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Theorem opsrtoslem2 19533
 Description: Lemma for opsrtos 19534. (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
opsrtoslem2 (𝜑𝑂 ∈ Toset)
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 opsrtoslem2
Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 opsrtoslem.d . . . . . . . 8 𝐷 = { ∈ (ℕ0𝑚 𝐼) ∣ ( “ ℕ) ∈ Fin}
2 ovex 6718 . . . . . . . 8 (ℕ0𝑚 𝐼) ∈ V
31, 2rabex2 4847 . . . . . . 7 𝐷 ∈ V
4 opsrtoslem.c . . . . . . . 8 𝐶 = (𝑇 <bag 𝐼)
5 opsrso.i . . . . . . . 8 (𝜑𝐼𝑉)
6 xpexg 7002 . . . . . . . . . 10 ((𝐼𝑉𝐼𝑉) → (𝐼 × 𝐼) ∈ V)
75, 5, 6syl2anc 694 . . . . . . . . 9 (𝜑 → (𝐼 × 𝐼) ∈ V)
8 opsrso.t . . . . . . . . 9 (𝜑𝑇 ⊆ (𝐼 × 𝐼))
97, 8ssexd 4838 . . . . . . . 8 (𝜑𝑇 ∈ V)
10 opsrso.w . . . . . . . 8 (𝜑𝑇 We 𝐼)
114, 1, 5, 9, 10ltbwe 19520 . . . . . . 7 (𝜑𝐶 We 𝐷)
12 opsrso.r . . . . . . . . 9 (𝜑𝑅 ∈ Toset)
13 eqid 2651 . . . . . . . . . . 11 (Base‘𝑅) = (Base‘𝑅)
14 eqid 2651 . . . . . . . . . . 11 (le‘𝑅) = (le‘𝑅)
15 opsrtoslem.q . . . . . . . . . . 11 < = (lt‘𝑅)
1613, 14, 15tosso 17083 . . . . . . . . . 10 (𝑅 ∈ Toset → (𝑅 ∈ Toset ↔ ( < Or (Base‘𝑅) ∧ ( I ↾ (Base‘𝑅)) ⊆ (le‘𝑅))))
1716ibi 256 . . . . . . . . 9 (𝑅 ∈ Toset → ( < Or (Base‘𝑅) ∧ ( I ↾ (Base‘𝑅)) ⊆ (le‘𝑅)))
1812, 17syl 17 . . . . . . . 8 (𝜑 → ( < Or (Base‘𝑅) ∧ ( I ↾ (Base‘𝑅)) ⊆ (le‘𝑅)))
1918simpld 474 . . . . . . 7 (𝜑< Or (Base‘𝑅))
20 opsrtoslem.ps . . . . . . . . 9 (𝜓 ↔ ∃𝑧𝐷 ((𝑥𝑧) < (𝑦𝑧) ∧ ∀𝑤𝐷 (𝑤𝐶𝑧 → (𝑥𝑤) = (𝑦𝑤))))
2120opabbii 4750 . . . . . . . 8 {⟨𝑥, 𝑦⟩ ∣ 𝜓} = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧𝐷 ((𝑥𝑧) < (𝑦𝑧) ∧ ∀𝑤𝐷 (𝑤𝐶𝑧 → (𝑥𝑤) = (𝑦𝑤)))}
2221wemapso 8497 . . . . . . 7 ((𝐷 ∈ V ∧ 𝐶 We 𝐷< Or (Base‘𝑅)) → {⟨𝑥, 𝑦⟩ ∣ 𝜓} Or ((Base‘𝑅) ↑𝑚 𝐷))
