag 𝐼)𝑧 → (𝑥‘𝑤) = (𝑦‘𝑤))) ∨ 𝑥 = 𝑦))}⟩))
51	13, 50	csbied 3935	. . . . . 6 ⊢ ((𝑖 = 𝐼 ∧ 𝑠 = 𝑅) → ⦋(𝑖 mPwSer 𝑠) / 𝑝⦌(𝑝 sSet ⟨(le‘ndx), {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ (Base‘𝑝) ∧ ([{ℎ ∈ (ℕ0 ↑m 𝑖) ∣ (◡ℎ “ ℕ) ∈ Fin} / 𝑑]∃𝑧 ∈ 𝑑 ((𝑥‘𝑧)(lt‘𝑠)(𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝑑 (𝑤(𝑟 <bag 𝑖)𝑧 → (𝑥‘𝑤) = (𝑦‘𝑤))) ∨ 𝑥 = 𝑦))}⟩) = (𝑆 sSet ⟨(le‘ndx), {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝐵 ∧ (∃𝑧 ∈ 𝐷 ((𝑥‘𝑧) < (𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐷 (𝑤(𝑟 <bag 𝐼)𝑧 → (𝑥‘𝑤) = (𝑦‘𝑤))) ∨ 𝑥 = 𝑦))}⟩))
52	12, 51	mpteq12dv 5233	. . . . 5 ⊢ ((𝑖 = 𝐼 ∧ 𝑠 = 𝑅) → (𝑟 ∈ 𝒫 (𝑖 × 𝑖) ↦ ⦋(𝑖 mPwSer 𝑠) / 𝑝⦌(𝑝 sSet ⟨(le‘ndx), {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ (Base‘𝑝) ∧ ([{ℎ ∈ (ℕ0 ↑m 𝑖) ∣ (◡ℎ “ ℕ) ∈ Fin} / 𝑑]∃𝑧 ∈ 𝑑 ((𝑥‘𝑧)(lt‘𝑠)(𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝑑 (𝑤(𝑟 <bag 𝑖)𝑧 → (𝑥‘𝑤) = (𝑦‘𝑤))) ∨ 𝑥 = 𝑦))}⟩)) = (𝑟 ∈ 𝒫 (𝐼 × 𝐼) ↦ (𝑆 sSet ⟨(le‘ndx), {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝐵 ∧ (∃𝑧 ∈ 𝐷 ((𝑥‘𝑧) < (𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐷 (𝑤(𝑟 <bag 𝐼)𝑧 → (𝑥‘𝑤) = (𝑦‘𝑤))) ∨ 𝑥 = 𝑦))}⟩)))
53		df-opsr 21933	. . . . 5 ⊢ ordPwSer = (𝑖 ∈ V, 𝑠 ∈ V ↦ (𝑟 ∈ 𝒫 (𝑖 × 𝑖) ↦ ⦋(𝑖 mPwSer 𝑠) / 𝑝⦌(𝑝 sSet ⟨(le‘ndx), {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ (Base‘𝑝) ∧ ([{ℎ ∈ (ℕ0 ↑m 𝑖) ∣ (◡ℎ “ ℕ) ∈ Fin} / 𝑑]∃𝑧 ∈ 𝑑 ((𝑥‘𝑧)(lt‘𝑠)(𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝑑 (𝑤(𝑟 <bag 𝑖)𝑧 → (𝑥‘𝑤) = (𝑦‘𝑤))) ∨ 𝑥 = 𝑦))}⟩)))
54	52, 53	ovmpoga 7587	. . . 4 ⊢ ((𝐼 ∈ V ∧ 𝑅 ∈ V ∧ (𝑟 ∈ 𝒫 (𝐼 × 𝐼) ↦ (𝑆 sSet ⟨(le‘ndx), {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝐵 ∧ (∃𝑧 ∈ 𝐷 ((𝑥‘𝑧) < (𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐷 (𝑤(𝑟 <bag 𝐼)𝑧 → (𝑥‘𝑤) = (𝑦‘𝑤))) ∨ 𝑥 = 𝑦))}⟩)) ∈ V) → (𝐼 ordPwSer 𝑅) = (𝑟 ∈ 𝒫 (𝐼 × 𝐼) ↦ (𝑆 sSet ⟨(le‘ndx), {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝐵 ∧ (∃𝑧 ∈ 𝐷 ((𝑥‘𝑧) < (𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐷 (𝑤(𝑟 <bag 𝐼)𝑧 → (𝑥‘𝑤) = (𝑦‘𝑤))) ∨ 𝑥 = 𝑦))}⟩)))
55	3, 5, 9, 54	syl3anc 1373	. . 3 ⊢ (𝜑 → (𝐼 ordPwSer 𝑅) = (𝑟 ∈ 𝒫 (𝐼 × 𝐼) ↦ (𝑆 sSet ⟨(le‘ndx), {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝐵 ∧ (∃𝑧 ∈ 𝐷 ((𝑥‘𝑧) < (𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐷 (𝑤(𝑟 <bag 𝐼)𝑧 → (𝑥‘𝑤) = (𝑦‘𝑤))) ∨ 𝑥 = 𝑦))}⟩)))
