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Theorem opsrval 19237
Description: The value of the "ordered power series" function. This is the same as mPwSer psrval 19125, 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 = { ∈ (ℕ0𝑚 𝐼) ∣ ( “ ℕ) ∈ Fin}
opsrval.l = {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝐵 ∧ (∃𝑧𝐷 ((𝑥𝑧) < (𝑦𝑧) ∧ ∀𝑤𝐷 (𝑤𝐶𝑧 → (𝑥𝑤) = (𝑦𝑤))) ∨ 𝑥 = 𝑦))}
opsrval.i (𝜑𝐼𝑉)
opsrval.r (𝜑𝑅𝑊)
opsrval.t (𝜑𝑇 ⊆ (𝐼 × 𝐼))
Assertion
Ref Expression
opsrval (𝜑𝑂 = (𝑆 sSet ⟨(le‘ndx), ⟩))
Distinct variable groups:   𝑤,,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()   𝐵(𝑥,𝑦,𝑧,𝑤,)   𝐶(𝑥,𝑦,𝑧,𝑤,)   𝐷(𝑥,𝑦,)   𝑅()   𝑆(𝑥,𝑦,𝑧,𝑤,)   < (𝑥,𝑦,𝑧,𝑤,)   𝑇()   (𝑥,𝑦,𝑧,𝑤,)   𝑂(𝑥,𝑦,𝑧,𝑤,)   𝑉(𝑥,𝑦,𝑧,𝑤,)   𝑊(𝑥,𝑦,𝑧,𝑤,)

Proof of Theorem opsrval
Dummy variables 𝑟 𝑖 𝑝 𝑠 𝑑 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 opsrval.o . 2 𝑂 = ((𝐼 ordPwSer 𝑅)‘𝑇)
2 opsrval.i . . . . 5 (𝜑𝐼𝑉)
3 elex 3180 . . . . 5 (𝐼𝑉𝐼 ∈ V)
42, 3syl 17 . . . 4 (𝜑𝐼 ∈ V)
5 opsrval.r . . . . 5 (𝜑𝑅𝑊)
6 elex 3180 . . . . 5 (𝑅𝑊𝑅 ∈ V)
75, 6syl 17 . . . 4 (𝜑𝑅 ∈ V)
8 xpexg 6831 . . . . . 6 ((𝐼𝑉𝐼𝑉) → (𝐼 × 𝐼) ∈ V)
92, 2, 8syl2anc 690 . . . . 5 (𝜑 → (𝐼 × 𝐼) ∈ V)
10 pwexg 4767 . . . . 5 ((𝐼 × 𝐼) ∈ V → 𝒫 (𝐼 × 𝐼) ∈ V)
11 mptexg 6363 . . . . 5 (𝒫 (𝐼 × 𝐼) ∈ V → (𝑟 ∈ 𝒫 (𝐼 × 𝐼) ↦ (𝑆 sSet ⟨(le‘ndx), {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝐵 ∧ (∃𝑧𝐷 ((𝑥𝑧) < (𝑦𝑧) ∧ ∀𝑤𝐷 (𝑤(𝑟 <bag 𝐼)𝑧 → (𝑥𝑤) = (𝑦𝑤))) ∨ 𝑥 = 𝑦))}⟩)) ∈ V)
129, 10, 113syl 18 . . . 4 (𝜑 → (𝑟 ∈ 𝒫 (𝐼 × 𝐼) ↦ (𝑆 sSet ⟨(le‘ndx), {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝐵 ∧ (∃𝑧𝐷 ((𝑥𝑧) < (𝑦𝑧) ∧ ∀𝑤𝐷 (𝑤(𝑟 <bag 𝐼)𝑧 → (𝑥𝑤) = (𝑦𝑤))) ∨ 𝑥 = 𝑦))}⟩)) ∈ V)
