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Theorem opsrval 21991
Description: The value of the "ordered power series" function. This is the same as mPwSer psrval 21855, 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)β€˜π‘‡)
opsrval.b 𝐡 = (Baseβ€˜π‘†)
opsrval.q < = (ltβ€˜π‘…)
opsrval.c 𝐢 = (𝑇 <bag 𝐼)
opsrval.d 𝐷 = {β„Ž ∈ (β„•0 ↑m 𝐼) ∣ (β—‘β„Ž β€œ β„•) ∈ Fin}
opsrval.l ≀ = {⟨π‘₯, π‘¦βŸ© ∣ ({π‘₯, 𝑦} βŠ† 𝐡 ∧ (βˆƒπ‘§ ∈ 𝐷 ((π‘₯β€˜π‘§) < (π‘¦β€˜π‘§) ∧ βˆ€π‘€ ∈ 𝐷 (𝑀𝐢𝑧 β†’ (π‘₯β€˜π‘€) = (π‘¦β€˜π‘€))) ∨ π‘₯ = 𝑦))}
opsrval.i (πœ‘ β†’ 𝐼 ∈ 𝑉)
opsrval.r (πœ‘ β†’ 𝑅 ∈ π‘Š)
opsrval.t (πœ‘ β†’ 𝑇 βŠ† (𝐼 Γ— 𝐼))
Assertion
Ref Expression
opsrval (πœ‘ β†’ 𝑂 = (𝑆 sSet ⟨(leβ€˜ndx), ≀ ⟩))
Distinct variable groups:   𝑀,β„Ž,π‘₯,𝑦,𝑧,𝐼   πœ‘,𝑀,π‘₯,𝑦,𝑧   𝑀,𝐷,𝑧   𝑀,𝑇,π‘₯,𝑦,𝑧   𝑀,𝑅,π‘₯,𝑦,𝑧
Allowed substitution hints:   πœ‘(β„Ž)   𝐡(π‘₯,𝑦,𝑧,𝑀,β„Ž)   𝐢(π‘₯,𝑦,𝑧,𝑀,β„Ž)   𝐷(π‘₯,𝑦,β„Ž)   𝑅(β„Ž)   𝑆(π‘₯,𝑦,𝑧,𝑀,β„Ž)   < (π‘₯,𝑦,𝑧,𝑀,β„Ž)   𝑇(β„Ž)   ≀ (π‘₯,𝑦,𝑧,𝑀,β„Ž)   𝑂(π‘₯,𝑦,𝑧,𝑀,β„Ž)   𝑉(π‘₯,𝑦,𝑧,𝑀,β„Ž)   π‘Š(π‘₯,𝑦,𝑧,𝑀,β„Ž)

Proof of Theorem opsrval
Dummy variables π‘Ÿ 𝑖 𝑝 𝑠 𝑑 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 opsrval.o . 2 𝑂 = ((𝐼 ordPwSer 𝑅)β€˜π‘‡)
2 opsrval.i . . . . 5 (πœ‘ β†’ 𝐼 ∈ 𝑉)
32elexd 3494 . . . 4 (πœ‘ β†’ 𝐼 ∈ V)
4 opsrval.r . . . . 5 (πœ‘ β†’ 𝑅 ∈ π‘Š)
54elexd 3494 . . . 4 (πœ‘ β†’ 𝑅 ∈ V)
62, 2xpexd 7759 . . . . 5 (πœ‘ β†’ (𝐼 Γ— 𝐼) ∈ V)
7 pwexg 5382 . . . . 5 ((𝐼 Γ— 𝐼) ∈ V β†’ 𝒫 (𝐼 Γ— 𝐼) ∈ V)
8 mptexg 7239 . . . . 5 (𝒫 (𝐼 Γ— 𝐼) ∈ V β†’ (π‘Ÿ ∈ 𝒫 (𝐼 Γ— 𝐼) ↦ (𝑆 sSet ⟨(leβ€˜ndx), {⟨π‘₯, π‘¦βŸ© ∣ ({π‘₯, 𝑦} βŠ† 𝐡 ∧ (βˆƒπ‘§ ∈ 𝐷 ((π‘₯β€˜π‘§) < (π‘¦β€˜π‘§) ∧ βˆ€π‘€ ∈ 𝐷 (𝑀(π‘Ÿ <bag 𝐼)𝑧 β†’ (π‘₯β€˜π‘€) = (π‘¦β€˜π‘€))) ∨ π‘₯ = 𝑦))}⟩)) ∈ V)
96, 7, 83syl 18 . . . 4 (πœ‘ β†’ (π‘Ÿ ∈ 𝒫 (𝐼 Γ— 𝐼) ↦ (𝑆 sSet ⟨(leβ€˜ndx), {⟨π‘₯, π‘¦βŸ© ∣ ({π‘₯, 𝑦} βŠ† 𝐡 ∧ (βˆƒπ‘§ ∈ 𝐷 ((π‘₯β€˜π‘§) < (π‘¦β€˜π‘§) ∧ βˆ€π‘€ ∈ 𝐷 (𝑀(π‘Ÿ <bag 𝐼)𝑧 β†’ (π‘₯β€˜π‘€) = (π‘¦β€˜π‘€))) ∨ π‘₯ = 𝑦))}⟩)) ∈ V)