233, 11, 19, 22mp3an2i 1469 . . . . . 6 (𝜑 → {⟨𝑥, 𝑦⟩ ∣ 𝜓} Or ((Base‘𝑅) ↑𝑚 𝐷))
24 opsrtoslem.s . . . . . . . 8 𝑆 = (𝐼 mPwSer 𝑅)
25 opsrtoslem.b . . . . . . . 8 𝐵 = (Base‘𝑆)
2624, 13, 1, 25, 5psrbas 19426 . . . . . . 7 (𝜑𝐵 = ((Base‘𝑅) ↑𝑚 𝐷))
27 soeq2 5084 . . . . . . 7 (𝐵 = ((Base‘𝑅) ↑𝑚 𝐷) → ({⟨𝑥, 𝑦⟩ ∣ 𝜓} Or 𝐵 ↔ {⟨𝑥, 𝑦⟩ ∣ 𝜓} Or ((Base‘𝑅) ↑𝑚 𝐷)))
2826, 27syl 17 . . . . . 6 (𝜑 → ({⟨𝑥, 𝑦⟩ ∣ 𝜓} Or 𝐵 ↔ {⟨𝑥, 𝑦⟩ ∣ 𝜓} Or ((Base‘𝑅) ↑𝑚 𝐷)))
2923, 28mpbird 247 . . . . 5 (𝜑 → {⟨𝑥, 𝑦⟩ ∣ 𝜓} Or 𝐵)
30 soinxp 5217 . . . . 5 ({⟨𝑥, 𝑦⟩ ∣ 𝜓} Or 𝐵 ↔ ({⟨𝑥, 𝑦⟩ ∣ 𝜓} ∩ (𝐵 × 𝐵)) Or 𝐵)
3129, 30sylib 208 . . . 4 (𝜑 → ({⟨𝑥, 𝑦⟩ ∣ 𝜓} ∩ (𝐵 × 𝐵)) Or 𝐵)
32 opsrso.o . . . . . . . 8 𝑂 = ((𝐼 ordPwSer 𝑅)‘𝑇)
33 fvex 6239 . . . . . . . 8 ((𝐼 ordPwSer 𝑅)‘𝑇) ∈ V
3432, 33eqeltri 2726 . . . . . . 7 𝑂 ∈ V
35 opsrtoslem.l . . . . . . . 8 = (le‘𝑂)
36 eqid 2651 . . . . . . . 8 (lt‘𝑂) = (lt‘𝑂)
3735, 36pltfval 17006 . . . . . . 7 (𝑂 ∈ V → (lt‘𝑂) = ( ∖ I ))
3834, 37ax-mp 5 . . . . . 6 (lt‘𝑂) = ( ∖ I )
39 difundir 3913 . . . . . . . 8 ((({⟨𝑥, 𝑦⟩ ∣ 𝜓} ∩ (𝐵 × 𝐵)) ∪ ( I ↾ 𝐵)) ∖ I ) = ((({⟨𝑥, 𝑦⟩ ∣ 𝜓} ∩ (𝐵 × 𝐵)) ∖ I ) ∪ (( I ↾ 𝐵) ∖ I ))
40 resss 5457 . . . . . . . . . 10 ( I ↾ 𝐵) ⊆ I
41 ssdif0 3975 . . . . . . . . . 10 (( I ↾ 𝐵) ⊆ I ↔ (( I ↾ 𝐵) ∖ I ) = ∅)
4240, 41mpbi 220 . . . . . . . . 9 (( I ↾ 𝐵) ∖ I ) = ∅
4342uneq2i 3797 . . . . . . . 8 ((({⟨𝑥, 𝑦⟩ ∣ 𝜓} ∩ (𝐵 × 𝐵)) ∖ I ) ∪ (( I ↾ 𝐵) ∖ I )) = ((({⟨𝑥, 𝑦⟩ ∣ 𝜓} ∩ (𝐵 × 𝐵)) ∖ I ) ∪ ∅)
44 un0 4000 . . . . . . . 8 ((({⟨𝑥, 𝑦⟩ ∣ 𝜓} ∩ (𝐵 × 𝐵)) ∖ I ) ∪ ∅) = (({⟨𝑥, 𝑦⟩ ∣ 𝜓} ∩ (𝐵 × 𝐵)) ∖ I )
4539, 43, 443eqtri 2677 . . . . . . 7 ((({⟨𝑥, 𝑦⟩ ∣ 𝜓} ∩ (𝐵 × 𝐵)) ∪ ( I ↾ 𝐵)) ∖ I ) = (({⟨𝑥, 𝑦⟩ ∣ 𝜓} ∩ (𝐵 × 𝐵)) ∖ I )