56		simpr 484	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑟 = 𝑇) → 𝑟 = 𝑇)
57	56	oveq1d 7446	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑟 = 𝑇) → (𝑟 <bag 𝐼) = (𝑇 <bag 𝐼))
58		opsrval.c	. . . . . . . . . . . . . . 15 ⊢ 𝐶 = (𝑇 <bag 𝐼)
59	57, 58	eqtr4di 2795	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑟 = 𝑇) → (𝑟 <bag 𝐼) = 𝐶)
60	59	breqd 5154	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑟 = 𝑇) → (𝑤(𝑟 <bag 𝐼)𝑧 ↔ 𝑤𝐶𝑧))
61	60	imbi1d 341	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑟 = 𝑇) → ((𝑤(𝑟 <bag 𝐼)𝑧 → (𝑥‘𝑤) = (𝑦‘𝑤)) ↔ (𝑤𝐶𝑧 → (𝑥‘𝑤) = (𝑦‘𝑤))))
62	61	ralbidv 3178	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑟 = 𝑇) → (∀𝑤 ∈ 𝐷 (𝑤(𝑟 <bag 𝐼)𝑧 → (𝑥‘𝑤) = (𝑦‘𝑤)) ↔ ∀𝑤 ∈ 𝐷 (𝑤𝐶𝑧 → (𝑥‘𝑤) = (𝑦‘𝑤))))
63	62	anbi2d 630	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑟 = 𝑇) → (((𝑥‘𝑧) < (𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐷 (𝑤(𝑟 <bag 𝐼)𝑧 → (𝑥‘𝑤) = (𝑦‘𝑤))) ↔ ((𝑥‘𝑧) < (𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐷 (𝑤𝐶𝑧 → (𝑥‘𝑤) = (𝑦‘𝑤)))))
64	63	rexbidv 3179	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑟 = 𝑇) → (∃𝑧 ∈ 𝐷 ((𝑥‘𝑧) < (𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐷 (𝑤(𝑟 <bag 𝐼)𝑧 → (𝑥‘𝑤) = (𝑦‘𝑤))) ↔ ∃𝑧 ∈ 𝐷 ((𝑥‘𝑧) < (𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐷 (𝑤𝐶𝑧 → (𝑥‘𝑤) = (𝑦‘𝑤)))))
65	64	orbi1d 917	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑟 = 𝑇) → ((∃𝑧 ∈ 𝐷 ((𝑥‘𝑧) < (𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐷 (𝑤(𝑟 <bag 𝐼)𝑧 → (𝑥‘𝑤) = (𝑦‘𝑤))) ∨ 𝑥 = 𝑦) ↔ (∃𝑧 ∈ 𝐷 ((𝑥‘𝑧) < (𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐷 (𝑤𝐶𝑧 → (𝑥‘𝑤) = (𝑦‘𝑤))) ∨ 𝑥 = 𝑦)))
66	65	anbi2d 630	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑟 = 𝑇) → (({𝑥, 𝑦} ⊆ 𝐵 ∧ (∃𝑧 ∈ 𝐷 ((𝑥‘𝑧) < (𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐷 (𝑤(𝑟 <bag 𝐼)𝑧 → (𝑥‘𝑤) = (𝑦‘𝑤))) ∨ 𝑥 = 𝑦)) ↔ ({𝑥, 𝑦} ⊆ 𝐵 ∧ (∃𝑧 ∈ 𝐷 ((𝑥‘𝑧) < (𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐷 (𝑤𝐶𝑧 → (𝑥‘𝑤) = (𝑦‘𝑤))) ∨ 𝑥 = 𝑦))))