13 simpl 471 . . . . . . . 8 ((𝑖 = 𝐼𝑠 = 𝑅) → 𝑖 = 𝐼)
1413sqxpeqd 5051 . . . . . . 7 ((𝑖 = 𝐼𝑠 = 𝑅) → (𝑖 × 𝑖) = (𝐼 × 𝐼))
1514pweqd 4108 . . . . . 6 ((𝑖 = 𝐼𝑠 = 𝑅) → 𝒫 (𝑖 × 𝑖) = 𝒫 (𝐼 × 𝐼))
16 ovex 6551 . . . . . . . 8 (𝑖 mPwSer 𝑠) ∈ V
1716a1i 11 . . . . . . 7 ((𝑖 = 𝐼𝑠 = 𝑅) → (𝑖 mPwSer 𝑠) ∈ V)
18 id 22 . . . . . . . . . 10 (𝑝 = (𝑖 mPwSer 𝑠) → 𝑝 = (𝑖 mPwSer 𝑠))
19 oveq12 6532 . . . . . . . . . 10 ((𝑖 = 𝐼𝑠 = 𝑅) → (𝑖 mPwSer 𝑠) = (𝐼 mPwSer 𝑅))
2018, 19sylan9eqr 2661 . . . . . . . . 9 (((𝑖 = 𝐼𝑠 = 𝑅) ∧ 𝑝 = (𝑖 mPwSer 𝑠)) → 𝑝 = (𝐼 mPwSer 𝑅))
21 opsrval.s . . . . . . . . 9 𝑆 = (𝐼 mPwSer 𝑅)
2220, 21syl6eqr 2657 . . . . . . . 8 (((𝑖 = 𝐼𝑠 = 𝑅) ∧ 𝑝 = (𝑖 mPwSer 𝑠)) → 𝑝 = 𝑆)
2322fveq2d 6088 . . . . . . . . . . . . 13 (((𝑖 = 𝐼𝑠 = 𝑅) ∧ 𝑝 = (𝑖 mPwSer 𝑠)) → (Base‘𝑝) = (Base‘𝑆))
24 opsrval.b . . . . . . . . . . . . 13 𝐵 = (Base‘𝑆)
2523, 24syl6eqr 2657 . . . . . . . . . . . 12 (((𝑖 = 𝐼𝑠 = 𝑅) ∧ 𝑝 = (𝑖 mPwSer 𝑠)) → (Base‘𝑝) = 𝐵)
2625sseq2d 3591 . . . . . . . . . . 11 (((𝑖 = 𝐼𝑠 = 𝑅) ∧ 𝑝 = (𝑖 mPwSer 𝑠)) → ({𝑥, 𝑦} ⊆ (Base‘𝑝) ↔ {𝑥, 𝑦} ⊆ 𝐵))
27 ovex 6551 . . . . . . . . . . . . . . 15 (ℕ0𝑚 𝑖) ∈ V
2827rabex 4731 . . . . . . . . . . . . . 14 { ∈ (ℕ0𝑚 𝑖) ∣ ( “ ℕ) ∈ Fin} ∈ V
2928a1i 11 . . . . . . . . . . . . 13 (((𝑖 = 𝐼𝑠 = 𝑅) ∧ 𝑝 = (𝑖 mPwSer 𝑠)) → { ∈ (ℕ0𝑚 𝑖) ∣ ( “ ℕ) ∈ Fin} ∈ V)
3013adantr 479 . . . . . . . . . . . . . . . 16 (((𝑖 = 𝐼𝑠 = 𝑅) ∧ 𝑝 = (𝑖 mPwSer 𝑠)) → 𝑖 = 𝐼)
3130oveq2d 6539 . . . . . . . . . . . . . . 15 (((𝑖 = 𝐼𝑠 = 𝑅) ∧ 𝑝 = (𝑖 mPwSer 𝑠)) → (ℕ0𝑚 𝑖) = (ℕ0𝑚 𝐼))
32 rabeq 3161 . . . . . . . . . . . . . . 15 ((ℕ0𝑚 𝑖) = (ℕ0𝑚 𝐼) → { ∈ (ℕ0𝑚 𝑖) ∣ ( “ ℕ) ∈ Fin} = { ∈ (ℕ0𝑚 𝐼) ∣ ( “ ℕ) ∈ Fin})
3331, 32syl 17 . . . . . . . . . . . . . 14 (((𝑖 = 𝐼𝑠 = 𝑅) ∧ 𝑝 = (𝑖 mPwSer 𝑠)) → { ∈ (ℕ0𝑚 𝑖) ∣ ( “ ℕ) ∈ Fin} = { ∈ (ℕ0𝑚 𝐼) ∣ ( “ ℕ) ∈ Fin})