4632, 5, 12, 8, 10, 24, 25, 15, 4, 1, 20, 35opsrtoslem1 19532 . . . . . . . 8 (𝜑 = (({⟨𝑥, 𝑦⟩ ∣ 𝜓} ∩ (𝐵 × 𝐵)) ∪ ( I ↾ 𝐵)))
4746difeq1d 3760 . . . . . . 7 (𝜑 → ( ∖ I ) = ((({⟨𝑥, 𝑦⟩ ∣ 𝜓} ∩ (𝐵 × 𝐵)) ∪ ( I ↾ 𝐵)) ∖ I ))
48 inss2 3867 . . . . . . . . . . . 12 ({⟨𝑥, 𝑦⟩ ∣ 𝜓} ∩ (𝐵 × 𝐵)) ⊆ (𝐵 × 𝐵)
49 relxp 5160 . . . . . . . . . . . 12 Rel (𝐵 × 𝐵)
50 relss 5240 . . . . . . . . . . . 12 (({⟨𝑥, 𝑦⟩ ∣ 𝜓} ∩ (𝐵 × 𝐵)) ⊆ (𝐵 × 𝐵) → (Rel (𝐵 × 𝐵) → Rel ({⟨𝑥, 𝑦⟩ ∣ 𝜓} ∩ (𝐵 × 𝐵))))
5148, 49, 50mp2 9 . . . . . . . . . . 11 Rel ({⟨𝑥, 𝑦⟩ ∣ 𝜓} ∩ (𝐵 × 𝐵))
5251a1i 11 . . . . . . . . . 10 (𝜑 → Rel ({⟨𝑥, 𝑦⟩ ∣ 𝜓} ∩ (𝐵 × 𝐵)))
53 df-br 4686 . . . . . . . . . . . . . 14 (𝑎 I 𝑏 ↔ ⟨𝑎, 𝑏⟩ ∈ I )
54 vex 3234 . . . . . . . . . . . . . . 15 𝑏 ∈ V
5554ideq 5307 . . . . . . . . . . . . . 14 (𝑎 I 𝑏𝑎 = 𝑏)
5653, 55bitr3i 266 . . . . . . . . . . . . 13 (⟨𝑎, 𝑏⟩ ∈ I ↔ 𝑎 = 𝑏)
57 brin 4737 . . . . . . . . . . . . . . . . . 18 (𝑎({⟨𝑥, 𝑦⟩ ∣ 𝜓} ∩ (𝐵 × 𝐵))𝑎 ↔ (𝑎{⟨𝑥, 𝑦⟩ ∣ 𝜓}𝑎𝑎(𝐵 × 𝐵)𝑎))
5857simprbi 479 . . . . . . . . . . . . . . . . 17 (𝑎({⟨𝑥, 𝑦⟩ ∣ 𝜓} ∩ (𝐵 × 𝐵))𝑎𝑎(𝐵 × 𝐵)𝑎)
59 brxp 5181 . . . . . . . . . . . . . . . . . 18 (𝑎(𝐵 × 𝐵)𝑎 ↔ (𝑎𝐵𝑎𝐵))
6059simprbi 479 . . . . . . . . . . . . . . . . 17 (𝑎(𝐵 × 𝐵)𝑎𝑎𝐵)
6158, 60syl 17 . . . . . . . . . . . . . . . 16 (𝑎({⟨𝑥, 𝑦⟩ ∣ 𝜓} ∩ (𝐵 × 𝐵))𝑎𝑎𝐵)
62 sonr 5085 . . . . . . . . . . . . . . . . 17 ((({⟨𝑥, 𝑦⟩ ∣ 𝜓} ∩ (𝐵 × 𝐵)) Or 𝐵𝑎𝐵) → ¬ 𝑎({⟨𝑥, 𝑦⟩ ∣ 𝜓} ∩ (𝐵 × 𝐵))𝑎)
6362ex 449 . . . . . . . . . . . . . . . 16 (({⟨𝑥, 𝑦⟩ ∣ 𝜓} ∩ (𝐵 × 𝐵)) Or 𝐵 → (𝑎𝐵 → ¬ 𝑎({⟨𝑥, 𝑦⟩ ∣ 𝜓} ∩ (𝐵 × 𝐵))𝑎))
6431, 61, 63syl2im 40 . . . . . . . . . . . . . . 15 (𝜑 → (𝑎({⟨𝑥, 𝑦⟩ ∣ 𝜓} ∩ (𝐵 × 𝐵))𝑎 → ¬ 𝑎({⟨𝑥, 𝑦⟩ ∣ 𝜓} ∩ (𝐵 × 𝐵))𝑎))
6564pm2.01d 181 . . . . . . . . . . . . . 14 (𝜑 → ¬ 𝑎({⟨𝑥, 𝑦⟩ ∣ 𝜓} ∩ (𝐵 × 𝐵))𝑎)