10 simpl 481 . . . . . . . 8 ((𝑖 = 𝐼 ∧ 𝑠 = 𝑅) β†’ 𝑖 = 𝐼)
1110sqxpeqd 5714 . . . . . . 7 ((𝑖 = 𝐼 ∧ 𝑠 = 𝑅) β†’ (𝑖 Γ— 𝑖) = (𝐼 Γ— 𝐼))
1211pweqd 4623 . . . . . 6 ((𝑖 = 𝐼 ∧ 𝑠 = 𝑅) β†’ 𝒫 (𝑖 Γ— 𝑖) = 𝒫 (𝐼 Γ— 𝐼))
13 ovexd 7461 . . . . . . 7 ((𝑖 = 𝐼 ∧ 𝑠 = 𝑅) β†’ (𝑖 mPwSer 𝑠) ∈ V)
14 id 22 . . . . . . . . . 10 (𝑝 = (𝑖 mPwSer 𝑠) β†’ 𝑝 = (𝑖 mPwSer 𝑠))
15 oveq12 7435 . . . . . . . . . 10 ((𝑖 = 𝐼 ∧ 𝑠 = 𝑅) β†’ (𝑖 mPwSer 𝑠) = (𝐼 mPwSer 𝑅))
1614, 15sylan9eqr 2790 . . . . . . . . 9 (((𝑖 = 𝐼 ∧ 𝑠 = 𝑅) ∧ 𝑝 = (𝑖 mPwSer 𝑠)) β†’ 𝑝 = (𝐼 mPwSer 𝑅))
17 opsrval.s . . . . . . . . 9 𝑆 = (𝐼 mPwSer 𝑅)
1816, 17eqtr4di 2786 . . . . . . . 8 (((𝑖 = 𝐼 ∧ 𝑠 = 𝑅) ∧ 𝑝 = (𝑖 mPwSer 𝑠)) β†’ 𝑝 = 𝑆)
1918fveq2d 6906 . . . . . . . . . . . . 13 (((𝑖 = 𝐼 ∧ 𝑠 = 𝑅) ∧ 𝑝 = (𝑖 mPwSer 𝑠)) β†’ (Baseβ€˜π‘) = (Baseβ€˜π‘†))
20 opsrval.b . . . . . . . . . . . . 13 𝐡 = (Baseβ€˜π‘†)
2119, 20eqtr4di 2786 . . . . . . . . . . . 12 (((𝑖 = 𝐼 ∧ 𝑠 = 𝑅) ∧ 𝑝 = (𝑖 mPwSer 𝑠)) β†’ (Baseβ€˜π‘) = 𝐡)
2221sseq2d 4014 . . . . . . . . . . 11 (((𝑖 = 𝐼 ∧ 𝑠 = 𝑅) ∧ 𝑝 = (𝑖 mPwSer 𝑠)) β†’ ({π‘₯, 𝑦} βŠ† (Baseβ€˜π‘) ↔ {π‘₯, 𝑦} βŠ† 𝐡))
23 ovex 7459 . . . . . . . . . . . . . . 15 (β„•0 ↑m 𝑖) ∈ V
2423rabex 5338 . . . . . . . . . . . . . 14 {β„Ž ∈ (β„•0 ↑m 𝑖) ∣ (β—‘β„Ž β€œ β„•) ∈ Fin} ∈ V
2524a1i 11 . . . . . . . . . . . . 13 (((𝑖 = 𝐼 ∧ 𝑠 = 𝑅) ∧ 𝑝 = (𝑖 mPwSer 𝑠)) β†’ {β„Ž ∈ (β„•0 ↑m 𝑖) ∣ (β—‘β„Ž β€œ β„•) ∈ Fin} ∈ V)
2610adantr 479 . . . . . . . . . . . . . . . 16 (((𝑖 = 𝐼 ∧ 𝑠 = 𝑅) ∧ 𝑝 = (𝑖 mPwSer 𝑠)) β†’ 𝑖 = 𝐼)
2726oveq2d 7442 . . . . . . . . . . . . . . 15 (((𝑖 = 𝐼 ∧ 𝑠 = 𝑅) ∧ 𝑝 = (𝑖 mPwSer 𝑠)) β†’ (β„•0 ↑m 𝑖) = (β„•0 ↑m 𝐼))
28 rabeq 3445 . . . . . . . . . . . . . . 15 ((β„•0 ↑m 𝑖) = (β„•0 ↑m 𝐼) β†’ {β„Ž ∈ (β„•0 ↑m 𝑖) ∣ (β—‘β„Ž β€œ β„•) ∈ Fin} = {β„Ž ∈ (β„•0 ↑m 𝐼) ∣ (β—‘β„Ž β€œ β„•) ∈ Fin})
2927, 28syl 17 . . . . . . . . . . . . . 14 (((𝑖 = 𝐼 ∧ 𝑠 = 𝑅) ∧ 𝑝 = (𝑖 mPwSer 𝑠)) β†’ {β„Ž ∈ (β„•0 ↑m 𝑖) ∣ (β—‘β„Ž β€œ β„•) ∈ Fin} = {β„Ž ∈ (β„•0 ↑m 𝐼) ∣ (β—‘β„Ž β€œ β„•) ∈ Fin})