34 opsrval.d . . . . . . . . . . . . . 14 𝐷 = { ∈ (ℕ0𝑚 𝐼) ∣ ( “ ℕ) ∈ Fin}
3533, 34syl6eqr 2657 . . . . . . . . . . . . 13 (((𝑖 = 𝐼𝑠 = 𝑅) ∧ 𝑝 = (𝑖 mPwSer 𝑠)) → { ∈ (ℕ0𝑚 𝑖) ∣ ( “ ℕ) ∈ Fin} = 𝐷)
36 simpr 475 . . . . . . . . . . . . . 14 ((((𝑖 = 𝐼𝑠 = 𝑅) ∧ 𝑝 = (𝑖 mPwSer 𝑠)) ∧ 𝑑 = 𝐷) → 𝑑 = 𝐷)
37 simpllr 794 . . . . . . . . . . . . . . . . . 18 ((((𝑖 = 𝐼𝑠 = 𝑅) ∧ 𝑝 = (𝑖 mPwSer 𝑠)) ∧ 𝑑 = 𝐷) → 𝑠 = 𝑅)
3837fveq2d 6088 . . . . . . . . . . . . . . . . 17 ((((𝑖 = 𝐼𝑠 = 𝑅) ∧ 𝑝 = (𝑖 mPwSer 𝑠)) ∧ 𝑑 = 𝐷) → (lt‘𝑠) = (lt‘𝑅))
39 opsrval.q . . . . . . . . . . . . . . . . 17 < = (lt‘𝑅)
4038, 39syl6eqr 2657 . . . . . . . . . . . . . . . 16 ((((𝑖 = 𝐼𝑠 = 𝑅) ∧ 𝑝 = (𝑖 mPwSer 𝑠)) ∧ 𝑑 = 𝐷) → (lt‘𝑠) = < )
4140breqd 4584 . . . . . . . . . . . . . . 15 ((((𝑖 = 𝐼𝑠 = 𝑅) ∧ 𝑝 = (𝑖 mPwSer 𝑠)) ∧ 𝑑 = 𝐷) → ((𝑥𝑧)(lt‘𝑠)(𝑦𝑧) ↔ (𝑥𝑧) < (𝑦𝑧)))
4230adantr 479 . . . . . . . . . . . . . . . . . . 19 ((((𝑖 = 𝐼𝑠 = 𝑅) ∧ 𝑝 = (𝑖 mPwSer 𝑠)) ∧ 𝑑 = 𝐷) → 𝑖 = 𝐼)
4342oveq2d 6539 . . . . . . . . . . . . . . . . . 18 ((((𝑖 = 𝐼𝑠 = 𝑅) ∧ 𝑝 = (𝑖 mPwSer 𝑠)) ∧ 𝑑 = 𝐷) → (𝑟 <bag 𝑖) = (𝑟 <bag 𝐼))
4443breqd 4584 . . . . . . . . . . . . . . . . 17 ((((𝑖 = 𝐼𝑠 = 𝑅) ∧ 𝑝 = (𝑖 mPwSer 𝑠)) ∧ 𝑑 = 𝐷) → (𝑤(𝑟 <bag 𝑖)𝑧𝑤(𝑟 <bag 𝐼)𝑧))
4544imbi1d 329 . . . . . . . . . . . . . . . 16 ((((𝑖 = 𝐼𝑠 = 𝑅) ∧ 𝑝 = (𝑖 mPwSer 𝑠)) ∧ 𝑑 = 𝐷) → ((𝑤(𝑟 <bag 𝑖)𝑧 → (𝑥𝑤) = (𝑦𝑤)) ↔ (𝑤(𝑟 <bag 𝐼)𝑧 → (𝑥𝑤) = (𝑦𝑤))))
4636, 45raleqbidv 3124 . . . . . . . . . . . . . . 15 ((((𝑖 = 𝐼𝑠 = 𝑅) ∧ 𝑝 = (𝑖 mPwSer 𝑠)) ∧ 𝑑 = 𝐷) → (∀𝑤𝑑 (𝑤(𝑟 <bag 𝑖)𝑧 → (𝑥𝑤) = (𝑦𝑤)) ↔ ∀𝑤𝐷 (𝑤(𝑟 <bag 𝐼)𝑧 → (𝑥𝑤) = (𝑦𝑤))))
