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Theorem opsrtoslem2 20251
 Description: Lemma for opsrtos 20252. (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 7171 . . . . . . . 8 (ℕ0m 𝐼) ∈ V
31, 2rabex2 5218 . . . . . . 7 𝐷 ∈ V
4 opsrtoslem.c . . . . . . . 8 𝐶 = (𝑇 <bag 𝐼)
5 opsrso.i . . . . . . . 8 (𝜑𝐼𝑉)
65, 5xpexd 7457 . . . . . . . . 9 (𝜑 → (𝐼 × 𝐼) ∈ V)
7 opsrso.t . . . . . . . . 9 (𝜑𝑇 ⊆ (𝐼 × 𝐼))
86, 7ssexd 5209 . . . . . . . 8 (𝜑𝑇 ∈ V)
9 opsrso.w . . . . . . . 8 (𝜑𝑇 We 𝐼)
104, 1, 5, 8, 9ltbwe 20239 . . . . . . 7 (𝜑𝐶 We 𝐷)
11 opsrso.r . . . . . . . . 9 (𝜑𝑅 ∈ Toset)
12 eqid 2824 . . . . . . . . . . 11 (Base‘𝑅) = (Base‘𝑅)
13 eqid 2824 . . . . . . . . . . 11 (le‘𝑅) = (le‘𝑅)
14 opsrtoslem.q . . . . . . . . . . 11 < = (lt‘𝑅)
1512, 13, 14tosso 17635 . . . . . . . . . 10 (𝑅 ∈ Toset → (𝑅 ∈ Toset ↔ ( < Or (Base‘𝑅) ∧ ( I ↾ (Base‘𝑅)) ⊆ (le‘𝑅))))
1615ibi 270 . . . . . . . . 9 (𝑅 ∈ Toset → ( < Or (Base‘𝑅) ∧ ( I ↾ (Base‘𝑅)) ⊆ (le‘𝑅)))
1711, 16syl 17 . . . . . . . 8 (𝜑 → ( < Or (Base‘𝑅) ∧ ( I ↾ (Base‘𝑅)) ⊆ (le‘𝑅)))
1817simpld 498 . . . . . . 7 (𝜑< Or (Base‘𝑅))
19 opsrtoslem.ps . . . . . . . . 9 (𝜓 ↔ ∃𝑧𝐷 ((𝑥𝑧) < (𝑦𝑧) ∧ ∀𝑤𝐷 (𝑤𝐶𝑧 → (𝑥𝑤) = (𝑦𝑤))))
2019opabbii 5114 . . . . . . . 8 {⟨𝑥, 𝑦⟩ ∣ 𝜓} = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧𝐷 ((𝑥𝑧) < (𝑦𝑧) ∧ ∀𝑤𝐷 (𝑤𝐶𝑧 → (𝑥𝑤) = (𝑦𝑤)))}
2120wemapso 8999 . . . . . . 7 ((𝐷 ∈ V ∧ 𝐶 We 𝐷< Or (Base‘𝑅)) → {⟨𝑥, 𝑦⟩ ∣ 𝜓} Or ((Base‘𝑅) ↑m 𝐷))
223, 10, 18, 21mp3an2i 1463 . . . . . 6 (𝜑 → {⟨𝑥, 𝑦⟩ ∣ 𝜓} Or ((Base‘𝑅) ↑m 𝐷))
23 opsrtoslem.s . . . . . . . 8 𝑆 = (𝐼 mPwSer 𝑅)