30 opsrval.d . . . . . . . . . . . . . 14 𝐷 = {β„Ž ∈ (β„•0 ↑m 𝐼) ∣ (β—‘β„Ž β€œ β„•) ∈ Fin}
3129, 30eqtr4di 2786 . . . . . . . . . . . . 13 (((𝑖 = 𝐼 ∧ 𝑠 = 𝑅) ∧ 𝑝 = (𝑖 mPwSer 𝑠)) β†’ {β„Ž ∈ (β„•0 ↑m 𝑖) ∣ (β—‘β„Ž β€œ β„•) ∈ Fin} = 𝐷)
32 simpr 483 . . . . . . . . . . . . . 14 ((((𝑖 = 𝐼 ∧ 𝑠 = 𝑅) ∧ 𝑝 = (𝑖 mPwSer 𝑠)) ∧ 𝑑 = 𝐷) β†’ 𝑑 = 𝐷)
33 simpllr 774 . . . . . . . . . . . . . . . . . 18 ((((𝑖 = 𝐼 ∧ 𝑠 = 𝑅) ∧ 𝑝 = (𝑖 mPwSer 𝑠)) ∧ 𝑑 = 𝐷) β†’ 𝑠 = 𝑅)
3433fveq2d 6906 . . . . . . . . . . . . . . . . 17 ((((𝑖 = 𝐼 ∧ 𝑠 = 𝑅) ∧ 𝑝 = (𝑖 mPwSer 𝑠)) ∧ 𝑑 = 𝐷) β†’ (ltβ€˜π‘ ) = (ltβ€˜π‘…))
35 opsrval.q . . . . . . . . . . . . . . . . 17 < = (ltβ€˜π‘…)
3634, 35eqtr4di 2786 . . . . . . . . . . . . . . . 16 ((((𝑖 = 𝐼 ∧ 𝑠 = 𝑅) ∧ 𝑝 = (𝑖 mPwSer 𝑠)) ∧ 𝑑 = 𝐷) β†’ (ltβ€˜π‘ ) = < )
3736breqd 5163 . . . . . . . . . . . . . . 15 ((((𝑖 = 𝐼 ∧ 𝑠 = 𝑅) ∧ 𝑝 = (𝑖 mPwSer 𝑠)) ∧ 𝑑 = 𝐷) β†’ ((π‘₯β€˜π‘§)(ltβ€˜π‘ )(π‘¦β€˜π‘§) ↔ (π‘₯β€˜π‘§) < (π‘¦β€˜π‘§)))
3826adantr 479 . . . . . . . . . . . . . . . . . . 19 ((((𝑖 = 𝐼 ∧ 𝑠 = 𝑅) ∧ 𝑝 = (𝑖 mPwSer 𝑠)) ∧ 𝑑 = 𝐷) β†’ 𝑖 = 𝐼)
3938oveq2d 7442 . . . . . . . . . . . . . . . . . 18 ((((𝑖 = 𝐼 ∧ 𝑠 = 𝑅) ∧ 𝑝 = (𝑖 mPwSer 𝑠)) ∧ 𝑑 = 𝐷) β†’ (π‘Ÿ <bag 𝑖) = (π‘Ÿ <bag 𝐼))
4039breqd 5163 . . . . . . . . . . . . . . . . 17 ((((𝑖 = 𝐼 ∧ 𝑠 = 𝑅) ∧ 𝑝 = (𝑖 mPwSer 𝑠)) ∧ 𝑑 = 𝐷) β†’ (𝑀(π‘Ÿ <bag 𝑖)𝑧 ↔ 𝑀(π‘Ÿ <bag 𝐼)𝑧))
4140imbi1d 340 . . . . . . . . . . . . . . . 16 ((((𝑖 = 𝐼 ∧ 𝑠 = 𝑅) ∧ 𝑝 = (𝑖 mPwSer 𝑠)) ∧ 𝑑 = 𝐷) β†’ ((𝑀(π‘Ÿ <bag 𝑖)𝑧 β†’ (π‘₯β€˜π‘€) = (π‘¦β€˜π‘€)) ↔ (𝑀(π‘Ÿ <bag 𝐼)𝑧 β†’ (π‘₯β€˜π‘€) = (π‘¦β€˜π‘€))))
4232, 41raleqbidv 3340 . . . . . . . . . . . . . . 15 ((((𝑖 = 𝐼 ∧ 𝑠 = 𝑅) ∧ 𝑝 = (𝑖 mPwSer 𝑠)) ∧ 𝑑 = 𝐷) β†’ (βˆ€π‘€ ∈ 𝑑 (𝑀(π‘Ÿ <bag 𝑖)𝑧 β†’ (π‘₯β€˜π‘€) = (π‘¦β€˜π‘€)) ↔ βˆ€π‘€ ∈ 𝐷 (𝑀(π‘Ÿ <bag 𝐼)𝑧 β†’ (π‘₯β€˜π‘€) = (π‘¦β€˜π‘€))))