4741, 46anbi12d 742 . . . . . . . . . . . . . 14 ((((𝑖 = 𝐼𝑠 = 𝑅) ∧ 𝑝 = (𝑖 mPwSer 𝑠)) ∧ 𝑑 = 𝐷) → (((𝑥𝑧)(lt‘𝑠)(𝑦𝑧) ∧ ∀𝑤𝑑 (𝑤(𝑟 <bag 𝑖)𝑧 → (𝑥𝑤) = (𝑦𝑤))) ↔ ((𝑥𝑧) < (𝑦𝑧) ∧ ∀𝑤𝐷 (𝑤(𝑟 <bag 𝐼)𝑧 → (𝑥𝑤) = (𝑦𝑤)))))
4836, 47rexeqbidv 3125 . . . . . . . . . . . . 13 ((((𝑖 = 𝐼𝑠 = 𝑅) ∧ 𝑝 = (𝑖 mPwSer 𝑠)) ∧ 𝑑 = 𝐷) → (∃𝑧𝑑 ((𝑥𝑧)(lt‘𝑠)(𝑦𝑧) ∧ ∀𝑤𝑑 (𝑤(𝑟 <bag 𝑖)𝑧 → (𝑥𝑤) = (𝑦𝑤))) ↔ ∃𝑧𝐷 ((𝑥𝑧) < (𝑦𝑧) ∧ ∀𝑤𝐷 (𝑤(𝑟 <bag 𝐼)𝑧 → (𝑥𝑤) = (𝑦𝑤)))))
4929, 35, 48sbcied2 3435 . . . . . . . . . . . 12 (((𝑖 = 𝐼𝑠 = 𝑅) ∧ 𝑝 = (𝑖 mPwSer 𝑠)) → ([{ ∈ (ℕ0𝑚 𝑖) ∣ ( “ ℕ) ∈ Fin} / 𝑑]𝑧𝑑 ((𝑥𝑧)(lt‘𝑠)(𝑦𝑧) ∧ ∀𝑤𝑑 (𝑤(𝑟 <bag 𝑖)𝑧 → (𝑥𝑤) = (𝑦𝑤))) ↔ ∃𝑧𝐷 ((𝑥𝑧) < (𝑦𝑧) ∧ ∀𝑤𝐷 (𝑤(𝑟 <bag 𝐼)𝑧 → (𝑥𝑤) = (𝑦𝑤)))))
5049orbi1d 734 . . . . . . . . . . 11 (((𝑖 = 𝐼𝑠 = 𝑅) ∧ 𝑝 = (𝑖 mPwSer 𝑠)) → (([{ ∈ (ℕ0𝑚 𝑖) ∣ ( “ ℕ) ∈ Fin} / 𝑑]𝑧𝑑 ((𝑥𝑧)(lt‘𝑠)(𝑦𝑧) ∧ ∀𝑤𝑑 (𝑤(𝑟 <bag 𝑖)𝑧 → (𝑥𝑤) = (𝑦𝑤))) ∨ 𝑥 = 𝑦) ↔ (∃𝑧𝐷 ((𝑥𝑧) < (𝑦𝑧) ∧ ∀𝑤𝐷 (𝑤(𝑟 <bag 𝐼)𝑧 → (𝑥𝑤) = (𝑦𝑤))) ∨ 𝑥 = 𝑦)))
5126, 50anbi12d 742 . . . . . . . . . 10 (((𝑖 = 𝐼𝑠 = 𝑅) ∧ 𝑝 = (𝑖 mPwSer 𝑠)) → (({𝑥, 𝑦} ⊆ (Base‘𝑝) ∧ ([{ ∈ (ℕ0𝑚 𝑖) ∣ ( “ ℕ) ∈ Fin} / 𝑑]𝑧𝑑 ((𝑥𝑧)(lt‘𝑠)(𝑦𝑧) ∧ ∀𝑤𝑑 (𝑤(𝑟 <bag 𝑖)𝑧 → (𝑥𝑤) = (𝑦𝑤))) ∨ 𝑥 = 𝑦)) ↔ ({𝑥, 𝑦} ⊆ 𝐵 ∧ (∃𝑧𝐷 ((𝑥𝑧) < (𝑦𝑧) ∧ ∀𝑤𝐷 (𝑤(𝑟 <bag 𝐼)𝑧 → (𝑥𝑤) = (𝑦𝑤))) ∨ 𝑥 = 𝑦))))
5251opabbidv 4638 . . . . . . . . 9 (((𝑖 = 𝐼𝑠 = 𝑅) ∧ 𝑝 = (𝑖 mPwSer 𝑠)) → {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ (Base‘𝑝) ∧ ([{ ∈ (ℕ0𝑚 𝑖) ∣ ( “ ℕ) ∈ Fin} / 𝑑]𝑧𝑑 ((𝑥𝑧)(lt‘𝑠)(𝑦𝑧) ∧ ∀𝑤𝑑 (𝑤(𝑟 <bag 𝑖)𝑧 → (𝑥𝑤) = (𝑦𝑤))) ∨ 𝑥 = 𝑦))} = {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝐵 ∧ (∃𝑧𝐷 ((𝑥𝑧) < (𝑦𝑧) ∧ ∀𝑤𝐷 (𝑤(𝑟 <bag 𝐼)𝑧 → (𝑥𝑤) = (𝑦𝑤))) ∨ 𝑥 = 𝑦))})