24 opsrtoslem.b . . . . . . . 8 𝐵 = (Base‘𝑆)
2523, 12, 1, 24, 5psrbas 20144 . . . . . . 7 (𝜑𝐵 = ((Base‘𝑅) ↑m 𝐷))
26 soeq2 5476 . . . . . . 7 (𝐵 = ((Base‘𝑅) ↑m 𝐷) → ({⟨𝑥, 𝑦⟩ ∣ 𝜓} Or 𝐵 ↔ {⟨𝑥, 𝑦⟩ ∣ 𝜓} Or ((Base‘𝑅) ↑m 𝐷)))
2725, 26syl 17 . . . . . 6 (𝜑 → ({⟨𝑥, 𝑦⟩ ∣ 𝜓} Or 𝐵 ↔ {⟨𝑥, 𝑦⟩ ∣ 𝜓} Or ((Base‘𝑅) ↑m 𝐷)))
2822, 27mpbird 260 . . . . 5 (𝜑 → {⟨𝑥, 𝑦⟩ ∣ 𝜓} Or 𝐵)
29 soinxp 5614 . . . . 5 ({⟨𝑥, 𝑦⟩ ∣ 𝜓} Or 𝐵 ↔ ({⟨𝑥, 𝑦⟩ ∣ 𝜓} ∩ (𝐵 × 𝐵)) Or 𝐵)
3028, 29sylib 221 . . . 4 (𝜑 → ({⟨𝑥, 𝑦⟩ ∣ 𝜓} ∩ (𝐵 × 𝐵)) Or 𝐵)
31 opsrso.o . . . . . . . 8 𝑂 = ((𝐼 ordPwSer 𝑅)‘𝑇)
3231fvexi 6665 . . . . . . 7 𝑂 ∈ V
33 opsrtoslem.l . . . . . . . 8 = (le‘𝑂)
34 eqid 2824 . . . . . . . 8 (lt‘𝑂) = (lt‘𝑂)
3533, 34pltfval 17558 . . . . . . 7 (𝑂 ∈ V → (lt‘𝑂) = ( ∖ I ))
3632, 35ax-mp 5 . . . . . 6 (lt‘𝑂) = ( ∖ I )
37 difundir 4240 . . . . . . . 8 ((({⟨𝑥, 𝑦⟩ ∣ 𝜓} ∩ (𝐵 × 𝐵)) ∪ ( I ↾ 𝐵)) ∖ I ) = ((({⟨𝑥, 𝑦⟩ ∣ 𝜓} ∩ (𝐵 × 𝐵)) ∖ I ) ∪ (( I ↾ 𝐵) ∖ I ))
38 resss 5859 . . . . . . . . . 10 ( I ↾ 𝐵) ⊆ I
39 ssdif0 4304 . . . . . . . . . 10 (( I ↾ 𝐵) ⊆ I ↔ (( I ↾ 𝐵) ∖ I ) = ∅)
4038, 39mpbi 233 . . . . . . . . 9 (( I ↾ 𝐵) ∖ I ) = ∅
4140uneq2i 4120 . . . . . . . 8 ((({⟨𝑥, 𝑦⟩ ∣ 𝜓} ∩ (𝐵 × 𝐵)) ∖ I ) ∪ (( I ↾ 𝐵) ∖ I )) = ((({⟨𝑥, 𝑦⟩ ∣ 𝜓} ∩ (𝐵 × 𝐵)) ∖ I ) ∪ ∅)
42 un0 4325 . . . . . . . 8 ((({⟨𝑥, 𝑦⟩ ∣ 𝜓} ∩ (𝐵 × 𝐵)) ∖ I ) ∪ ∅) = (({⟨𝑥, 𝑦⟩ ∣ 𝜓} ∩ (𝐵 × 𝐵)) ∖ I )
4337, 41, 423eqtri 2851 . . . . . . 7 ((({⟨𝑥, 𝑦⟩ ∣ 𝜓} ∩ (𝐵 × 𝐵)) ∪ ( I ↾ 𝐵)) ∖ I ) = (({⟨𝑥, 𝑦⟩ ∣ 𝜓} ∩ (𝐵 × 𝐵)) ∖ I )
4431, 5, 11, 7, 9, 23, 24, 14, 4, 1, 19, 33opsrtoslem1 20250 . . . . . . . 8 (𝜑 = (({⟨𝑥, 𝑦⟩ ∣ 𝜓} ∩ (𝐵 × 𝐵)) ∪ ( I ↾ 𝐵)))