4337, 42anbi12d 630 . . . . . . . . . . . . . 14 ((((𝑖 = 𝐼 ∧ 𝑠 = 𝑅) ∧ 𝑝 = (𝑖 mPwSer 𝑠)) ∧ 𝑑 = 𝐷) β†’ (((π‘₯β€˜π‘§)(ltβ€˜π‘ )(π‘¦β€˜π‘§) ∧ βˆ€π‘€ ∈ 𝑑 (𝑀(π‘Ÿ <bag 𝑖)𝑧 β†’ (π‘₯β€˜π‘€) = (π‘¦β€˜π‘€))) ↔ ((π‘₯β€˜π‘§) < (π‘¦β€˜π‘§) ∧ βˆ€π‘€ ∈ 𝐷 (𝑀(π‘Ÿ <bag 𝐼)𝑧 β†’ (π‘₯β€˜π‘€) = (π‘¦β€˜π‘€)))))
4432, 43rexeqbidv 3341 . . . . . . . . . . . . 13 ((((𝑖 = 𝐼 ∧ 𝑠 = 𝑅) ∧ 𝑝 = (𝑖 mPwSer 𝑠)) ∧ 𝑑 = 𝐷) β†’ (βˆƒπ‘§ ∈ 𝑑 ((π‘₯β€˜π‘§)(ltβ€˜π‘ )(π‘¦β€˜π‘§) ∧ βˆ€π‘€ ∈ 𝑑 (𝑀(π‘Ÿ <bag 𝑖)𝑧 β†’ (π‘₯β€˜π‘€) = (π‘¦β€˜π‘€))) ↔ βˆƒπ‘§ ∈ 𝐷 ((π‘₯β€˜π‘§) < (π‘¦β€˜π‘§) ∧ βˆ€π‘€ ∈ 𝐷 (𝑀(π‘Ÿ <bag 𝐼)𝑧 β†’ (π‘₯β€˜π‘€) = (π‘¦β€˜π‘€)))))
4525, 31, 44sbcied2 3826 . . . . . . . . . . . 12 (((𝑖 = 𝐼 ∧ 𝑠 = 𝑅) ∧ 𝑝 = (𝑖 mPwSer 𝑠)) β†’ ([{β„Ž ∈ (β„•0 ↑m 𝑖) ∣ (β—‘β„Ž β€œ β„•) ∈ Fin} / 𝑑]βˆƒπ‘§ ∈ 𝑑 ((π‘₯β€˜π‘§)(ltβ€˜π‘ )(π‘¦β€˜π‘§) ∧ βˆ€π‘€ ∈ 𝑑 (𝑀(π‘Ÿ <bag 𝑖)𝑧 β†’ (π‘₯β€˜π‘€) = (π‘¦β€˜π‘€))) ↔ βˆƒπ‘§ ∈ 𝐷 ((π‘₯β€˜π‘§) < (π‘¦β€˜π‘§) ∧ βˆ€π‘€ ∈ 𝐷 (𝑀(π‘Ÿ <bag 𝐼)𝑧 β†’ (π‘₯β€˜π‘€) = (π‘¦β€˜π‘€)))))
4645orbi1d 914 . . . . . . . . . . 11 (((𝑖 = 𝐼 ∧ 𝑠 = 𝑅) ∧ 𝑝 = (𝑖 mPwSer 𝑠)) β†’ (([{β„Ž ∈ (β„•0 ↑m 𝑖) ∣ (β—‘β„Ž β€œ β„•) ∈ Fin} / 𝑑]βˆƒπ‘§ ∈ 𝑑 ((π‘₯β€˜π‘§)(ltβ€˜π‘ )(π‘¦β€˜π‘§) ∧ βˆ€π‘€ ∈ 𝑑 (𝑀(π‘Ÿ <bag 𝑖)𝑧 β†’ (π‘₯β€˜π‘€) = (π‘¦β€˜π‘€))) ∨ π‘₯ = 𝑦) ↔ (βˆƒπ‘§ ∈ 𝐷 ((π‘₯β€˜π‘§) < (π‘¦β€˜π‘§) ∧ βˆ€π‘€ ∈ 𝐷 (𝑀(π‘Ÿ <bag 𝐼)𝑧 β†’ (π‘₯β€˜π‘€) = (π‘¦β€˜π‘€))) ∨ π‘₯ = 𝑦)))
4722, 46anbi12d 630 . . . . . . . . . 10 (((𝑖 = 𝐼 ∧ 𝑠 = 𝑅) ∧ 𝑝 = (𝑖 mPwSer 𝑠)) β†’ (({π‘₯, 𝑦} βŠ† (Baseβ€˜π‘) ∧ ([{β„Ž ∈ (β„•0 ↑m 𝑖) ∣ (β—‘β„Ž β€œ β„•) ∈ Fin} / 𝑑]βˆƒπ‘§ ∈ 𝑑 ((π‘₯β€˜π‘§)(ltβ€˜π‘ )(π‘¦β€˜π‘§) ∧ βˆ€π‘€ ∈ 𝑑 (𝑀(π‘Ÿ <bag 𝑖)𝑧 β†’ (π‘₯β€˜π‘€) = (π‘¦β€˜π‘€))) ∨ π‘₯ = 𝑦)) ↔ ({π‘₯, 𝑦} βŠ† 𝐡 ∧ (βˆƒπ‘§ ∈ 𝐷 ((π‘₯β€˜π‘§) < (π‘¦β€˜π‘§) ∧ βˆ€π‘€ ∈ 𝐷 (𝑀(π‘Ÿ <bag 𝐼)𝑧 β†’ (π‘₯β€˜π‘€) = (π‘¦β€˜π‘€))) ∨ π‘₯ = 𝑦))))