5352opeq2d 4337 . . . . . . . 8 (((𝑖 = 𝐼𝑠 = 𝑅) ∧ 𝑝 = (𝑖 mPwSer 𝑠)) → ⟨(le‘ndx), {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ (Base‘𝑝) ∧ ([{ ∈ (ℕ0𝑚 𝑖) ∣ ( “ ℕ) ∈ Fin} / 𝑑]𝑧𝑑 ((𝑥𝑧)(lt‘𝑠)(𝑦𝑧) ∧ ∀𝑤𝑑 (𝑤(𝑟 <bag 𝑖)𝑧 → (𝑥𝑤) = (𝑦𝑤))) ∨ 𝑥 = 𝑦))}⟩ = ⟨(le‘ndx), {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝐵 ∧ (∃𝑧𝐷 ((𝑥𝑧) < (𝑦𝑧) ∧ ∀𝑤𝐷 (𝑤(𝑟 <bag 𝐼)𝑧 → (𝑥𝑤) = (𝑦𝑤))) ∨ 𝑥 = 𝑦))}⟩)
5422, 53oveq12d 6541 . . . . . . 7 (((𝑖 = 𝐼𝑠 = 𝑅) ∧ 𝑝 = (𝑖 mPwSer 𝑠)) → (𝑝 sSet ⟨(le‘ndx), {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ (Base‘𝑝) ∧ ([{ ∈ (ℕ0𝑚 𝑖) ∣ ( “ ℕ) ∈ Fin} / 𝑑]𝑧𝑑 ((𝑥𝑧)(lt‘𝑠)(𝑦𝑧) ∧ ∀𝑤𝑑 (𝑤(𝑟 <bag 𝑖)𝑧 → (𝑥𝑤) = (𝑦𝑤))) ∨ 𝑥 = 𝑦))}⟩) = (𝑆 sSet ⟨(le‘ndx), {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝐵 ∧ (∃𝑧𝐷 ((𝑥𝑧) < (𝑦𝑧) ∧ ∀𝑤𝐷 (𝑤(𝑟 <bag 𝐼)𝑧 → (𝑥𝑤) = (𝑦𝑤))) ∨ 𝑥 = 𝑦))}⟩))
5517, 54csbied 3521 . . . . . 6 ((𝑖 = 𝐼𝑠 = 𝑅) → (𝑖 mPwSer 𝑠) / 𝑝(𝑝 sSet ⟨(le‘ndx), {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ (Base‘𝑝) ∧ ([{ ∈ (ℕ0𝑚 𝑖) ∣ ( “ ℕ) ∈ Fin} / 𝑑]𝑧𝑑 ((𝑥𝑧)(lt‘𝑠)(𝑦𝑧) ∧ ∀𝑤𝑑 (𝑤(𝑟 <bag 𝑖)𝑧 → (𝑥𝑤) = (𝑦𝑤))) ∨ 𝑥 = 𝑦))}⟩) = (𝑆 sSet ⟨(le‘ndx), {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝐵 ∧ (∃𝑧𝐷 ((𝑥𝑧) < (𝑦𝑧) ∧ ∀𝑤𝐷 (𝑤(𝑟 <bag 𝐼)𝑧 → (𝑥𝑤) = (𝑦𝑤))) ∨ 𝑥 = 𝑦))}⟩))
5615, 55mpteq12dv 4653 . . . . 5 ((𝑖 = 𝐼𝑠 = 𝑅) → (𝑟 ∈ 𝒫 (𝑖 × 𝑖) ↦ (𝑖 mPwSer 𝑠) / 𝑝(𝑝 sSet ⟨(le‘ndx), {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ (Base‘𝑝) ∧ ([{ ∈ (ℕ0𝑚 𝑖) ∣ ( “ ℕ) ∈ Fin} / 𝑑]𝑧𝑑 ((𝑥𝑧)(lt‘𝑠)(𝑦𝑧) ∧ ∀𝑤𝑑 (𝑤(𝑟 <bag 𝑖)𝑧 → (𝑥𝑤) = (𝑦𝑤))) ∨ 𝑥 = 𝑦))}⟩)) = (𝑟 ∈ 𝒫 (𝐼 × 𝐼) ↦ (𝑆 sSet ⟨(le‘ndx), {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝐵 ∧ (∃𝑧𝐷 ((𝑥𝑧) < (𝑦𝑧) ∧ ∀𝑤𝐷 (𝑤(𝑟 <bag 𝐼)𝑧 → (𝑥𝑤) = (𝑦𝑤))) ∨ 𝑥 = 𝑦))}⟩)))
57 df-opsr 19123 . . . . 5 ordPwSer = (𝑖 ∈ V, 𝑠 ∈ V ↦ (𝑟 ∈ 𝒫 (𝑖 × 𝑖) ↦ (𝑖 mPwSer 𝑠) / 𝑝(𝑝 sSet ⟨(le‘ndx), {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ (Base‘𝑝) ∧ ([{ ∈ (ℕ0𝑚 𝑖) ∣ ( “ ℕ) ∈ Fin} / 𝑑]𝑧𝑑 ((𝑥𝑧)(lt‘𝑠)(𝑦𝑧) ∧ ∀𝑤𝑑 (𝑤(𝑟 <bag 𝑖)𝑧 → (𝑥𝑤) = (𝑦𝑤))) ∨ 𝑥 = 𝑦))}⟩)))
