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Theorem opsrtoslem1 19305
 Description: Lemma for opsrtos 19307. (Contributed by Mario Carneiro, 8-Feb-2015.)
Hypotheses
Ref Expression
opsrso.o 𝑂 = ((𝐼 ordPwSer 𝑅)‘𝑇)
opsrso.i (𝜑𝐼𝑉)
opsrso.r (𝜑𝑅 ∈ Toset)
opsrso.t (𝜑𝑇 ⊆ (𝐼 × 𝐼))
opsrso.w (𝜑𝑇 We 𝐼)
opsrtoslem.s 𝑆 = (𝐼 mPwSer 𝑅)
opsrtoslem.b 𝐵 = (Base‘𝑆)
opsrtoslem.q < = (lt‘𝑅)
opsrtoslem.c 𝐶 = (𝑇 <bag 𝐼)
opsrtoslem.d 𝐷 = { ∈ (ℕ0𝑚 𝐼) ∣ ( “ ℕ) ∈ Fin}
opsrtoslem.ps (𝜓 ↔ ∃𝑧𝐷 ((𝑥𝑧) < (𝑦𝑧) ∧ ∀𝑤𝐷 (𝑤𝐶𝑧 → (𝑥𝑤) = (𝑦𝑤))))
opsrtoslem.l = (le‘𝑂)
Assertion
Ref Expression
opsrtoslem1 (𝜑 = (({⟨𝑥, 𝑦⟩ ∣ 𝜓} ∩ (𝐵 × 𝐵)) ∪ ( I ↾ 𝐵)))
Distinct variable groups:   𝑥,𝑦,𝐵   𝑥,𝑤,𝑦,𝑧,𝐶   𝑤,,𝑥,𝑦,𝑧,𝐼   𝜑,,𝑤,𝑥,𝑦,𝑧   𝑤,𝐷,𝑥,𝑦,𝑧   𝑤, < ,𝑥,𝑦,𝑧   𝑤,𝑅,𝑥,𝑦,𝑧   𝑤,𝑇,𝑥,𝑦,𝑧
Allowed substitution hints:   𝜓(𝑥,𝑦,𝑧,𝑤,)   𝐵(𝑧,𝑤,)   𝐶()   𝐷()   𝑅()   𝑆(𝑥,𝑦,𝑧,𝑤,)   < ()   𝑇()   (𝑥,𝑦,𝑧,𝑤,)   𝑂(𝑥,𝑦,𝑧,𝑤,)   𝑉(𝑥,𝑦,𝑧,𝑤,)

Proof of Theorem opsrtoslem1
StepHypRef Expression
1 opsrtoslem.s . . 3 𝑆 = (𝐼 mPwSer 𝑅)
2 opsrso.o . . 3 𝑂 = ((𝐼 ordPwSer 𝑅)‘𝑇)
3 opsrtoslem.b . . 3 𝐵 = (Base‘𝑆)
4 opsrtoslem.q . . 3 < = (lt‘𝑅)
5 opsrtoslem.c . . 3 𝐶 = (𝑇 <bag 𝐼)
6 opsrtoslem.d . . 3 𝐷 = { ∈ (ℕ0𝑚 𝐼) ∣ ( “ ℕ) ∈ Fin}
7 opsrtoslem.l . . 3 = (le‘𝑂)
8 opsrso.t . . 3 (𝜑𝑇 ⊆ (𝐼 × 𝐼))
91, 2, 3, 4, 5, 6, 7, 8opsrle 19296 . 2 (𝜑 = {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝐵 ∧ (∃𝑧𝐷 ((𝑥𝑧) < (𝑦𝑧) ∧ ∀𝑤𝐷 (𝑤𝐶𝑧 → (𝑥𝑤) = (𝑦𝑤))) ∨ 𝑥 = 𝑦))})
10 unopab 4660 . . 3 ({⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝐵𝜓)} ∪ {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝐵𝑥 = 𝑦)}) = {⟨𝑥, 𝑦⟩ ∣ (({𝑥, 𝑦} ⊆ 𝐵𝜓) ∨ ({𝑥, 𝑦} ⊆ 𝐵𝑥 = 𝑦))}
11 inopab 5174 . . . . 5 ({⟨𝑥, 𝑦⟩ ∣ 𝜓} ∩ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐵𝑦𝐵)}) = {⟨𝑥, 𝑦⟩ ∣ (𝜓 ∧ (𝑥𝐵𝑦𝐵))}