67	66	opabbidv 5209	. . . . . 6 ⊢ ((𝜑 ∧ 𝑟 = 𝑇) → {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝐵 ∧ (∃𝑧 ∈ 𝐷 ((𝑥‘𝑧) < (𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐷 (𝑤(𝑟 <bag 𝐼)𝑧 → (𝑥‘𝑤) = (𝑦‘𝑤))) ∨ 𝑥 = 𝑦))} = {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝐵 ∧ (∃𝑧 ∈ 𝐷 ((𝑥‘𝑧) < (𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐷 (𝑤𝐶𝑧 → (𝑥‘𝑤) = (𝑦‘𝑤))) ∨ 𝑥 = 𝑦))})
68		opsrval.l	. . . . . 6 ⊢ ≤ = {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝐵 ∧ (∃𝑧 ∈ 𝐷 ((𝑥‘𝑧) < (𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐷 (𝑤𝐶𝑧 → (𝑥‘𝑤) = (𝑦‘𝑤))) ∨ 𝑥 = 𝑦))}
69	67, 68	eqtr4di 2795	. . . . 5 ⊢ ((𝜑 ∧ 𝑟 = 𝑇) → {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝐵 ∧ (∃𝑧 ∈ 𝐷 ((𝑥‘𝑧) < (𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐷 (𝑤(𝑟 <bag 𝐼)𝑧 → (𝑥‘𝑤) = (𝑦‘𝑤))) ∨ 𝑥 = 𝑦))} = ≤ )
70	69	opeq2d 4880	. . . 4 ⊢ ((𝜑 ∧ 𝑟 = 𝑇) → ⟨(le‘ndx), {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝐵 ∧ (∃𝑧 ∈ 𝐷 ((𝑥‘𝑧) < (𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐷 (𝑤(𝑟 <bag 𝐼)𝑧 → (𝑥‘𝑤) = (𝑦‘𝑤))) ∨ 𝑥 = 𝑦))}⟩ = ⟨(le‘ndx), ≤ ⟩)
71	70	oveq2d 7447	. . 3 ⊢ ((𝜑 ∧ 𝑟 = 𝑇) → (𝑆 sSet ⟨(le‘ndx), {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝐵 ∧ (∃𝑧 ∈ 𝐷 ((𝑥‘𝑧) < (𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐷 (𝑤(𝑟 <bag 𝐼)𝑧 → (𝑥‘𝑤) = (𝑦‘𝑤))) ∨ 𝑥 = 𝑦))}⟩) = (𝑆 sSet ⟨(le‘ndx), ≤ ⟩))
72		opsrval.t	. . . 4 ⊢ (𝜑 → 𝑇 ⊆ (𝐼 × 𝐼))
73	6, 72	sselpwd 5328	. . 3 ⊢ (𝜑 → 𝑇 ∈ 𝒫 (𝐼 × 𝐼))
74		ovexd 7466	. . 3 ⊢ (𝜑 → (𝑆 sSet ⟨(le‘ndx), ≤ ⟩) ∈ V)
75	55, 71, 73, 74	fvmptd 7023	. 2 ⊢ (𝜑 → ((𝐼 ordPwSer 𝑅)‘𝑇) = (𝑆 sSet ⟨(le‘ndx), ≤ ⟩))
76	1, 75	eqtrid 2789	1 ⊢ (𝜑 → 𝑂 = (𝑆 sSet ⟨(le‘ndx), ≤ ⟩))

Syntax hints:   → wi 4   ∧ wa 395   ∨ wo 848   = wceq 1540   ∈ wcel 2108  ∀wral 3061  ∃wrex 3070  {crab 3436  Vcvv 3480  [wsbc 3788  ⦋csb 3899   ⊆ wss 3951  𝒫 cpw 4600  {cpr 4628  ⟨cop 4632   class class class wbr 5143  {copab 5205   ↦ cmpt 5225   × cxp 5683  ^◡ccnv 5684   “ cima 5688  ‘cfv 6561  (class class class)co 7431   ↑_m cmap 8866  Fincfn 8985  ℕcn 12266  ℕ₀cn0 12526   sSet csts 17200  ndxcnx 17230  Basecbs 17247  lecple 17304  ltcplt 18354   mPwSer cmps 21924   <_bag cltb 21927   ordPwSer copws 21928

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-opsr 21933

This theorem is referenced by:  opsrle  22065  opsrval2  22066