4544difeq1d 4082 . . . . . . 7 (𝜑 → ( ∖ I ) = ((({⟨𝑥, 𝑦⟩ ∣ 𝜓} ∩ (𝐵 × 𝐵)) ∪ ( I ↾ 𝐵)) ∖ I ))
46 relinxp 5668 . . . . . . . . . . 11 Rel ({⟨𝑥, 𝑦⟩ ∣ 𝜓} ∩ (𝐵 × 𝐵))
4746a1i 11 . . . . . . . . . 10 (𝜑 → Rel ({⟨𝑥, 𝑦⟩ ∣ 𝜓} ∩ (𝐵 × 𝐵)))
48 df-br 5048 . . . . . . . . . . . . . 14 (𝑎 I 𝑏 ↔ ⟨𝑎, 𝑏⟩ ∈ I )
49 vex 3482 . . . . . . . . . . . . . . 15 𝑏 ∈ V
5049ideq 5704 . . . . . . . . . . . . . 14 (𝑎 I 𝑏𝑎 = 𝑏)
5148, 50bitr3i 280 . . . . . . . . . . . . 13 (⟨𝑎, 𝑏⟩ ∈ I ↔ 𝑎 = 𝑏)
52 brin 5099 . . . . . . . . . . . . . . . . . 18 (𝑎({⟨𝑥, 𝑦⟩ ∣ 𝜓} ∩ (𝐵 × 𝐵))𝑎 ↔ (𝑎{⟨𝑥, 𝑦⟩ ∣ 𝜓}𝑎𝑎(𝐵 × 𝐵)𝑎))
5352simprbi 500 . . . . . . . . . . . . . . . . 17 (𝑎({⟨𝑥, 𝑦⟩ ∣ 𝜓} ∩ (𝐵 × 𝐵))𝑎𝑎(𝐵 × 𝐵)𝑎)
54 brxp 5582 . . . . . . . . . . . . . . . . . 18 (𝑎(𝐵 × 𝐵)𝑎 ↔ (𝑎𝐵𝑎𝐵))
5554simprbi 500 . . . . . . . . . . . . . . . . 17 (𝑎(𝐵 × 𝐵)𝑎𝑎𝐵)
5653, 55syl 17 . . . . . . . . . . . . . . . 16 (𝑎({⟨𝑥, 𝑦⟩ ∣ 𝜓} ∩ (𝐵 × 𝐵))𝑎𝑎𝐵)
57 sonr 5477 . . . . . . . . . . . . . . . . 17 ((({⟨𝑥, 𝑦⟩ ∣ 𝜓} ∩ (𝐵 × 𝐵)) Or 𝐵𝑎𝐵) → ¬ 𝑎({⟨𝑥, 𝑦⟩ ∣ 𝜓} ∩ (𝐵 × 𝐵))𝑎)
5857ex 416 . . . . . . . . . . . . . . . 16 (({⟨𝑥, 𝑦⟩ ∣ 𝜓} ∩ (𝐵 × 𝐵)) Or 𝐵 → (𝑎𝐵 → ¬ 𝑎({⟨𝑥, 𝑦⟩ ∣ 𝜓} ∩ (𝐵 × 𝐵))𝑎))
5930, 56, 58syl2im 40 . . . . . . . . . . . . . . 15 (𝜑 → (𝑎({⟨𝑥, 𝑦⟩ ∣ 𝜓} ∩ (𝐵 × 𝐵))𝑎 → ¬ 𝑎({⟨𝑥, 𝑦⟩ ∣ 𝜓} ∩ (𝐵 × 𝐵))𝑎))
6059pm2.01d 193 . . . . . . . . . . . . . 14 (𝜑 → ¬ 𝑎({⟨𝑥, 𝑦⟩ ∣ 𝜓} ∩ (𝐵 × 𝐵))𝑎)
61 breq2 5051 . . . . . . . . . . . . . . . 16 (𝑎 = 𝑏 → (𝑎({⟨𝑥, 𝑦⟩ ∣ 𝜓} ∩ (𝐵 × 𝐵))𝑎𝑎({⟨𝑥, 𝑦⟩ ∣ 𝜓} ∩ (𝐵 × 𝐵))𝑏))
62 df-br 5048 . . . . . . . . . . . . . . . 16 (𝑎({⟨𝑥, 𝑦⟩ ∣ 𝜓} ∩ (𝐵 × 𝐵))𝑏 ↔ ⟨𝑎, 𝑏⟩ ∈ ({⟨𝑥, 𝑦⟩ ∣ 𝜓} ∩ (𝐵 × 𝐵)))