66 breq2 4689 . . . . . . . . . . . . . . . 16 (𝑎 = 𝑏 → (𝑎({⟨𝑥, 𝑦⟩ ∣ 𝜓} ∩ (𝐵 × 𝐵))𝑎𝑎({⟨𝑥, 𝑦⟩ ∣ 𝜓} ∩ (𝐵 × 𝐵))𝑏))
67 df-br 4686 . . . . . . . . . . . . . . . 16 (𝑎({⟨𝑥, 𝑦⟩ ∣ 𝜓} ∩ (𝐵 × 𝐵))𝑏 ↔ ⟨𝑎, 𝑏⟩ ∈ ({⟨𝑥, 𝑦⟩ ∣ 𝜓} ∩ (𝐵 × 𝐵)))
6866, 67syl6bb 276 . . . . . . . . . . . . . . 15 (𝑎 = 𝑏 → (𝑎({⟨𝑥, 𝑦⟩ ∣ 𝜓} ∩ (𝐵 × 𝐵))𝑎 ↔ ⟨𝑎, 𝑏⟩ ∈ ({⟨𝑥, 𝑦⟩ ∣ 𝜓} ∩ (𝐵 × 𝐵))))
6968notbid 307 . . . . . . . . . . . . . 14 (𝑎 = 𝑏 → (¬ 𝑎({⟨𝑥, 𝑦⟩ ∣ 𝜓} ∩ (𝐵 × 𝐵))𝑎 ↔ ¬ ⟨𝑎, 𝑏⟩ ∈ ({⟨𝑥, 𝑦⟩ ∣ 𝜓} ∩ (𝐵 × 𝐵))))
7065, 69syl5ibcom 235 . . . . . . . . . . . . 13 (𝜑 → (𝑎 = 𝑏 → ¬ ⟨𝑎, 𝑏⟩ ∈ ({⟨𝑥, 𝑦⟩ ∣ 𝜓} ∩ (𝐵 × 𝐵))))
7156, 70syl5bi 232 . . . . . . . . . . . 12 (𝜑 → (⟨𝑎, 𝑏⟩ ∈ I → ¬ ⟨𝑎, 𝑏⟩ ∈ ({⟨𝑥, 𝑦⟩ ∣ 𝜓} ∩ (𝐵 × 𝐵))))
7271con2d 129 . . . . . . . . . . 11 (𝜑 → (⟨𝑎, 𝑏⟩ ∈ ({⟨𝑥, 𝑦⟩ ∣ 𝜓} ∩ (𝐵 × 𝐵)) → ¬ ⟨𝑎, 𝑏⟩ ∈ I ))
73 opex 4962 . . . . . . . . . . . 12 𝑎, 𝑏⟩ ∈ V
74 eldif 3617 . . . . . . . . . . . 12 (⟨𝑎, 𝑏⟩ ∈ (V ∖ I ) ↔ (⟨𝑎, 𝑏⟩ ∈ V ∧ ¬ ⟨𝑎, 𝑏⟩ ∈ I ))
7573, 74mpbiran 973 . . . . . . . . . . 11 (⟨𝑎, 𝑏⟩ ∈ (V ∖ I ) ↔ ¬ ⟨𝑎, 𝑏⟩ ∈ I )
7672, 75syl6ibr 242 . . . . . . . . . 10 (𝜑 → (⟨𝑎, 𝑏⟩ ∈ ({⟨𝑥, 𝑦⟩ ∣ 𝜓} ∩ (𝐵 × 𝐵)) → ⟨𝑎, 𝑏⟩ ∈ (V ∖ I )))
7752, 76relssdv 5246 . . . . . . . . 9 (𝜑 → ({⟨𝑥, 𝑦⟩ ∣ 𝜓} ∩ (𝐵 × 𝐵)) ⊆ (V ∖ I ))
78 disj2 4057 . . . . . . . . 9 ((({⟨𝑥, 𝑦⟩ ∣ 𝜓} ∩ (𝐵 × 𝐵)) ∩ I ) = ∅ ↔ ({⟨𝑥, 𝑦⟩ ∣ 𝜓} ∩ (𝐵 × 𝐵)) ⊆ (V ∖ I ))
7977, 78sylibr 224 . . . . . . . 8 (𝜑 → (({⟨𝑥, 𝑦⟩ ∣ 𝜓} ∩ (𝐵 × 𝐵)) ∩ I ) = ∅)
80 disj3 4054 . . . . . . . 8 ((({⟨𝑥, 𝑦⟩ ∣ 𝜓} ∩ (𝐵 × 𝐵)) ∩ I ) = ∅ ↔ ({⟨𝑥, 𝑦⟩ ∣ 𝜓} ∩ (𝐵 × 𝐵)) = (({⟨𝑥, 𝑦⟩ ∣ 𝜓} ∩ (𝐵 × 𝐵)) ∖ I ))
8179, 80sylib 208 . . . . . . 7 (𝜑 → ({⟨𝑥, 𝑦⟩ ∣ 𝜓} ∩ (𝐵 × 𝐵)) = (({⟨𝑥, 𝑦⟩ ∣ 𝜓} ∩ (𝐵 × 𝐵)) ∖ I ))
8245, 47, 813eqtr4a 2711 . . . . . 6 (𝜑 → ( ∖ I ) = ({⟨𝑥, 𝑦⟩ ∣ 𝜓} ∩ (𝐵 × 𝐵)))