4847opabbidv 5218 . . . . . . . . 9 (((𝑖 = 𝐼 ∧ 𝑠 = 𝑅) ∧ 𝑝 = (𝑖 mPwSer 𝑠)) β†’ {⟨π‘₯, π‘¦βŸ© ∣ ({π‘₯, 𝑦} βŠ† (Baseβ€˜π‘) ∧ ([{β„Ž ∈ (β„•0 ↑m 𝑖) ∣ (β—‘β„Ž β€œ β„•) ∈ Fin} / 𝑑]βˆƒπ‘§ ∈ 𝑑 ((π‘₯β€˜π‘§)(ltβ€˜π‘ )(π‘¦β€˜π‘§) ∧ βˆ€π‘€ ∈ 𝑑 (𝑀(π‘Ÿ <bag 𝑖)𝑧 β†’ (π‘₯β€˜π‘€) = (π‘¦β€˜π‘€))) ∨ π‘₯ = 𝑦))} = {⟨π‘₯, π‘¦βŸ© ∣ ({π‘₯, 𝑦} βŠ† 𝐡 ∧ (βˆƒπ‘§ ∈ 𝐷 ((π‘₯β€˜π‘§) < (π‘¦β€˜π‘§) ∧ βˆ€π‘€ ∈ 𝐷 (𝑀(π‘Ÿ <bag 𝐼)𝑧 β†’ (π‘₯β€˜π‘€) = (π‘¦β€˜π‘€))) ∨ π‘₯ = 𝑦))})
4948opeq2d 4885 . . . . . . . 8 (((𝑖 = 𝐼 ∧ 𝑠 = 𝑅) ∧ 𝑝 = (𝑖 mPwSer 𝑠)) β†’ ⟨(leβ€˜ndx), {⟨π‘₯, π‘¦βŸ© ∣ ({π‘₯, 𝑦} βŠ† (Baseβ€˜π‘) ∧ ([{β„Ž ∈ (β„•0 ↑m 𝑖) ∣ (β—‘β„Ž β€œ β„•) ∈ Fin} / 𝑑]βˆƒπ‘§ ∈ 𝑑 ((π‘₯β€˜π‘§)(ltβ€˜π‘ )(π‘¦β€˜π‘§) ∧ βˆ€π‘€ ∈ 𝑑 (𝑀(π‘Ÿ <bag 𝑖)𝑧 β†’ (π‘₯β€˜π‘€) = (π‘¦β€˜π‘€))) ∨ π‘₯ = 𝑦))}⟩ = ⟨(leβ€˜ndx), {⟨π‘₯, π‘¦βŸ© ∣ ({π‘₯, 𝑦} βŠ† 𝐡 ∧ (βˆƒπ‘§ ∈ 𝐷 ((π‘₯β€˜π‘§) < (π‘¦β€˜π‘§) ∧ βˆ€π‘€ ∈ 𝐷 (𝑀(π‘Ÿ <bag 𝐼)𝑧 β†’ (π‘₯β€˜π‘€) = (π‘¦β€˜π‘€))) ∨ π‘₯ = 𝑦))}⟩)
5018, 49oveq12d 7444 . . . . . . 7 (((𝑖 = 𝐼 ∧ 𝑠 = 𝑅) ∧ 𝑝 = (𝑖 mPwSer 𝑠)) β†’ (𝑝 sSet ⟨(leβ€˜ndx), {⟨π‘₯, π‘¦βŸ© ∣ ({π‘₯, 𝑦} βŠ† (Baseβ€˜π‘) ∧ ([{β„Ž ∈ (β„•0 ↑m 𝑖) ∣ (β—‘β„Ž β€œ β„•) ∈ Fin} / 𝑑]βˆƒπ‘§ ∈ 𝑑 ((π‘₯β€˜π‘§)(ltβ€˜π‘ )(π‘¦β€˜π‘§) ∧ βˆ€π‘€ ∈ 𝑑 (𝑀(π‘Ÿ <bag 𝑖)𝑧 β†’ (π‘₯β€˜π‘€) = (π‘¦β€˜π‘€))) ∨ π‘₯ = 𝑦))}⟩) = (𝑆 sSet ⟨(leβ€˜ndx), {⟨π‘₯, π‘¦βŸ© ∣ ({π‘₯, 𝑦} βŠ† 𝐡 ∧ (βˆƒπ‘§ ∈ 𝐷 ((π‘₯β€˜π‘§) < (π‘¦β€˜π‘§) ∧ βˆ€π‘€ ∈ 𝐷 (𝑀(π‘Ÿ <bag 𝐼)𝑧 β†’ (π‘₯β€˜π‘€) = (π‘¦β€˜π‘€))) ∨ π‘₯ = 𝑦))}⟩))
5113, 50csbied 3932 . . . . . 6 ((𝑖 = 𝐼 ∧ 𝑠 = 𝑅) β†’ ⦋(𝑖 mPwSer 𝑠) / π‘β¦Œ(𝑝 sSet ⟨(leβ€˜ndx), {⟨π‘₯, π‘¦βŸ© ∣ ({π‘₯, 𝑦} βŠ† (Baseβ€˜π‘) ∧ ([{β„Ž ∈ (β„•0 ↑m 𝑖) ∣ (β—‘β„Ž β€œ β„•) ∈ Fin} / 𝑑]βˆƒπ‘§ ∈ 𝑑 ((π‘₯β€˜π‘§)(ltβ€˜π‘ )(π‘¦β€˜π‘§) ∧ βˆ€π‘€ ∈ 𝑑 (𝑀(π‘Ÿ <bag 𝑖)𝑧 β†’ (π‘₯β€˜π‘€) = (π‘¦β€˜π‘€))) ∨ π‘₯ = 𝑦))}⟩) = (𝑆 sSet ⟨(leβ€˜ndx), {⟨π‘₯, π‘¦βŸ© ∣ ({π‘₯, 𝑦} βŠ† 𝐡 ∧ (βˆƒπ‘§ ∈ 𝐷 ((π‘₯β€˜π‘§) < (π‘¦β€˜π‘§) ∧ βˆ€π‘€ ∈ 𝐷 (𝑀(π‘Ÿ <bag 𝐼)𝑧 β†’ (π‘₯β€˜π‘€) = (π‘¦β€˜π‘€))) ∨ π‘₯ = 𝑦))}⟩))