5856, 57ovmpt2ga 6662 . . . 4 ((𝐼 ∈ V ∧ 𝑅 ∈ V ∧ (𝑟 ∈ 𝒫 (𝐼 × 𝐼) ↦ (𝑆 sSet ⟨(le‘ndx), {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝐵 ∧ (∃𝑧𝐷 ((𝑥𝑧) < (𝑦𝑧) ∧ ∀𝑤𝐷 (𝑤(𝑟 <bag 𝐼)𝑧 → (𝑥𝑤) = (𝑦𝑤))) ∨ 𝑥 = 𝑦))}⟩)) ∈ V) → (𝐼 ordPwSer 𝑅) = (𝑟 ∈ 𝒫 (𝐼 × 𝐼) ↦ (𝑆 sSet ⟨(le‘ndx), {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝐵 ∧ (∃𝑧𝐷 ((𝑥𝑧) < (𝑦𝑧) ∧ ∀𝑤𝐷 (𝑤(𝑟 <bag 𝐼)𝑧 → (𝑥𝑤) = (𝑦𝑤))) ∨ 𝑥 = 𝑦))}⟩)))
594, 7, 12, 58syl3anc 1317 . . 3 (𝜑 → (𝐼 ordPwSer 𝑅) = (𝑟 ∈ 𝒫 (𝐼 × 𝐼) ↦ (𝑆 sSet ⟨(le‘ndx), {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝐵 ∧ (∃𝑧𝐷 ((𝑥𝑧) < (𝑦𝑧) ∧ ∀𝑤𝐷 (𝑤(𝑟 <bag 𝐼)𝑧 → (𝑥𝑤) = (𝑦𝑤))) ∨ 𝑥 = 𝑦))}⟩)))
60 simpr 475 . . . . . . . . . . . . . . . 16 ((𝜑𝑟 = 𝑇) → 𝑟 = 𝑇)
6160oveq1d 6538 . . . . . . . . . . . . . . 15 ((𝜑𝑟 = 𝑇) → (𝑟 <bag 𝐼) = (𝑇 <bag 𝐼))
62 opsrval.c . . . . . . . . . . . . . . 15 𝐶 = (𝑇 <bag 𝐼)
6361, 62syl6eqr 2657 . . . . . . . . . . . . . 14 ((𝜑𝑟 = 𝑇) → (𝑟 <bag 𝐼) = 𝐶)
6463breqd 4584 . . . . . . . . . . . . 13 ((𝜑𝑟 = 𝑇) → (𝑤(𝑟 <bag 𝐼)𝑧𝑤𝐶𝑧))
6564imbi1d 329 . . . . . . . . . . . 12 ((𝜑𝑟 = 𝑇) → ((𝑤(𝑟 <bag 𝐼)𝑧 → (𝑥𝑤) = (𝑦𝑤)) ↔ (𝑤𝐶𝑧 → (𝑥𝑤) = (𝑦𝑤))))
6665ralbidv 2964 . . . . . . . . . . 11 ((𝜑𝑟 = 𝑇) → (∀𝑤𝐷 (𝑤(𝑟 <bag 𝐼)𝑧 → (𝑥𝑤) = (𝑦𝑤)) ↔ ∀𝑤𝐷 (𝑤𝐶𝑧 → (𝑥𝑤) = (𝑦𝑤))))
6766anbi2d 735 . . . . . . . . . 10 ((𝜑𝑟 = 𝑇) → (((𝑥𝑧) < (𝑦𝑧) ∧ ∀𝑤𝐷 (𝑤(𝑟 <bag 𝐼)𝑧 → (𝑥𝑤) = (𝑦𝑤))) ↔ ((𝑥𝑧) < (𝑦𝑧) ∧ ∀𝑤𝐷 (𝑤𝐶𝑧 → (𝑥𝑤) = (𝑦𝑤)))))
6867rexbidv 3029 . . . . . . . . 9 ((𝜑𝑟 = 𝑇) → (∃𝑧𝐷 ((𝑥𝑧) < (𝑦𝑧) ∧ ∀𝑤𝐷 (𝑤(𝑟 <bag 𝐼)𝑧 → (𝑥𝑤) = (𝑦𝑤))) ↔ ∃𝑧𝐷 ((𝑥𝑧) < (𝑦𝑧) ∧ ∀𝑤𝐷 (𝑤𝐶𝑧 → (𝑥𝑤) = (𝑦𝑤)))))