12 df-xp 5044 . . . . . 6 (𝐵 × 𝐵) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐵𝑦𝐵)}
1312ineq2i 3773 . . . . 5 ({⟨𝑥, 𝑦⟩ ∣ 𝜓} ∩ (𝐵 × 𝐵)) = ({⟨𝑥, 𝑦⟩ ∣ 𝜓} ∩ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐵𝑦𝐵)})
14 vex 3176 . . . . . . . . 9 𝑥 ∈ V
15 vex 3176 . . . . . . . . 9 𝑦 ∈ V
1614, 15prss 4291 . . . . . . . 8 ((𝑥𝐵𝑦𝐵) ↔ {𝑥, 𝑦} ⊆ 𝐵)
1716anbi1i 727 . . . . . . 7 (((𝑥𝐵𝑦𝐵) ∧ 𝜓) ↔ ({𝑥, 𝑦} ⊆ 𝐵𝜓))
18 ancom 465 . . . . . . 7 (((𝑥𝐵𝑦𝐵) ∧ 𝜓) ↔ (𝜓 ∧ (𝑥𝐵𝑦𝐵)))
1917, 18bitr3i 265 . . . . . 6 (({𝑥, 𝑦} ⊆ 𝐵𝜓) ↔ (𝜓 ∧ (𝑥𝐵𝑦𝐵)))
2019opabbii 4649 . . . . 5 {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝐵𝜓)} = {⟨𝑥, 𝑦⟩ ∣ (𝜓 ∧ (𝑥𝐵𝑦𝐵))}
2111, 13, 203eqtr4i 2642 . . . 4 ({⟨𝑥, 𝑦⟩ ∣ 𝜓} ∩ (𝐵 × 𝐵)) = {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝐵𝜓)}
22 opabresid 5374 . . . . 5 {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐵𝑦 = 𝑥)} = ( I ↾ 𝐵)
23 equcom 1932 . . . . . . . . 9 (𝑥 = 𝑦𝑦 = 𝑥)
2423anbi2i 726 . . . . . . . 8 ((𝑥𝐵𝑥 = 𝑦) ↔ (𝑥𝐵𝑦 = 𝑥))
25 eleq1 2676 . . . . . . . . . 10 (𝑥 = 𝑦 → (𝑥𝐵𝑦𝐵))
2625biimpac 502 . . . . . . . . 9 ((𝑥𝐵𝑥 = 𝑦) → 𝑦𝐵)
2726pm4.71i 662 . . . . . . . 8 ((𝑥𝐵𝑥 = 𝑦) ↔ ((𝑥𝐵𝑥 = 𝑦) ∧ 𝑦𝐵))
2824, 27bitr3i 265 . . . . . . 7 ((𝑥𝐵𝑦 = 𝑥) ↔ ((𝑥𝐵𝑥 = 𝑦) ∧ 𝑦𝐵))
29 an32 835 . . . . . . 7 (((𝑥𝐵𝑥 = 𝑦) ∧ 𝑦𝐵) ↔ ((𝑥𝐵𝑦𝐵) ∧ 𝑥 = 𝑦))
3016anbi1i 727 . . . . . . 7 (((𝑥𝐵𝑦𝐵) ∧ 𝑥 = 𝑦) ↔ ({𝑥, 𝑦} ⊆ 𝐵𝑥 = 𝑦))
3128, 29, 303bitri 285 . . . . . 6 ((𝑥𝐵𝑦 = 𝑥) ↔ ({𝑥, 𝑦} ⊆ 𝐵𝑥 = 𝑦))
3231opabbii 4649 . . . . 5 {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐵𝑦 = 𝑥)} = {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝐵𝑥 = 𝑦)}
3322, 32eqtr3i 2634 . . . 4 ( I ↾ 𝐵) = {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝐵𝑥 = 𝑦)}