5212, 51mpteq12dv 5243 . . . . 5 ((𝑖 = 𝐼 ∧ 𝑠 = 𝑅) β†’ (π‘Ÿ ∈ 𝒫 (𝑖 Γ— 𝑖) ↦ ⦋(𝑖 mPwSer 𝑠) / π‘β¦Œ(𝑝 sSet ⟨(leβ€˜ndx), {⟨π‘₯, π‘¦βŸ© ∣ ({π‘₯, 𝑦} βŠ† (Baseβ€˜π‘) ∧ ([{β„Ž ∈ (β„•0 ↑m 𝑖) ∣ (β—‘β„Ž β€œ β„•) ∈ Fin} / 𝑑]βˆƒπ‘§ ∈ 𝑑 ((π‘₯β€˜π‘§)(ltβ€˜π‘ )(π‘¦β€˜π‘§) ∧ βˆ€π‘€ ∈ 𝑑 (𝑀(π‘Ÿ <bag 𝑖)𝑧 β†’ (π‘₯β€˜π‘€) = (π‘¦β€˜π‘€))) ∨ π‘₯ = 𝑦))}⟩)) = (π‘Ÿ ∈ 𝒫 (𝐼 Γ— 𝐼) ↦ (𝑆 sSet ⟨(leβ€˜ndx), {⟨π‘₯, π‘¦βŸ© ∣ ({π‘₯, 𝑦} βŠ† 𝐡 ∧ (βˆƒπ‘§ ∈ 𝐷 ((π‘₯β€˜π‘§) < (π‘¦β€˜π‘§) ∧ βˆ€π‘€ ∈ 𝐷 (𝑀(π‘Ÿ <bag 𝐼)𝑧 β†’ (π‘₯β€˜π‘€) = (π‘¦β€˜π‘€))) ∨ π‘₯ = 𝑦))}⟩)))
53 df-opsr 21853 . . . . 5 ordPwSer = (𝑖 ∈ V, 𝑠 ∈ V ↦ (π‘Ÿ ∈ 𝒫 (𝑖 Γ— 𝑖) ↦ ⦋(𝑖 mPwSer 𝑠) / π‘β¦Œ(𝑝 sSet ⟨(leβ€˜ndx), {⟨π‘₯, π‘¦βŸ© ∣ ({π‘₯, 𝑦} βŠ† (Baseβ€˜π‘) ∧ ([{β„Ž ∈ (β„•0 ↑m 𝑖) ∣ (β—‘β„Ž β€œ β„•) ∈ Fin} / 𝑑]βˆƒπ‘§ ∈ 𝑑 ((π‘₯β€˜π‘§)(ltβ€˜π‘ )(π‘¦β€˜π‘§) ∧ βˆ€π‘€ ∈ 𝑑 (𝑀(π‘Ÿ <bag 𝑖)𝑧 β†’ (π‘₯β€˜π‘€) = (π‘¦β€˜π‘€))) ∨ π‘₯ = 𝑦))}⟩)))
5452, 53ovmpoga 7581 . . . 4 ((𝐼 ∈ V ∧ 𝑅 ∈ V ∧ (π‘Ÿ ∈ 𝒫 (𝐼 Γ— 𝐼) ↦ (𝑆 sSet ⟨(leβ€˜ndx), {⟨π‘₯, π‘¦βŸ© ∣ ({π‘₯, 𝑦} βŠ† 𝐡 ∧ (βˆƒπ‘§ ∈ 𝐷 ((π‘₯β€˜π‘§) < (π‘¦β€˜π‘§) ∧ βˆ€π‘€ ∈ 𝐷 (𝑀(π‘Ÿ <bag 𝐼)𝑧 β†’ (π‘₯β€˜π‘€) = (π‘¦β€˜π‘€))) ∨ π‘₯ = 𝑦))}⟩)) ∈ V) β†’ (𝐼 ordPwSer 𝑅) = (π‘Ÿ ∈ 𝒫 (𝐼 Γ— 𝐼) ↦ (𝑆 sSet ⟨(leβ€˜ndx), {⟨π‘₯, π‘¦βŸ© ∣ ({π‘₯, 𝑦} βŠ† 𝐡 ∧ (βˆƒπ‘§ ∈ 𝐷 ((π‘₯β€˜π‘§) < (π‘¦β€˜π‘§) ∧ βˆ€π‘€ ∈ 𝐷 (𝑀(π‘Ÿ <bag 𝐼)𝑧 β†’ (π‘₯β€˜π‘€) = (π‘¦β€˜π‘€))) ∨ π‘₯ = 𝑦))}⟩)))
553, 5, 9, 54syl3anc 1368 . . 3 (πœ‘ β†’ (𝐼 ordPwSer 𝑅) = (π‘Ÿ ∈ 𝒫 (𝐼 Γ— 𝐼) ↦ (𝑆 sSet ⟨(leβ€˜ndx), {⟨π‘₯, π‘¦βŸ© ∣ ({π‘₯, 𝑦} βŠ† 𝐡 ∧ (βˆƒπ‘§ ∈ 𝐷 ((π‘₯β€˜π‘§) < (π‘¦β€˜π‘§) ∧ βˆ€π‘€ ∈ 𝐷 (𝑀(π‘Ÿ <bag 𝐼)𝑧 β†’ (π‘₯β€˜π‘€) = (π‘¦β€˜π‘€))) ∨ π‘₯ = 𝑦))}⟩)))
56 simpr 483 . . . . . . . . . . . . . . . 16 ((πœ‘ ∧ π‘Ÿ = 𝑇) β†’ π‘Ÿ = 𝑇)
5756oveq1d 7441 . . . . . . . . . . . . . . 15 ((πœ‘ ∧ π‘Ÿ = 𝑇) β†’ (π‘Ÿ <bag 𝐼) = (𝑇 <bag 𝐼))
58 opsrval.c . . . . . . . . . . . . . . 15 𝐢 = (𝑇 <bag 𝐼)
5957, 58eqtr4di 2786 . . . . . . . . . . . . . 14 ((πœ‘ ∧ π‘Ÿ = 𝑇) β†’ (π‘Ÿ <bag 𝐼) = 𝐢)
6059breqd 5163 . . . . . . . . . . . . 13 ((πœ‘ ∧ π‘Ÿ = 𝑇) β†’ (𝑀(π‘Ÿ <bag 𝐼)𝑧 ↔ 𝑀𝐢𝑧))