6968orbi1d 734 . . . . . . . 8 ((𝜑𝑟 = 𝑇) → ((∃𝑧𝐷 ((𝑥𝑧) < (𝑦𝑧) ∧ ∀𝑤𝐷 (𝑤(𝑟 <bag 𝐼)𝑧 → (𝑥𝑤) = (𝑦𝑤))) ∨ 𝑥 = 𝑦) ↔ (∃𝑧𝐷 ((𝑥𝑧) < (𝑦𝑧) ∧ ∀𝑤𝐷 (𝑤𝐶𝑧 → (𝑥𝑤) = (𝑦𝑤))) ∨ 𝑥 = 𝑦)))
7069anbi2d 735 . . . . . . 7 ((𝜑𝑟 = 𝑇) → (({𝑥, 𝑦} ⊆ 𝐵 ∧ (∃𝑧𝐷 ((𝑥𝑧) < (𝑦𝑧) ∧ ∀𝑤𝐷 (𝑤(𝑟 <bag 𝐼)𝑧 → (𝑥𝑤) = (𝑦𝑤))) ∨ 𝑥 = 𝑦)) ↔ ({𝑥, 𝑦} ⊆ 𝐵 ∧ (∃𝑧𝐷 ((𝑥𝑧) < (𝑦𝑧) ∧ ∀𝑤𝐷 (𝑤𝐶𝑧 → (𝑥𝑤) = (𝑦𝑤))) ∨ 𝑥 = 𝑦))))
7170opabbidv 4638 . . . . . 6 ((𝜑𝑟 = 𝑇) → {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝐵 ∧ (∃𝑧𝐷 ((𝑥𝑧) < (𝑦𝑧) ∧ ∀𝑤𝐷 (𝑤(𝑟 <bag 𝐼)𝑧 → (𝑥𝑤) = (𝑦𝑤))) ∨ 𝑥 = 𝑦))} = {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝐵 ∧ (∃𝑧𝐷 ((𝑥𝑧) < (𝑦𝑧) ∧ ∀𝑤𝐷 (𝑤𝐶𝑧 → (𝑥𝑤) = (𝑦𝑤))) ∨ 𝑥 = 𝑦))})
72 opsrval.l . . . . . 6 = {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝐵 ∧ (∃𝑧𝐷 ((𝑥𝑧) < (𝑦𝑧) ∧ ∀𝑤𝐷 (𝑤𝐶𝑧 → (𝑥𝑤) = (𝑦𝑤))) ∨ 𝑥 = 𝑦))}
7371, 72syl6eqr 2657 . . . . 5 ((𝜑𝑟 = 𝑇) → {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝐵 ∧ (∃𝑧𝐷 ((𝑥𝑧) < (𝑦𝑧) ∧ ∀𝑤𝐷 (𝑤(𝑟 <bag 𝐼)𝑧 → (𝑥𝑤) = (𝑦𝑤))) ∨ 𝑥 = 𝑦))} = )
7473opeq2d 4337 . . . 4 ((𝜑𝑟 = 𝑇) → ⟨(le‘ndx), {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝐵 ∧ (∃𝑧𝐷 ((𝑥𝑧) < (𝑦𝑧) ∧ ∀𝑤𝐷 (𝑤(𝑟 <bag 𝐼)𝑧 → (𝑥𝑤) = (𝑦𝑤))) ∨ 𝑥 = 𝑦))}⟩ = ⟨(le‘ndx), ⟩)
7574oveq2d 6539 . . 3 ((𝜑𝑟 = 𝑇) → (𝑆 sSet ⟨(le‘ndx), {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝐵 ∧ (∃𝑧𝐷 ((𝑥𝑧) < (𝑦𝑧) ∧ ∀𝑤𝐷 (𝑤(𝑟 <bag 𝐼)𝑧 → (𝑥𝑤) = (𝑦𝑤))) ∨ 𝑥 = 𝑦))}⟩) = (𝑆 sSet ⟨(le‘ndx), ⟩))
76 opsrval.t . . . 4 (𝜑𝑇 ⊆ (𝐼 × 𝐼))
77 elpw2g 4745 . . . . 5 ((𝐼 × 𝐼) ∈ V → (𝑇 ∈ 𝒫 (𝐼 × 𝐼) ↔ 𝑇 ⊆ (𝐼 × 𝐼)))
789, 77syl 17 . . . 4 (𝜑 → (𝑇 ∈ 𝒫 (𝐼 × 𝐼) ↔ 𝑇 ⊆ (𝐼 × 𝐼)))
7976, 78mpbird 245 . . 3 (𝜑𝑇 ∈ 𝒫 (𝐼 × 𝐼))
80 ovex 6551 . . . 4 (𝑆 sSet ⟨(le‘ndx), ⟩) ∈ V
8180a1i 11 . . 3 (𝜑 → (𝑆 sSet ⟨(le‘ndx), ⟩) ∈ V)