8338, 82syl5eq 2697 . . . . 5 (𝜑 → (lt‘𝑂) = ({⟨𝑥, 𝑦⟩ ∣ 𝜓} ∩ (𝐵 × 𝐵)))
84 soeq1 5083 . . . . 5 ((lt‘𝑂) = ({⟨𝑥, 𝑦⟩ ∣ 𝜓} ∩ (𝐵 × 𝐵)) → ((lt‘𝑂) Or 𝐵 ↔ ({⟨𝑥, 𝑦⟩ ∣ 𝜓} ∩ (𝐵 × 𝐵)) Or 𝐵))
8583, 84syl 17 . . . 4 (𝜑 → ((lt‘𝑂) Or 𝐵 ↔ ({⟨𝑥, 𝑦⟩ ∣ 𝜓} ∩ (𝐵 × 𝐵)) Or 𝐵))
8631, 85mpbird 247 . . 3 (𝜑 → (lt‘𝑂) Or 𝐵)
8724, 32, 8opsrbas 19527 . . . . 5 (𝜑 → (Base‘𝑆) = (Base‘𝑂))
8825, 87syl5eq 2697 . . . 4 (𝜑𝐵 = (Base‘𝑂))
89 soeq2 5084 . . . 4 (𝐵 = (Base‘𝑂) → ((lt‘𝑂) Or 𝐵 ↔ (lt‘𝑂) Or (Base‘𝑂)))
9088, 89syl 17 . . 3 (𝜑 → ((lt‘𝑂) Or 𝐵 ↔ (lt‘𝑂) Or (Base‘𝑂)))
9186, 90mpbid 222 . 2 (𝜑 → (lt‘𝑂) Or (Base‘𝑂))
9288reseq2d 5428 . . . 4 (𝜑 → ( I ↾ 𝐵) = ( I ↾ (Base‘𝑂)))
93 ssun2 3810 . . . 4 ( I ↾ 𝐵) ⊆ (({⟨𝑥, 𝑦⟩ ∣ 𝜓} ∩ (𝐵 × 𝐵)) ∪ ( I ↾ 𝐵))
9492, 93syl6eqssr 3689 . . 3 (𝜑 → ( I ↾ (Base‘𝑂)) ⊆ (({⟨𝑥, 𝑦⟩ ∣ 𝜓} ∩ (𝐵 × 𝐵)) ∪ ( I ↾ 𝐵)))
9594, 46sseqtr4d 3675 . 2 (𝜑 → ( I ↾ (Base‘𝑂)) ⊆ )
96 eqid 2651 . . . 4 (Base‘𝑂) = (Base‘𝑂)
9796, 35, 36tosso 17083 . . 3 (𝑂 ∈ V → (𝑂 ∈ Toset ↔ ((lt‘𝑂) Or (Base‘𝑂) ∧ ( I ↾ (Base‘𝑂)) ⊆ )))
9834, 97ax-mp 5 . 2 (𝑂 ∈ Toset ↔ ((lt‘𝑂) Or (Base‘𝑂) ∧ ( I ↾ (Base‘𝑂)) ⊆ ))
9991, 95, 98sylanbrc 699 1 (𝜑𝑂 ∈ Toset)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∧ wa 383   = wceq 1523   ∈ wcel 2030  ∀wral 2941  ∃wrex 2942  {crab 2945  Vcvv 3231   ∖ cdif 3604   ∪ cun 3605   ∩ cin 3606   ⊆ wss 3607  ∅c0 3948  ⟨cop 4216   class class class wbr 4685  {copab 4745   I cid 5052   Or wor 5063   We wwe 5101   × cxp 5141  ◡ccnv 5142   ↾ cres 5145   “ cima 5146  Rel wrel 5148  ‘cfv 5926  (class class class)co 6690   ↑𝑚 cmap 7899  Fincfn 7997  ℕcn 11058  ℕ0cn0 11330  Basecbs 15904  lecple 15995  ltcplt 16988  Tosetctos 17080   mPwSer cmps 19399
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