6160imbi1d 340 . . . . . . . . . . . 12 ((πœ‘ ∧ π‘Ÿ = 𝑇) β†’ ((𝑀(π‘Ÿ <bag 𝐼)𝑧 β†’ (π‘₯β€˜π‘€) = (π‘¦β€˜π‘€)) ↔ (𝑀𝐢𝑧 β†’ (π‘₯β€˜π‘€) = (π‘¦β€˜π‘€))))
6261ralbidv 3175 . . . . . . . . . . 11 ((πœ‘ ∧ π‘Ÿ = 𝑇) β†’ (βˆ€π‘€ ∈ 𝐷 (𝑀(π‘Ÿ <bag 𝐼)𝑧 β†’ (π‘₯β€˜π‘€) = (π‘¦β€˜π‘€)) ↔ βˆ€π‘€ ∈ 𝐷 (𝑀𝐢𝑧 β†’ (π‘₯β€˜π‘€) = (π‘¦β€˜π‘€))))
6362anbi2d 628 . . . . . . . . . 10 ((πœ‘ ∧ π‘Ÿ = 𝑇) β†’ (((π‘₯β€˜π‘§) < (π‘¦β€˜π‘§) ∧ βˆ€π‘€ ∈ 𝐷 (𝑀(π‘Ÿ <bag 𝐼)𝑧 β†’ (π‘₯β€˜π‘€) = (π‘¦β€˜π‘€))) ↔ ((π‘₯β€˜π‘§) < (π‘¦β€˜π‘§) ∧ βˆ€π‘€ ∈ 𝐷 (𝑀𝐢𝑧 β†’ (π‘₯β€˜π‘€) = (π‘¦β€˜π‘€)))))
6463rexbidv 3176 . . . . . . . . 9 ((πœ‘ ∧ π‘Ÿ = 𝑇) β†’ (βˆƒπ‘§ ∈ 𝐷 ((π‘₯β€˜π‘§) < (π‘¦β€˜π‘§) ∧ βˆ€π‘€ ∈ 𝐷 (𝑀(π‘Ÿ <bag 𝐼)𝑧 β†’ (π‘₯β€˜π‘€) = (π‘¦β€˜π‘€))) ↔ βˆƒπ‘§ ∈ 𝐷 ((π‘₯β€˜π‘§) < (π‘¦β€˜π‘§) ∧ βˆ€π‘€ ∈ 𝐷 (𝑀𝐢𝑧 β†’ (π‘₯β€˜π‘€) = (π‘¦β€˜π‘€)))))
6564orbi1d 914 . . . . . . . 8 ((πœ‘ ∧ π‘Ÿ = 𝑇) β†’ ((βˆƒπ‘§ ∈ 𝐷 ((π‘₯β€˜π‘§) < (π‘¦β€˜π‘§) ∧ βˆ€π‘€ ∈ 𝐷 (𝑀(π‘Ÿ <bag 𝐼)𝑧 β†’ (π‘₯β€˜π‘€) = (π‘¦β€˜π‘€))) ∨ π‘₯ = 𝑦) ↔ (βˆƒπ‘§ ∈ 𝐷 ((π‘₯β€˜π‘§) < (π‘¦β€˜π‘§) ∧ βˆ€π‘€ ∈ 𝐷 (𝑀𝐢𝑧 β†’ (π‘₯β€˜π‘€) = (π‘¦β€˜π‘€))) ∨ π‘₯ = 𝑦)))
6665anbi2d 628 . . . . . . 7 ((πœ‘ ∧ π‘Ÿ = 𝑇) β†’ (({π‘₯, 𝑦} βŠ† 𝐡 ∧ (βˆƒπ‘§ ∈ 𝐷 ((π‘₯β€˜π‘§) < (π‘¦β€˜π‘§) ∧ βˆ€π‘€ ∈ 𝐷 (𝑀(π‘Ÿ <bag 𝐼)𝑧 β†’ (π‘₯β€˜π‘€) = (π‘¦β€˜π‘€))) ∨ π‘₯ = 𝑦)) ↔ ({π‘₯, 𝑦} βŠ† 𝐡 ∧ (βˆƒπ‘§ ∈ 𝐷 ((π‘₯β€˜π‘§) < (π‘¦β€˜π‘§) ∧ βˆ€π‘€ ∈ 𝐷 (𝑀𝐢𝑧 β†’ (π‘₯β€˜π‘€) = (π‘¦β€˜π‘€))) ∨ π‘₯ = 𝑦))))
6766opabbidv 5218 . . . . . 6 ((πœ‘ ∧ π‘Ÿ = 𝑇) β†’ {⟨π‘₯, π‘¦βŸ© ∣ ({π‘₯, 𝑦} βŠ† 𝐡 ∧ (βˆƒπ‘§ ∈ 𝐷 ((π‘₯β€˜π‘§) < (π‘¦β€˜π‘§) ∧ βˆ€π‘€ ∈ 𝐷 (𝑀(π‘Ÿ <bag 𝐼)𝑧 β†’ (π‘₯β€˜π‘€) = (π‘¦β€˜π‘€))) ∨ π‘₯ = 𝑦))} = {⟨π‘₯, π‘¦βŸ© ∣ ({π‘₯, 𝑦} βŠ† 𝐡 ∧ (βˆƒπ‘§ ∈ 𝐷 ((π‘₯β€˜π‘§) < (π‘¦β€˜π‘§) ∧ βˆ€π‘€ ∈ 𝐷 (𝑀𝐢𝑧 β†’ (π‘₯β€˜π‘€) = (π‘¦β€˜π‘€))) ∨ π‘₯ = 𝑦))})
68 opsrval.l . . . . . 6 ≀ = {⟨π‘₯, π‘¦βŸ© ∣ ({π‘₯, 𝑦} βŠ† 𝐡 ∧ (βˆƒπ‘§ ∈ 𝐷 ((π‘₯β€˜π‘§) < (π‘¦β€˜π‘§) ∧ βˆ€π‘€ ∈ 𝐷 (𝑀𝐢𝑧 β†’ (π‘₯β€˜π‘€) = (π‘¦β€˜π‘€))) ∨ π‘₯ = 𝑦))}