8259, 75, 79, 81fvmptd 6178 . 2 (𝜑 → ((𝐼 ordPwSer 𝑅)‘𝑇) = (𝑆 sSet ⟨(le‘ndx), ⟩))
831, 82syl5eq 2651 1 (𝜑𝑂 = (𝑆 sSet ⟨(le‘ndx), ⟩))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wo 381  wa 382   = wceq 1474  wcel 1975  wral 2891  wrex 2892  {crab 2895  Vcvv 3168  [wsbc 3397  csb 3494  wss 3535  𝒫 cpw 4103  {cpr 4122  cop 4126   class class class wbr 4573  {copab 4632  cmpt 4633   × cxp 5022  ccnv 5023  cima 5027  cfv 5786  (class class class)co 6523  𝑚 cmap 7717  Fincfn 7814  cn 10863  0cn0 11135  ndxcnx 15634   sSet csts 15635  Basecbs 15637  lecple 15717  ltcplt 16706   mPwSer cmps 19114   <bag cltb 19117   ordPwSer copws 19118
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-8 1977  ax-9 1984  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2228  ax-ext 2585  ax-rep 4689  ax-sep 4699  ax-nul 4708  ax-pow 4760  ax-pr 4824  ax-un 6820
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1866  df-eu 2457  df-mo 2458  df-clab 2592  df-cleq 2598  df-clel 2601  df-nfc 2735  df-ne 2777  df-ral 2896  df-rex 2897  df-reu 2898  df-rab 2900  df-v 3170  df-sbc 3398  df-csb 3495  df-dif 3538  df-un 3540  df-in 3542  df-ss 3549  df-nul 3870  df-if 4032  df-pw 4105  df-sn 4121  df-pr 4123  df-op 4127  df-uni 4363  df-iun 4447  df-br 4574  df-opab 4634  df-mpt 4635  df-id 4939  df-xp 5030  df-rel 5031  df-cnv 5032  df-co 5033  df-dm 5034  df-rn 5035  df-res 5036  df-ima 5037  df-iota 5750  df-fun 5788  df-fn 5789  df-f 5790  df-f1 5791  df-fo 5792  df-f1o 5793  df-fv 5794  df-ov 6526  df-oprab 6527  df-mpt2 6528  df-opsr 19123
This theorem is referenced by:  opsrle  19238  opsrval2  19239
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