3421, 33uneq12i 3727 . . 3 (({⟨𝑥, 𝑦⟩ ∣ 𝜓} ∩ (𝐵 × 𝐵)) ∪ ( I ↾ 𝐵)) = ({⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝐵𝜓)} ∪ {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝐵𝑥 = 𝑦)})
35 opsrtoslem.ps . . . . . . 7 (𝜓 ↔ ∃𝑧𝐷 ((𝑥𝑧) < (𝑦𝑧) ∧ ∀𝑤𝐷 (𝑤𝐶𝑧 → (𝑥𝑤) = (𝑦𝑤))))
3635orbi1i 541 . . . . . 6 ((𝜓𝑥 = 𝑦) ↔ (∃𝑧𝐷 ((𝑥𝑧) < (𝑦𝑧) ∧ ∀𝑤𝐷 (𝑤𝐶𝑧 → (𝑥𝑤) = (𝑦𝑤))) ∨ 𝑥 = 𝑦))
3736anbi2i 726 . . . . 5 (({𝑥, 𝑦} ⊆ 𝐵 ∧ (𝜓𝑥 = 𝑦)) ↔ ({𝑥, 𝑦} ⊆ 𝐵 ∧ (∃𝑧𝐷 ((𝑥𝑧) < (𝑦𝑧) ∧ ∀𝑤𝐷 (𝑤𝐶𝑧 → (𝑥𝑤) = (𝑦𝑤))) ∨ 𝑥 = 𝑦)))
38 andi 907 . . . . 5 (({𝑥, 𝑦} ⊆ 𝐵 ∧ (𝜓𝑥 = 𝑦)) ↔ (({𝑥, 𝑦} ⊆ 𝐵𝜓) ∨ ({𝑥, 𝑦} ⊆ 𝐵𝑥 = 𝑦)))
3937, 38bitr3i 265 . . . 4 (({𝑥, 𝑦} ⊆ 𝐵 ∧ (∃𝑧𝐷 ((𝑥𝑧) < (𝑦𝑧) ∧ ∀𝑤𝐷 (𝑤𝐶𝑧 → (𝑥𝑤) = (𝑦𝑤))) ∨ 𝑥 = 𝑦)) ↔ (({𝑥, 𝑦} ⊆ 𝐵𝜓) ∨ ({𝑥, 𝑦} ⊆ 𝐵𝑥 = 𝑦)))
4039opabbii 4649 . . 3 {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝐵 ∧ (∃𝑧𝐷 ((𝑥𝑧) < (𝑦𝑧) ∧ ∀𝑤𝐷 (𝑤𝐶𝑧 → (𝑥𝑤) = (𝑦𝑤))) ∨ 𝑥 = 𝑦))} = {⟨𝑥, 𝑦⟩ ∣ (({𝑥, 𝑦} ⊆ 𝐵𝜓) ∨ ({𝑥, 𝑦} ⊆ 𝐵𝑥 = 𝑦))}
4110, 34, 403eqtr4ri 2643 . 2 {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝐵 ∧ (∃𝑧𝐷 ((𝑥𝑧) < (𝑦𝑧) ∧ ∀𝑤𝐷 (𝑤𝐶𝑧 → (𝑥𝑤) = (𝑦𝑤))) ∨ 𝑥 = 𝑦))} = (({⟨𝑥, 𝑦⟩ ∣ 𝜓} ∩ (𝐵 × 𝐵)) ∪ ( I ↾ 𝐵))
429, 41syl6eq 2660 1 (𝜑 = (({⟨𝑥, 𝑦⟩ ∣ 𝜓} ∩ (𝐵 × 𝐵)) ∪ ( I ↾ 𝐵)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∨ wo 382   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896  ∃wrex 2897  {crab 2900   ∪ cun 3538   ∩ cin 3539   ⊆ wss 3540  {cpr 4127   class class class wbr 4583  {copab 4642   I cid 4948   We wwe 4996   × cxp 5036  ◡ccnv 5037   ↾ cres 5040   “ cima 5041  ‘cfv 5804  (class class class)co 6549   ↑𝑚 cmap 7744  Fincfn 7841  ℕcn 10897  ℕ0cn0 11169  Basecbs 15695  lecple 15775  ltcplt 16764  Tosetctos 16856   mPwSer cmps 19172
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