6967, 68eqtr4di 2786 . . . . 5 ((πœ‘ ∧ π‘Ÿ = 𝑇) β†’ {⟨π‘₯, π‘¦βŸ© ∣ ({π‘₯, 𝑦} βŠ† 𝐡 ∧ (βˆƒπ‘§ ∈ 𝐷 ((π‘₯β€˜π‘§) < (π‘¦β€˜π‘§) ∧ βˆ€π‘€ ∈ 𝐷 (𝑀(π‘Ÿ <bag 𝐼)𝑧 β†’ (π‘₯β€˜π‘€) = (π‘¦β€˜π‘€))) ∨ π‘₯ = 𝑦))} = ≀ )
7069opeq2d 4885 . . . 4 ((πœ‘ ∧ π‘Ÿ = 𝑇) β†’ ⟨(leβ€˜ndx), {⟨π‘₯, π‘¦βŸ© ∣ ({π‘₯, 𝑦} βŠ† 𝐡 ∧ (βˆƒπ‘§ ∈ 𝐷 ((π‘₯β€˜π‘§) < (π‘¦β€˜π‘§) ∧ βˆ€π‘€ ∈ 𝐷 (𝑀(π‘Ÿ <bag 𝐼)𝑧 β†’ (π‘₯β€˜π‘€) = (π‘¦β€˜π‘€))) ∨ π‘₯ = 𝑦))}⟩ = ⟨(leβ€˜ndx), ≀ ⟩)
7170oveq2d 7442 . . 3 ((πœ‘ ∧ π‘Ÿ = 𝑇) β†’ (𝑆 sSet ⟨(leβ€˜ndx), {⟨π‘₯, π‘¦βŸ© ∣ ({π‘₯, 𝑦} βŠ† 𝐡 ∧ (βˆƒπ‘§ ∈ 𝐷 ((π‘₯β€˜π‘§) < (π‘¦β€˜π‘§) ∧ βˆ€π‘€ ∈ 𝐷 (𝑀(π‘Ÿ <bag 𝐼)𝑧 β†’ (π‘₯β€˜π‘€) = (π‘¦β€˜π‘€))) ∨ π‘₯ = 𝑦))}⟩) = (𝑆 sSet ⟨(leβ€˜ndx), ≀ ⟩))
72 opsrval.t . . . 4 (πœ‘ β†’ 𝑇 βŠ† (𝐼 Γ— 𝐼))
736, 72sselpwd 5332 . . 3 (πœ‘ β†’ 𝑇 ∈ 𝒫 (𝐼 Γ— 𝐼))
74 ovexd 7461 . . 3 (πœ‘ β†’ (𝑆 sSet ⟨(leβ€˜ndx), ≀ ⟩) ∈ V)
7555, 71, 73, 74fvmptd 7017 . 2 (πœ‘ β†’ ((𝐼 ordPwSer 𝑅)β€˜π‘‡) = (𝑆 sSet ⟨(leβ€˜ndx), ≀ ⟩))
761, 75eqtrid 2780 1 (πœ‘ β†’ 𝑂 = (𝑆 sSet ⟨(leβ€˜ndx), ≀ ⟩))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 394   ∨ wo 845   = wceq 1533   ∈ wcel 2098  βˆ€wral 3058  βˆƒwrex 3067  {crab 3430  Vcvv 3473  [wsbc 3778  β¦‹csb 3894   βŠ† wss 3949  π’« cpw 4606  {cpr 4634  βŸ¨cop 4638   class class class wbr 5152  {copab 5214   ↦ cmpt 5235   Γ— cxp 5680  β—‘ccnv 5681   β€œ cima 5685  β€˜cfv 6553  (class class class)co 7426   ↑m cmap 8851  Fincfn 8970  β„•cn 12250  β„•0cn0 12510   sSet csts 17139  ndxcnx 17169  Basecbs 17187  lecple 17247  ltcplt 18307   mPwSer cmps 21844   <bag cltb 21847   ordPwSer copws 21848
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-rep 5289  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7746
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-ov 7429  df-oprab 7430  df-mpo 7431  df-opsr 21853
This theorem is referenced by:  opsrle  21992  opsrval2  21993
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