6361, 62syl6bb 290 . . . . . . . . . . . . . . 15 (𝑎 = 𝑏 → (𝑎({⟨𝑥, 𝑦⟩ ∣ 𝜓} ∩ (𝐵 × 𝐵))𝑎 ↔ ⟨𝑎, 𝑏⟩ ∈ ({⟨𝑥, 𝑦⟩ ∣ 𝜓} ∩ (𝐵 × 𝐵))))
6463notbid 321 . . . . . . . . . . . . . 14 (𝑎 = 𝑏 → (¬ 𝑎({⟨𝑥, 𝑦⟩ ∣ 𝜓} ∩ (𝐵 × 𝐵))𝑎 ↔ ¬ ⟨𝑎, 𝑏⟩ ∈ ({⟨𝑥, 𝑦⟩ ∣ 𝜓} ∩ (𝐵 × 𝐵))))
6560, 64syl5ibcom 248 . . . . . . . . . . . . 13 (𝜑 → (𝑎 = 𝑏 → ¬ ⟨𝑎, 𝑏⟩ ∈ ({⟨𝑥, 𝑦⟩ ∣ 𝜓} ∩ (𝐵 × 𝐵))))
6651, 65syl5bi 245 . . . . . . . . . . . 12 (𝜑 → (⟨𝑎, 𝑏⟩ ∈ I → ¬ ⟨𝑎, 𝑏⟩ ∈ ({⟨𝑥, 𝑦⟩ ∣ 𝜓} ∩ (𝐵 × 𝐵))))
6766con2d 136 . . . . . . . . . . 11 (𝜑 → (⟨𝑎, 𝑏⟩ ∈ ({⟨𝑥, 𝑦⟩ ∣ 𝜓} ∩ (𝐵 × 𝐵)) → ¬ ⟨𝑎, 𝑏⟩ ∈ I ))
68 opex 5337 . . . . . . . . . . . 12 𝑎, 𝑏⟩ ∈ V
69 eldif 3928 . . . . . . . . . . . 12 (⟨𝑎, 𝑏⟩ ∈ (V ∖ I ) ↔ (⟨𝑎, 𝑏⟩ ∈ V ∧ ¬ ⟨𝑎, 𝑏⟩ ∈ I ))
7068, 69mpbiran 708 . . . . . . . . . . 11 (⟨𝑎, 𝑏⟩ ∈ (V ∖ I ) ↔ ¬ ⟨𝑎, 𝑏⟩ ∈ I )
7167, 70syl6ibr 255 . . . . . . . . . 10 (𝜑 → (⟨𝑎, 𝑏⟩ ∈ ({⟨𝑥, 𝑦⟩ ∣ 𝜓} ∩ (𝐵 × 𝐵)) → ⟨𝑎, 𝑏⟩ ∈ (V ∖ I )))
7247, 71relssdv 5642 . . . . . . . . 9 (𝜑 → ({⟨𝑥, 𝑦⟩ ∣ 𝜓} ∩ (𝐵 × 𝐵)) ⊆ (V ∖ I ))
73 disj2 4388 . . . . . . . . 9 ((({⟨𝑥, 𝑦⟩ ∣ 𝜓} ∩ (𝐵 × 𝐵)) ∩ I ) = ∅ ↔ ({⟨𝑥, 𝑦⟩ ∣ 𝜓} ∩ (𝐵 × 𝐵)) ⊆ (V ∖ I ))
7472, 73sylibr 237 . . . . . . . 8 (𝜑 → (({⟨𝑥, 𝑦⟩ ∣ 𝜓} ∩ (𝐵 × 𝐵)) ∩ I ) = ∅)
75 disj3 4384 . . . . . . . 8 ((({⟨𝑥, 𝑦⟩ ∣ 𝜓} ∩ (𝐵 × 𝐵)) ∩ I ) = ∅ ↔ ({⟨𝑥, 𝑦⟩ ∣ 𝜓} ∩ (𝐵 × 𝐵)) = (({⟨𝑥, 𝑦⟩ ∣ 𝜓} ∩ (𝐵 × 𝐵)) ∖ I ))
7674, 75sylib 221 . . . . . . 7 (𝜑 → ({⟨𝑥, 𝑦⟩ ∣ 𝜓} ∩ (𝐵 × 𝐵)) = (({⟨𝑥, 𝑦⟩ ∣ 𝜓} ∩ (𝐵 × 𝐵)) ∖ I ))
7743, 45, 763eqtr4a 2885 . . . . . 6 (𝜑 → ( ∖ I ) = ({⟨𝑥, 𝑦⟩ ∣ 𝜓} ∩ (𝐵 × 𝐵)))
7836, 77syl5eq 2871 . . . . 5 (𝜑 → (lt‘𝑂) = ({⟨𝑥, 𝑦⟩ ∣ 𝜓} ∩ (𝐵 × 𝐵)))
79 soeq1 5475 . . . . 5 ((lt‘𝑂) = ({⟨𝑥, 𝑦⟩ ∣ 𝜓} ∩ (𝐵 × 𝐵)) → ((lt‘𝑂) Or 𝐵 ↔ ({⟨𝑥, 𝑦⟩ ∣ 𝜓} ∩ (𝐵 × 𝐵)) Or 𝐵))
8078, 79syl 17 . . . 4 (𝜑 → ((lt‘𝑂) Or 𝐵 ↔ ({⟨𝑥, 𝑦⟩ ∣ 𝜓} ∩ (𝐵 × 𝐵)) Or 𝐵))
8130, 80mpbird 260 . . 3 (𝜑 → (lt‘𝑂) Or 𝐵)
8223, 31, 7opsrbas 20245 . . . . 5 (𝜑 → (Base‘𝑆) = (Base‘𝑂))
8324, 82syl5eq 2871 . . . 4 (𝜑𝐵 = (Base‘𝑂))
84 soeq2 5476 . . . 4 (𝐵 = (Base‘𝑂) → ((lt‘𝑂) Or 𝐵 ↔ (lt‘𝑂) Or (Base‘𝑂)))
8583, 84syl 17 . . 3 (𝜑 → ((lt‘𝑂) Or 𝐵 ↔ (lt‘𝑂) Or (Base‘𝑂)))
8681, 85mpbid 235 . 2 (𝜑 → (lt‘𝑂) Or (Base‘𝑂))
8783reseq2d 5834 . . . 4 (𝜑 → ( I ↾ 𝐵) = ( I ↾ (Base‘𝑂)))
88 ssun2 4133 . . . 4 ( I ↾ 𝐵) ⊆ (({⟨𝑥, 𝑦⟩ ∣ 𝜓} ∩ (𝐵 × 𝐵)) ∪ ( I ↾ 𝐵))
8987, 88eqsstrrdi 4006 . . 3 (𝜑 → ( I ↾ (Base‘𝑂)) ⊆ (({⟨𝑥, 𝑦⟩ ∣ 𝜓} ∩ (𝐵 × 𝐵)) ∪ ( I ↾ 𝐵)))
9089, 44sseqtrrd 3992 . 2 (𝜑 → ( I ↾ (Base‘𝑂)) ⊆ )
91 eqid 2824 . . . 4 (Base‘𝑂) = (Base‘𝑂)
9291, 33, 34tosso 17635 . . 3 (𝑂 ∈ V → (𝑂 ∈ Toset ↔ ((lt‘𝑂) Or (Base‘𝑂) ∧ ( I ↾ (Base‘𝑂)) ⊆ )))
9332, 92ax-mp 5 . 2 (𝑂 ∈ Toset ↔ ((lt‘𝑂) Or (Base‘𝑂) ∧ ( I ↾ (Base‘𝑂)) ⊆ ))
9486, 90, 93sylanbrc 586 1 (𝜑𝑂 ∈ Toset)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2115  ∀wral 3132  ∃wrex 3133  {crab 3136  Vcvv 3479   ∖ cdif 3915   ∪ cun 3916   ∩ cin 3917   ⊆ wss 3918  ∅c0 4274  ⟨cop 4554   class class class wbr 5047  {copab 5109   I cid 5440   Or wor 5454   We wwe 5494   × cxp 5534  ◡ccnv 5535   ↾ cres 5538   “ cima 5539  Rel wrel 5541  ‘cfv 6336  (class class class)co 7138   ↑m cmap 8389  Fincfn 8492  ℕcn 11623  ℕ0cn0 11883  Basecbs 16472  lecple 16561  ltcplt 17540  Tosetctos 17632   mPwSer cmps 20117
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